Topological equivalence of linear representations\\ for cyclic groups: \partone
Ian Hambleton
In the two parts of this paper we prove that the Reidemeister torsion invariants determine topological equivalence of $G${--}representations, for $G$ a finite cyclic group.
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\begin{document}


  \title[Similarities of cyclic groups]{Topological equivalence of linear representations\\
 for cyclic groups: \partone}
\author{Ian Hambleton}
\address{Department of Mathematics \& Statistics,
McMaster University\newline
\indent Hamilton, Ont.,  Canada,  L8S 4K1}
\email{ ian@math.mcmaster.ca}

\author{Erik K. Pedersen}
\address{Department\ of\ Mathematical Sciences\\
SUNY at Binghamton\newline
\indent Binghamton,  NY, 13901}
\email{ erik@math.binghamton.edu}
\thanks{Partially supported by NSERC grant A4000
and  NSF grant  DMS 9104026. The authors also
wish to thank the Max Planck Institut f\"ur Mathematik, Bonn, for
its hospitality and support.}
\begin{abstract} In  the two parts of this paper we prove that the Reidemeister torsion invariants
determine  topological equivalence of $G${--}representations,
for $G$ a finite cyclic group.
\end{abstract}
\date{March 15, 2004}
\maketitle

\section{ Introduction\label{intro}}
Let $G$ be a finite group and $V$, $V'$ finite dimensional
real orthogonal representations of $G$.
Then $V $ is said to be
\emph{topologically equivalent} to $V'$
(denoted $V \sim_t V'$) if there exists a homeomorphism $h\colon
V \to V'$ which is $G$-equivariant.
If $V$, $V'$ are topologically equivalent, but not linearly isomorphic,
then such a homeomorphism is called a non-linear similarity.
These notions were introduced and studied by de Rham \cite{drh2},
\cite{drh1}, and developed extensively in
 \cite{cs1}, \cite{cs2}, \cite{hsp1},  \cite{mr1}, and
\cite{cssw1}.
In the two parts of this paper, referred to as \pone\
and \ptwo, we complete de Rham's program by showing that
Reidemeister torsion invariants and number theory determine
non-linear similarity for finite cyclic groups.

A  $G$-representation is called \emph{free} if
each element $1\neq g \in G$ fixes only the zero vector.
Every representation of a finite cyclic group has a
unique maximal free subrepresentation.

\begin{thmmain} Let $G$ be a finite cyclic group
and $V_1$, $V_2$ be free
$G${--}representations. For any $G${--}representation $W$, the
existence of a non-linear similarity
$V_1\oplus W \sim_t V_2\oplus W$ is entirely 
determined by explicit
congruences in the weights of the free summands $V_1$, $V_2$,
 and the ratio
$\Delta(V_1)/\Delta(V_2)$ of their Reidemeister torsions, 
up to an algebraically described indeterminacy.
\end{thmmain}
\noindent
The notation and the  indeterminacy are given in Section
\ref{results} and a detailed statement of results in
Theorems A{--}E. For cyclic groups of $2$-power order, 
 we obtain a complete classification of non-linear similarities
(see Section \ref{unstable}). 

In \cite{cs1}, Cappell and Shaneson  showed that non-linear
similarities $V \sim_t V'$ 
exist for  cyclic groups $G=\Cy{4q}$ of every
order $4q \geqq 8$.
On the other hand, if $G=\Cy q$  or $G=\Cy{2q}$, for $q$ odd,
 Hsiang{--}Pardon \cite{hsp1} and
Madsen{--}Rothenberg \cite{mr1} proved that
topological equivalence of $G${--}representations implies
linear equivalence (the case $G=\Cy 4$ is trivial).
Since linear $G$-equivalence for general finite groups $G$
is detected by restriction to cyclic subgroups, it is reasonable
to study this case first.
For the rest of the paper, unless otherwise mentioned,
$G$ denotes a finite cyclic group.

Further positive results can be obtained by imposing
assumptions on the isotropy subgroups allowed in $V$ and $V'$.
For example, 
de Rham \cite{drh2} proved in 1935 that
 piecewise linear similarity implies
linear equivalence for free $G$-representations,
 by using Reidemeister torsion and the Franz
Independence Lemma.
Topological invariance of Whitehead torsion
shows that his method also rules out non{--}linear similarity
in this case.
In \cite[Thm.A]{hp2} we studied ``first{--}time"
similarities, where $\Res_K V \cong \Res_K V'$ for
all proper subgroups $K\subsetneqq G$, and
showed that topological equivalence implies linear equivalence
if $V$, $V'$ have no isotropy subgroup of index $2$.
This result is an application of  bounded surgery
theory (see \cite{hp1}, \cite[\S 4]{hp2}),
and provides a  more conceptual proof of the Odd
Order Theorem.
These techniques are
extended here to provide a necessary and sufficient
condition for non-linear similarity in terms of the
vanishing of a bounded transfer map (see Theorem \ref{maintest}).
This gives a new approach to de Rham's
problem.
The main  work of the present paper is to establish methods
for effective calculation of the bounded transfer in the
presence of isotropy groups of arbitrary index. 


An interesting question in non{--}linear similarity
concerns the minimum
possible dimension for examples.
It is easy to see that the existence of a non{--}linear
similarity $V \sim_t V'$ implies $\dim V=\dim V' \geqq 5$.
Cappell, Shaneson, Steinberger and West \cite{cssw1}, proved
that $6$-dimensional similarities
 exist for $G= \Cy{2^r}$, $r\geqq 4$ and referred to the 1981
Cappell-Shaneson preprint (now published \cite{cs3}) for the
complete proof that $5${--}dimensional similarities do not exist
for any finite group. See Corollary \ref{eightteenfive} for a
direct argument using the criterion of Theorem A in the special
case of cyclic $2$-groups.


In \cite{cs2}, Cappell and Shaneson  initiated the study
of \emph{stable} topological equvalence for 
$G${--}representations.
We say that $V_1$ and $V_2$ are stably topologically similar
($V_1 \approx_t V_2$) if there exists a $G${--}representation
$W$ such that $V_1 \oplus W \sim_t V_2\oplus W$.
Let  $\rtop{G} = R(G)/R_t(G)$ denote the quotient group of
 the real representation ring of $G$ by the subgroup
$R_t(G) = \{ [V_1] -[V_2] \,\mid\, V_1 \approx_t V_2\}$.
In \cite{cs2}, $\rtop{G}\otimes \mathbf Z[1/2]$ was computed,
and the torsion subgroup was shown to be $2${--}primary.
As an application of our general results,  we determine the
structure of the torsion in $\rtop{G}$, for $G$ any cyclic group
(see \ptwo, Section \ref{sD}). 
In Theorem E we give the  calculation of $\rtop{G}$
for $G = \Cy{2^r}$. This is the first complete calculation of 
 $\rtop{G}$ for any group that admits non{--}linear similarities.




\tableofcontents
\section{ Statement of Results\label{results}}
We first introduce some notation, and then give the main
results. 
Let $G = \Cy{4q}$, where $q >1$, and let $H = \Cy{2q}$ denote
the subgroup of index 2 in $G$.
The maximal odd order subgroup of $G$ is denoted $\Godd$.
We fix a generator $G=\langle t \rangle$ and a primitive $4q^{th}$-root of
unity $\zeta = \exp{2\pi i/{4q}}$.
The group $G$ has both a trivial $1$-dimensional real representation,
denoted $\rplus$, and a non-trivial $1$-dimensional
real representation, denoted $\rminus$.

A \emph{free} $G${--}representation is a sum of faithful
1-dimensional complex representations.
Let $t^a$, $a \in \mathbf Z$, denote the complex numbers
 $\bc$ with action
$t\cdot z = \zeta^az$ for all $z \in \bc$.
This representation is free if and only if $(a, 4q) = 1$, and
 the coefficient $a$ is well{--}defined only modulo $4q$.
Since $t^a \cong t^{-a}$ as real $G${--}representations,  we
can  always choose the weights $a \equiv 1 \Mod{4}$.
This will be assumed unless otherwise mentioned.

Now suppose that $V_1 = t^{a_1} + \dots + t^{a_k}$
is a free $G${--}representation.
The Reidemeister torsion invariant of $V_1$ is defined
as
\[\Delta(V_1) = \prod_{i=1}^{k} (t^{a_i} -1) \in \mathbf Z [t]/\{\pm t^m\} \ .\]
Let $V_2 = t^{b_1} + \dots + t^{b_k}$ be another free representation,
such that $S(V_1)$ and $S(V_2)$ are $G$-homotopy equivalent. 
This just means that the products of the weights $\prod a_i \equiv \prod b_i
\Mod{4q}$.
Then the Whitehead torsion of any $G$-homotopy equivalence  is
determined by the element
\[\Delta(V_1)/\Delta(V_2) = \frac{\prod (t^{a_i}-1)}{\prod(t^{b_i}-1)}
\]
since $\wh(\ZG)\to \wh(\QG)$ is monic
 \cite[p.14]{o1}.  
When there exists a $G$-homotopy equivalence
 $f\colon S(V_2) \to S(V_1)$
which is freely $G$-normally cobordant to the identity map on $S(V_1)$,
we say that $S(V_1)$ and $S(V_2)$ are freely \emph{$G$-normally cobordant}.
More generally, we say that
 $S(V_1)$ and $S(V_2)$ are \emph{$s${--}normally cobordant}
if $S(V_1\oplus U)$ and $S(V_2\oplus U)$ are freely $G$-normally cobordant
for all free $G${--}representations $U$.
This is a necessary condition for non-linear similarity, which can
be decided by explicit congruences in the weights
(see \cite[Thm. 1.2]{yo1} and \ptwo, Section \ref{normalinvariant}).

This quantity, $\Delta(V_1)/\Delta(V_2)$ is the basic invariant
determining non-linear similarity. It represents a unit in the group
ring $\ZG$, explicitly described for $G = \Cy{2^r}$ by Cappell and
Shaneson in
\cite[\S 1]{cs5} using a pull-back square of rings. 
To state concrete results we need to evaluate this invariant
modulo suitable indeterminacy. 

The involution $t \mapsto t^{-1}$ induces the identity on $\wh(\ZG)$,
so we get an element
\[\left\{ \Delta(V_1)/\Delta(V_2) \right\}
\in H^0(\wh(\ZG))\]
where we use $H^i(A)$ to denote the Tate cohomology
$H^i(\cy2;A)$ of $\cy2$ with coefficients in $A$.

Let $\wh(\ZG^-)$ denote the Whitehead group $\wh(\ZG)$
together with the involution induced by $t \mapsto -t^{-1}$.
Then for  $\tau(t) = \frac{\prod(t^{a_i}-1)}{\prod(t^{b_i}-1)}$,
we compute
\[\tau(t)\tau(-t)=
\frac{\prod(t^{a_i}-1)\prod((-t)^{a_i}-1)}
{\prod(t^{b_i}-1)\prod((-t)^{b_i}-1)}
= \prod\frac{(t^2)^{a_i}-1}{((t^2)^{b_i}-1)}\]
which is clearly induced from $\wh(\ZH)$.
Hence we also get a well defined
element
 \[\left\{ \Delta(V_1)/\Delta(V_2) \right\}
\in H^1(\wh(\ZG^-)/\wh(\ZH))\ .\]
This calculation takes place over the ring
 $\Lambda_{2q} =\bz [t]/(1 + t^2 + \dots + t^{4q-2})$,
 but the result holds
over
$\ZG$ via the involution{--}invariant pull-back square
\[
 \begin{matrix} \ZG  & \to &\Lambda_{2q}\cr
\downarrow &&\downarrow\cr
\bz[\cy 2]& \to & \cy{2q}[\cy 2]
\end{matrix}
\]


Consider the exact sequence of modules with involution:
\eqncount
\begin{equation}\label{okseq}
K_1(\ZH) \to K_1(\ZG)\to
 K_1(\rel{\ZH}{\ZG}) \to \widetilde K_0(\ZH)\to \widetilde K_0(\ZG)
\end{equation}
and define
$\wh(\rel{\ZH}{\ZG}) =K_1(\rel{\ZH}{\ZG})/{\scriptstyle \{\pm
G\}}$ . We then have a short exact sequence
\[0\to \wh(\ZG)/\wh(\ZH) \to \wh(\rel{\ZH}{\ZG}) \to \ck\to 0\]
where $\ck =\ker(\widetilde K_0(\ZH) \to \widetilde K_0(\ZG))$.
Such an exact sequence of $\cy 2$-modules induces a long exact
sequence in Tate cohomology.
In particular, we have a coboundary map
\[\delta\colon H^0(\ck) \to H^1(\wh(\ZG^-)/\wh(\ZH))\ .\]
Our first result deals with isotropy groups of index 2, as
is the case  for all the non{--}linear similarities
constructed in \cite{cs1}.
\begin{thma}
Let $V_1=t^{a_1} + \dots + t^{a_k} $
and $V_2 = t^{b_1} + \dots + t^{b_k}$ be free 
$G${--}representations,
with $a_i \equiv b_i\equiv 1 \Mod{4}$.
There exists a topological similarity $V_1\oplus \rminus \sim_t
V_2\oplus \rminus$ if and only if
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $\prod a_i \equiv \prod b_i \Mod{4q}$,
\item $\Res_H V_1 \cong \Res_H V_2$, and
\item the element
$\left\{ \Delta(V_1)/\Delta(V_2) \right\}
\in H^1(\wh(\ZG^-)/\wh(\ZH))$
is in the image of the coboundary
$\delta\colon H^0(\ck) \to H^1(\wh(\ZG^-)/\wh(\ZH))$.
\end{enumerate}
\end{thma}
\begin{remark} The condition (iii) simplifies for $G$ a cyclic $2$-group
since $H^0(\ck) = 0$ in that case (see Lemma \ref{vanB}).
Theorem A  should be compared with \cite[Cor.1]{cs1},
where more explicit conditions are given for ``first-time" similarities
of this kind under the assumption
that $q$ is odd, or a $2$-power, or $4q$ is a ``tempered" number. 
See also \ptwo, Theorem \ref{Cprime} for a more general result concerning
similarities without $\rplus$ summands.
The case $\dim V_1 = \dim V_2 = 4$ gives a reduction to
number theory for the existence of $5$-dimensional
similarities (see Remark \ref{Arem}).
\end{remark}

Our next result uses a more elaborate setting for the invariant.
Let
\[
\Phi = \begin{pmatrix} \ZH  & \to &\Zadictwo{H}\cr
\downarrow &&\downarrow\cr
\ZG & \to & \Zadictwo{G}
\end{pmatrix}
\]
and consider the exact sequence
\eqncount
\begin{equation}0 \to K_1(\rel{\ZH}{\ZG}) \to K_1(\rel{\Zadictwo{H}}{\Zadictwo{G}})
\to K_1(\Phi) \to \widetilde K_0(\rel{\ZH}{\ZG}) \to 0\ .
\end{equation}
Again we can define the Whitehead group versions by dividing
out trivial units $\{\pm G\}$, and get a double coboundary
\[\delta^2\colon H^1(\widetilde K_0(\rel{\ZH}{\ZG^{-}}))
\to H^1( \wh(\rel{\ZH}{\ZG^{-}}))\ .\]
There is a natural map
$H^1(\wh(\ZG^-)/\wh(\ZH))\to H^1( \wh(\rel{\ZH}{\ZG^{-}}))$,
and we will use the same notation $\left\{ \Delta(V_1)/\Delta(V_2) \right\}$ for the image of the
Reidemeister torsion invariant in this new domain.
The non-linear similarities handled by the next result have
isotropy of index $\leqq 2$.
\begin{thmb}
Let $V_1=t^{a_1} + \dots + t^{a_k} $
and $V_2 = t^{b_1} + \dots + t^{b_k}$ be free $G${--}representations.
There exists a topological similarity
 $V_1\oplus \rminus\oplus \rplus \sim_t
V_2\oplus \rminus\oplus \rplus$ if and only if
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $\prod a_i \equiv \prod b_i \Mod{4q}$,
\item $\Res_H V_1 \cong \Res_H V_2$, and
\item the element
$\left\{ \Delta(V_1)/\Delta(V_2) \right\}$
is in the image of the double coboundary
\[\delta^2\colon H^1(\widetilde K_0(\rel{\ZH}{\ZG^{-}}))
\to H^1(\wh(\rel{\ZH}{\ZG^{-}}))\ .\]
\end{enumerate}
\end{thmb}
This result can be applied to $6$-dimensional similarities.
\begin{corollary}\label{sixdim}
Let $G=\Cy{4q}$, with $q$ odd, and suppose that the
fields $\bq(\zeta_{d})$ have odd class number for
all $d \mid 4q$.
Then $G$ has no $6$-dimensional non-linear similarities.
\end{corollary}
\begin{remark} For example, the class number condition is satisfied for $q\leqq 11$, but not for $q=29$. 
The proof  is given in \ptwo, Section \ref{sixdiml}.
This result corrects \cite[Thm.1(i)]{cssw1}, and shows that the
computations of $\rtop{G}$ given in \cite[Thm. 2]{cssw1}
are incorrect. 
We explain the the source of these
 mistakes in Remark \ref{cssw-wrong}.
\end{remark}
Our final example of the computation of  bounded transfers is suitable
for determining stable non{--}linear similarities inductively,
with only a minor assumption on the isotropy subgroups.
To state the algebraic conditions, we must again generalize the
indeterminacy for the Reidemeister torsion invariant
to include bounded $K${--}groups (see \ptwo, Section \ref{excision}).
In this setting $\widetilde K_0(\rel{\ZH}{\ZG}) = \widetilde K_0(\ccat{\rminus,G}{\bz})$
and $\wh(\rel{\ZH}{\ZG})=\wh(\ccat{\rminus,G}{\bz})$.
We consider the analogous double coboundary
\[\delta^2\colon
H^1(\widetilde K_0(\ccat{W\times \rminus,G}{\bz})) \to
H^1(\wh(\ccat{W\times \rminus,G}{\bz}))\]
and note that there is a map
$\wh(\ccat{\rminus,G}{\bz} \to
\wh(\ccat{W\times \rminus,G}{\bz})$
induced by the inclusion on the control spaces. We will  use the same notation 
$\{\Delta(V_1)/\Delta(V_2)\}$ for the image of  our Reidemeister torsion invariant in  this new domain.

\begin{thmc}
Let $V_1=t^{a_1} + \dots + t^{a_k} $
and $V_2 = t^{b_1} + \dots + t^{b_k}$ be
 free $G${--}representations.
Let $W$ be a  complex $G${--}representation
with no $\rplus$ summands.
Then there exists a topological similarity
 $V_1\oplus W \oplus \rminus\oplus \rplus \sim_t
V_2\oplus W \oplus \rminus\oplus \rplus$ if and only if
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $S(V_1)$ is $s${--}normally cobordant to $S(V_2)$,
\item $\Res_H (V_1\oplus W) \oplus \rplus
 \sim_t \Res_H (V_2\oplus W)\oplus \rplus$, and
\item the element
$\left\{ \Delta(V_1)/\Delta(V_2) \right\}$
is in the image of the double coboundary
\[\delta^2\colon
H^1(\widetilde K_0(\ccat{\Wmax\times \rminus,G}{\bz})) \to
H^1(\wh(\ccat{\Wmax\times \rminus,G}{\bz}))\ , \]
where $0\subseteq \Wmax \subseteq W$ is a
complex subrepresentation of real dimension $\leqq 2$,  with  maximal
isotropy group among the isotropy groups
of $W$ with 2-power index.
\end{enumerate}
\end{thmc}
\begin{remark}\label{unstablenormal} The existence of a
 similarity  implies that
 $S(V_1)$ and $S(V_2)$ are  $s$-normally cobordant.
In particular, $S(V_1)$ must be freely $G$-normally cobordant to
$S(V_2)$ and
 this unstable normal invariant condition is enough
to give us a surgery problem. The computation of
the bounded transfer  in $L${--}theory leads  to condition (iii),
and an expression of the obstruction to the existence of a similarity purely in terms of bounded $K$-theory. To carry out this computation we may need  to stabilize in the free part,
and this uses the $s$-normal cobordism condition.
\end{remark}
\begin{remark} Theorem C  is proved in 
\ptwo, Section \ref{C}.
Note that $\Wmax=0$ in condition 
(iii) if $W$ has no isotropy
subgroups of $2$-power index.
Theorem C suffices to handle stable topological similarities,
but leaves out cases where $W$ has an
odd number of $\rminus$ summands (handled in \ptwo, Theorem
\ref{Cprime} and the results of \ptwo, Section
\ref{rminustwo}). 
Simpler conditions can be given when $G = \Cy{2^r}$
 (see Section \ref{twogroups} in this part).

The double coboundary in (iii) can also be expressed in more
``classical" terms by using the short exact sequence
\eqncount
\begin{equation} 0 \to \wh(\bCatrminus) \to 
\wh(\bcatrminus{\Wmax}) \to
K_1(\brelCatrminus{\Wmax}) \to 0       \label{ses}
\end{equation}
derived in \ptwo, Corollary \ref{kersimple}.
We have $K_1(\brelCatrminus{\Wmax}) = \Kminus{1}(\ZK)$, where
$K$ is the isotropy group of $\Wmax$, and
$\wh(\bCatrminus)=\wh(\rel{\ZH}{\ZG})$.
The indeterminacy in Theorem C
is then generated by the double coboundary
\[\delta^2\colon H^1(\widetilde K_0(\rel{\ZH}{\ZG^{-}}))
\to H^1(\wh(\rel{\ZH}{\ZG^{-}}))\]
used in Theorem B and the coboundary
\[\delta\colon \Coh{0}{\Kminus{1}(\ZK)} \to \Coh{1}{\wh(\rel{\ZH}{\ZG^{-}})}\]
from the Tate cohomology sequence of (\ref{ses}).

\end{remark}

Finally, we will  apply these results to $\rtop{G}$.
In Part \parttwo, Section \ref{splitting}, we will define a subgroup filtration
\eqncount
\begin{equation}R_t(G) \subseteq R_n(G) \subseteq R_h(G) \subseteq R(G)
\label{tnh}
\end{equation}
on the real representation ring
$R(G)$, inducing a filtration on
 \[\rtop{G} =  R(G)/R_t(G)\ .\]
Here $R_h(G)$ consists of those virtual elements with no
homotopy obstruction to similarity,
and $R_n(G)$ the virtual elements with no
normal invariant obstruction to similarity
(see \ptwo, Section \ref{splitting} for more precise definitions).
Note that $R(G)$ has the nice basis
$\{ t^i, \delta, \epsilon\,\mid\, 1\leqq i \leqq 2q-1\}$,
where $\delta=[\rminus]$ and $\epsilon=[\rplus]$.

Let $\rfree{G} = \{ t^a \mid (a, 4q) = 1\} \subset R(G)$
 be the subgroup
generated by the free representations.
To complete the definition, we let $\rfree{\Cy 2} = \{\rminus\}$
and $\rfree{e} = \{\rplus\}$.
Then
\[R(G) = \bigoplus\limits_{K \subseteq G} \rfree{G/K}\]
 and this
direct sum splitting intersected with the
filtration above gives the subgroups $\Rfree_h(G)$, $\Rfree_n(G)$ and
$\Rfree_t(G)$.
In addition, we can divide out $\Rfree_t(G)$ and
obtain subgroups $\Rfree_{h,\Top}(G)$ and $\Rfree_{n,\Top}(G)$
of $\rtopfree{G} = \rfree{G}/\Rfree_t(G)$.
By induction on the order of $G$, we see that it suffices to
study the  summand $\rtopfree{G}$.

Let $\rtildefree{G} = \ker(\Res\colon\rfree{G} \to \rfree{\Godd})$,
 and then project into $\rtop{G}$ to define
\[\rtildetopfree{G} = \rtildefree{G}/\Rfree_t(G)\ .\]
In Part \parttwo, Section \ref{ratcomp} we prove that $\rtildetopfree{G}$ 
is \emph{precisely}
the torsion subgroup of $\rtopfree{G}$, and
in \ptwo, Section \ref{sD} we show that the subquotient
$\Rtildefree_{n,\Top}(G)=\Rtildefree_n(G)/\Rfree_t(G)$
always has exponent two.

Here is a specific computation (correcting \cite[Thm. 2]{cssw1}),
proved in \ptwo, Section \ref{sD}.
\begin{thmd}
Let $G=\Cy{4q}$, with $q>1$ odd, and  suppose that the
fields $\bq(\zeta_{d})$ have odd class number for
all $d \mid 4q$.
Then $\rtildetopfree{G} = \cy{4}$ generated by
$(t-t^{1+2q})$.
\end{thmd}
For any cyclic group $G$,
both $\rfree{G}/\Rfree_h(G)$ and
$\Rfree_h(G)/\Rfree_n(G)$
are torsion groups  which can be explicitly
determined by congruences in the weights
 (see  \ptwo, Section \ref{normalinvariant} and \cite[Thm.1.2]{yo1}).


We conclude this list of sample results with a calculation of
$\rtop{G}$ for cyclic $2$-groups.

\begin{thme}
Let $G=\Cy{2^r}$, with $r\geqq 4$.
Then
\[\rtildetopfree{G} = \big\langle \alpha_1, \alpha_2, \dots, 
\alpha_{r-2}, \beta_1, \beta_2, \dots,  \beta_{r-3} 
\big\rangle\]
subject to the relations $2^s \alpha_s = 0$ for $1\leqq s \leqq r-2$,
and $2^{s-1}(\alpha_s + \beta_s) =0$ for $2\leqq s \leqq r-3$, 
together with $2(\alpha_1 + \beta_1) =0$.
\end{thme}
\noindent
The generators for $r\geqq 4$ are given by the elements
\[\begin{matrix} \alpha_s = t - t^{5^{2^{r-s-2}}} & \text{and}&
\beta_s =t^5 - t^{5^{2^{r-s-2}+1}}.\end{matrix} \]
We remark that  $\rtildetopfree{\Cy{8}} = \cy 4$ generated by
$t-t^5$. 
In Theorem \ref{unstablefour}
we use this information to give a complete
topological classification of linear representations for cyclic
$2$-groups. 
\begin{acknowledgement} The authors would like to express their
appreciation to the referee for many constructive comments and suggestions.
\end{acknowledgement}

\section{ A criterion for non{--}linear similarity}
\label{basics}
Our approach to the non-linear similarity problem is
through bounded surgery
theory (see \cite{fp2}\cite{hp1}, \cite{hp2}).
First an elementary observation about topological equivalences for
cyclic groups.
\begin{lemma}\label{LsameW}
If $V_1\oplus W \sim_t V_2\oplus W^{\prime}$,
where $V_1$, $V_2$ are free $G${--}representations, and $W$
and $W^{\prime}$ have  no free summands,
then there is a $G$-homeomorphism $h\colon V_1\oplus W \to
V_2\oplus W $ such that
\[h\big\vert\,\bigcup_{1\neq H \leq G}W^H\]
is the identity.
\end{lemma}
\begin{proof} Let $h$ be the homeomorphism given by
$V_1\oplus W \sim_t V_2\oplus W^{\prime}$.
 We will successively change $h$,  stratum by stratum.
For every subgroup $K$ of $G$, consider the homeomorphism of $K$-fixed sets
\[f^K\colon   W^K\to W^{\prime K}.\]
This is a homeomorphism of $G/K$,  hence of $G${--}representations.
As $G${--}representations we can split
\[V_2\oplus W' = U\oplus W^{\prime K} \sim_t U\oplus W^K = V_2\oplus W''\]
where the similarity uses the product of the identity and $(f^K)^{-1}$.
Notice that the composition of $f$ with this similarity is the
\emph{identity} on the $K$-fixed set.
Rename $W''$ as $W'$ and repeat this successively for all subgroups.
We end up with
$W=W'$ and a $G${--}homeomorphism inducing the identity
on the singular set.
\end{proof}
One  consequence is
\begin{lemma}If $V_1\oplus W \sim_t V_2\oplus W$,  then there exists
a $G$-homotopy equivalence $ S(V_2) \to S(V_1)$.
\end{lemma}
\begin{proof} We may assume that $W$ contains no free summand, since a $G$-homotopy equivalence $S(V_2\oplus U) \to S(V_1\oplus U)$, with $U$ a free $G$-representation,  is $G$-homotopic to $f\times 1$, where 
$f\colon S(V_2) \to S(V_1)$ is a $G$-homotopy equivalence.
If we $1$-point compactify $h$,  we obtain a $G$-homeomorphism
\[h^{\scriptscriptstyle +}
\colon S(V_1\oplus W\oplus \br) \to S(V_2\oplus W\oplus\br).\]
After an isotopy, the image of the free $G$-sphere
$S(V_1)$ may be assumed to lie  in the complement
$S(V_2\oplus W\oplus\br) - S(W\oplus\br)$
of $S(W\oplus\br)$ which is $G$-homotopy equivalent to $S(V_2)$.
\end{proof}

Any homotopy equivalence $f\colon S(V_2)/G \to S(V_1)/G$ defines an element
$[f]$ in the structure set $\St^h(S(V_1)/G)$.
We may assume that $\dim V_i \geqq 4$.
This element must be non-trivial: otherwise
$S(V_2)/G$ would be topologically $h$-cobordant to $S(V_1)/G$,
and Stallings infinite
repetition of $h$-cobordisms trick
would produce a homeomorphism $V_1\to V_2$
contradicting  \cite[7.27]{atb2} (see also \cite[12.12]{mi5}), since $V_1$ and $V_2$ are free
representations.
More precisely,  we  use Wall's extension of the Atiyah{--}Singer
equivariant index formula to the topological
locally linear case \cite{w2}.
If $\dim V_i =4$, we can cross with ${\mathbf{CP}}^2$ to avoid
low{--}dimensional difficulties.
Crossing with $W$ and parameterising by projection
on $W$ defines a map  from
the classical surgery sequence to the
bounded surgery exact sequence
\eqncount
\begin{equation}
\begin{matrix}
\xymatrix{
L^h_n(\ZG)\ar[r]\ar[d] &
{\mathcal S}^h(S(V_1)/G)\ar[r]\ar[d]&
[S(V_1)/G,\text{F/Top}]\ar[d] \\
L^h_{n+k}(\bcat{W}) \ar[r] &
\clunk\ar[r] &
[S(V_1)\times_G W,\text{F/Top}]
}
\end{matrix}
\label{transferdiagram}
\end{equation}
The $L${--}groups in the upper row are the ordinary surgery obstruction
groups for oriented manifolds and surgery up to homotopy equivalence.
In the lower row, we have bounded $L${--}groups (see 
\ptwo, Section \ref{excision})
corresponding to an orthogonal action $\rho_W\colon G \to O(W)$,
with orientation character given by $\det(\rho_W)$.
 Our main criterion for non{--}linear similarities is:

\begin{theorem}\label{maincriterion}
Let $V_1$ and $V_2$ be free $G${--}representations
 with $\dim V_i \geqq 2$.
Then, there is a topological equivalence $V_1 \oplus W
\sim_t V_2 \oplus W$
if and only if there exists a $G${--}homotopy equivalence
$f\colon S(V_2) \to S(V_1)$ such that the
element $[f] \in \St^h (S(V_1)/G)$
 is in the kernel of the bounded transfer map
\[\trf{W}\colon\,  \St^h (S(V_1)/G)  \to
\St^h_b {\left(\begin{smallmatrix} S(V_1)\times_G W\\
 \downarrow\\ W/G\end{smallmatrix}\right)}\ .\]
\end{theorem}
\begin{proof}
For necessity, we refer the reader to
 \cite{hp2} where this is proved using a
version of equivariant engulfing.
For sufficiency, we notice that
crossing with $\br$ gives an isomorphism
of the bounded surgery exact sequences parameterized
by $W$ to simple bounded
surgery exact sequence parameterized by $W\times \br$.
By the bounded
$s$-cobordism theorem, this means that the vanishing of the
bounded transfer implies that
\[
\xymatrix{ S(V_2)\times W\times \br \ar[r]^{f\times 1}
&S(V_1)\times W\times \br \ar[d]\\ & W\times \br }
\]
is within a bounded
distance of an equivariant homeomorphism $h$, where distances
are measured in
$W\times \br$.
We can  obviously complete $f\times 1$   to the map
 \[f*1 \colon  S(V_2)*S(W\times \br) \to S(V_1)* S(W\times \br)\]
 and since bounded in
$W\times \br$ means small near the subset
\[S(W\times \br)\subset
S(V_i)*S(W\times\br)= S(V_i\oplus W \oplus \br),\]
 we can complete $h$ by the
identity to get a $G$-homeomorphism
\[S(V_2\oplus W\oplus \br) \to S(V_1\oplus W\oplus \br)\]
 and taking a point out we have a $G$-homeomorphism $V_2\oplus W
\to V_2\oplus W$
\end{proof}

By comparing the ordinary and bounded
surgery exact sequences (\ref{transferdiagram}),
and noting that the bounded transfer induces the
identity on the normal invariant term, we see that
a necessary condition for the existence of any stable
similarity $f\colon V_2 \approx_t V_1$ is
 that $f\colon S(V_2) \to S(V_1)$ has
 $s${--}normal invariant zero.
Assuming this, under the natural map
\[L_n^h(\ZG) \to \St^h (S(V_1)/G),\]
where $n = \dim V_1$,
the element $[f]$ is the image of $\sigma(f) \in L_n^h(\ZG)$,
obtained as the surgery obstruction (relative to the
boundary) of a normal cobordism from $f$ to the identity.
The element $\sigma(f)$ is well-defined in
$\tilde  L_n^h(\ZG) = \coker (L^h_n(\bz) \to L_n^h(\ZG))$.
Since the image of the normal invariants
\[ [S(V_1)/G\times I,S(V_1)/G\times \partial I ,\text{F/Top}]
\to L_n^h(\ZG)\]
factors through $L_n^h(\bz)$ (see \cite[Thm.A, 7.4]{hmtw1}
for the image of the assembly map), we may apply the criterion
of \ref{maincriterion} to any lift $\sigma(f)$ of $[f]$.
This reduces the evaluation of the bounded transfer on 
structure sets to a bounded $L${--}theory calculation.
\begin{theorem}\label{maintest}
Let $V_1$ and $V_2$ be free $G${--}representations 
with $\dim V_i \geqq 2$.
Then, there is a topological equivalence $V_1 \oplus W
\sim_t V_2 \oplus W$
if and only if there exists a $G${--}homotopy equivalence
$f\colon S(V_2) \to S(V_1)$, which is $G${--}normally
cobordant to the identity, such that
$\trf{W}(\sigma(f))=0$, where
$\trf{W}\colon L_n^h(\ZG) \to L_{n+k}^h(\ccat{W,G}{\bz})$
is the bounded transfer.
\end{theorem}
\medskip
The rest of the paper is about the computation
of these bounded transfer homomorphisms in $L${--}theory.
We will need the following result (proved for $K_0$ in \cite[6.3]{hp2}).
\begin{theorem}\label{ktrans}
Let $W$ be a $G${--}representation with $W^G=0$.
For all $i \in \bz$, the  bounded transfer
$\trf{W}\colon K_i(\ZG) \to K_i(\bcat{W})$ is equal to
 the cone point inclusion
$c_{\ast}\colon K_i(\ZG) = K_i(\ccat{pt,G}{\bz}) \to K_i(\bcat{W})$.
\end{theorem}

\begin{proof}
Let $G$ be a finite group and $V$ a representation.
Crossing with $V$ defines a transfer map in $K${--}theory
$K_i(RG)\to K_i(\ccat{V,G}{R})$ for all $i$,
where $R$ is any ring with unit \cite[p.117]{hp1}.
To show that it is equal to the map
$K_i(\ccat{0,G}{R})\to K_i(\ccat{V,G}{R})$
induced by the inclusion $0\subset V$,
 we need to choose models for $K${--}theory.

For $RG$ we choose the category of finitely generated free $RG$
modules, but we think of it as a category with cofibrations and weak
equivalences with weak equivalences isomorphisms and cofibrations split
inclusions.
For $\ccat{V,G}{R}$ we use the category of finite length chain
complexes, with weak equivalences chain homotopy equivalences and cofibrations
sequences that are split short exact at each level.
The $K${--}theory of this
category is the same as the $K$-theory of $\ccat{V,G}{R}$.
For an argument working
in this generality see \cite{cap1}.

Tensoring with the chain complex of $(V,G)$ induces a map of categories with
cofibrations and weak equivalences hence a map on $K${--}theory.
It is elementary
to see this agrees with the geometric definition in low dimensions, since the
identification of the $K${--}theory of chain complexes of an additive category with
the $K${--}theory of the additive category is an Euler characteristic
 (see e. g. \cite{cap1}).

By abuse of notation we denote the category of finite chain complexes in
$\ccat{V,G}{R}$ simply by $\ccat{V,G}{R}$.
We need to study various related
categories.
First there is $\cc^{iso}_{V,G}(R)$ where we have replaced the weak
equivalences by isomorphisms.
Obviously the transfer map, tensoring with the
chains of $(V,G)$ factors through this category.
Also the transfer factors
through the category $\cd^{iso}_{V,G}(R)$ with the same objects, and
isomorphisms as weak equivalences but the control condition is $0$-control
instead of bounded control.
The category $\cd^{iso}_{V,G}(R)$ is the product of
the full subcategories on objects with support at $0$ and the full subcategory
on objects with support on $V-0$, $\cd^{iso}_{0,G}(R) \times
\cd^{iso}_{V-0,G}(R)$,
and the transfer factors through chain  complexes concentrated in degree $0$ in
$\cd^{iso}_{0,G}(R)$ crossed with chain complexes in the other factor.

But the
subcategory of chain complexes concentrated in degree
zero of $\cd^{iso}_{0,G}(R)$
is precisely the same as $\cc_{0,G}(R)$
and the map to $\cc_{V,G}(R)$ is induced by inclusion.
So to finish the proof
we have to show that the other factor $\cd^{iso}_{V-0,G}(R)$ maps to zero.
For this we construct an intermediate category
 $\ce^{iso}_{V-0,G}(R)$ with the same objects,
but where the morphisms are bounded radially and $0$-controlled otherwise
(i.~e. a nontrivial map between objects at different points is only allowed if
the points are on the same radial line, and there is a  bound on the
distance independent of the points).
This category has trivial $K$-theory since we can make a radial
Eilenberg swindle toward infinity.
Since the other factor $\cd^{iso}_{V-0,G}(R)$ maps through this
category, we find that the transfer maps through the corner inclusion as
claimed.
\end{proof}

\begin{remark}
It is an easy consequence of the filtering arguments
based on \cite[Thm.3.12]{hp1} that the bounded $L${--}groups
are finitely generated abelian groups with $2${--}primary
torsion subgroups.
We will therefore localize all the $L${--}groups by tensoring
with $\bz_{(2)}$ (without changing the notation), and this loses
no information for computing bounded transfers.
One concrete advantage of working with the $2${--}local
$L${--}groups is that we can use the idempotent
decomposition \cite[\S 6]{hm1} and the direct sum
splitting $ L_n^h(\ccat{W,G}{\bz}) =
\oplus_{d\mid q}  L_n^h(\ccat{W,G}{\bz})(d)$.
Since the ``top component" $L_n^h(\ccat{W,G}{\bz})(q)$
is just the kernel of the restriction map to all odd index
subgroups of $G$, the use of components is well-adapted
to inductive calculations.
\end{remark}
A first application of these techniques was given in
\cite[5.1]{hp2}.
\begin{theorem}\label{twopoweriso}
For any $G${--}representation $W$,
let $W=W_1\oplus W_2$ where $W_1$ is the direct sum of the
irreducible summands of $W$ with
isotropy subgroups of $2${--}power index.
If $W^G=0$, then
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item the  inclusion
$L_n^h(\ccat{W_1,G}{\bz})(q) \to
L_n^h(\ccat{W,G}{\bz})(q)$
is an isomorphism on the top component,
\item the bounded transfer
\[\trf{W_2}\colon L_n^h(\ccat{W_1,G}{\bz})(q) \to
L_n^h(\ccat{W,G}{\bz})(q)\]
is an injection on the top component, and
\item $\ker(\trf{W}) =\ker (\trf{W_1}) \subseteq L^h_n(\ZG)(q)$.
\end{enumerate}
\end{theorem}
\begin{proof}
In \cite{hp2} we localized at an odd prime $p \nmid |G|$ in order to
use the Burnside idempotents for all cyclic subgroups of $G$.
The same proof works for the $L${--}groups localized at $2$,
to show that $\trf{W_2}$ is injective on the top component.
\end{proof}


\begin{lemma}
For any choice of normal cobordism between $f$ and
the identity, the surgery obstruction $\sigma(f)$
is a nonzero element of infinite order in $\tilde L_n^h(\ZG)$.
\end{lemma}
\begin{proof}
See \cite[4.5]{hp2}
\end{proof}
The following result (combined with Theorem \ref{maintest}) shows
that there are no non-linear similarities between
semi-free $G${--}representations, since
$L_{n+1}^h(\ccat{\br,G}{\bz}) = L^p_n(\ZG)$ and the
natural map $L^h_n(\ZG ) \to L^p_n(\ZG)$
may be identified with the bounded transfer
$\trf{\br}\colon L^h_n(\ZG ) \to L^h_{n+1}(\bcat{\br})$ \cite[\S 15]{ra6}.
\begin{corollary}\label{semifree}
Under the natural map $L^h_n(\ZG ) \to L^p_n(\ZG)$,
the image of $\sigma(f)$ is nonzero.
\end{corollary}
\begin{proof}
The kernel of the map $L^h_n(\ZG ) \to L^p_n(\ZG)$
is the image of $H^n(\widetilde K_0(\ZG))$ which is a torsion
group.
\end{proof}

\section{ Bounded $\rminus $ transfers}
\label{rmin}
 Let $G$ denote a finite group of even order,
with a subgroup $H < G$ of index 2.
We first describe the connection between the
bounded $\rminus $ transfer
and the compact line bundle transfer  of \cite[12C]{w2}
by means of the following diagram:
\[
\xymatrix{
L^h_n(\ZG,w)\ar[d]_{\trfrminus}\ar[r]^{\trf{I_{-}}} &
L^{h}_{n+1}(\rel{\ZH}{\ZG},w\phi)\ar[d]^{j_{\ast}}\\
L^h_{n+1}(\bCatrminus,w\phi)&
 L^{k,h}_{n+1}(\rel{\ZH}{\ZG},w\phi)\ar[l]^{r_{\ast}}
}
\]
where $w\colon G \to \{\pm 1\}$ is the orientation
character for $G$ and $\phi\colon G \to \{ \pm 1\}$
has kernel $H$.
On $\bCatrminus$ we start with the standard orientation defined
in \ptwo, Example \ref{standard},  and then twist by $w$ or $w\phi$.
Note that the  (untwisted) orientation induced
on $\ccat{pt}{\ZG}$ via the cone point
inclusion $c\colon \ccat{pt}{\ZG} \to \bCatrminus $
is \emph{non{--}trivial}.
The homomorphism
\[r_*\colon L^{k,h}_{n+1}(\rel{\ZH}{\ZG},w\phi)
\to L^h_{n+1}(\bCatrminus,w\phi)\]
is obtained by adding a ray $[1,\infty)$ to each point
of the boundary double cover in domain and
range of a surgery problem.
Here $\ck$ in the decoration means that
we are allowing  projective $\ZH$-modules that become
free when induced up to $\ZG$
\begin{theorem}\label{rays} The
map $r_*\colon L^{k,h}_{n+1}(\rel{\ZH}{\ZG},w\phi)
\to L^h_{n+1}(\bCatrminus,w\phi)$ is an isomorphism,
and under this identification, the bounded
$\rminus$ transfer corresponds to the
line bundle transfer, followed by the relaxation of projectivity
map $j_{\ast}$ given by $\ck$.
\end{theorem}
\begin{proof}
Let $\ca$ be the full subcategory of $\cu=\bcat{\rminus}$ with objects
that are only nontrivial in a bounded neighborhood of $0$.
Then $\bcat{\rminus}$ is $\ca$-filtered.
The category $\ca$ is
equivalent to the category of free $\ZG$-modules (with the non-orientable
involution).
The quotient category $\cu/\ca$ is equivalent to
 $\mathcal C_{[0,\infty),H}^{>0}(\bz)$, which has the same $L$-theory as
 $\mathcal C_{\br}(\ZH)$, so we get a fibration of spectra
\[\BL^k(\ZH) \to \BL^h(\ZG) \to \BL^h(\bCatrminus)\ .\]
This shows that
 \[\BL^h(\bCatrminus) \simeq \BL^{k,h}(\ZH \to \ZG)\ \]
\end{proof}
The  line bundle transfer can be studied by the long exact sequence
\eqncount
\begin{multline}
\dots\to LN_n(\rel{\ZH}{\ZG},w\phi) \to
L^h_n(\ZG,w) \to  L^h_{n+1}(\rel{\ZH}{\ZG},w\phi)\\
 \to LN_{n-1}(\rel{\ZH}{\ZG},w\phi)\to
 L^h_{n-1}(\ZG,w) \to\dots
\label{LNseq}
\end{multline}
given in \cite[11.6]{w2}.
The obstruction groups $ LN_n(\rel{\ZH}{\ZG},w\phi)$
for  codimension 1 surgery have an algebraic description
\eqncount
\begin{equation} LN_n(\rel{\ZH}{\ZG},w\phi)\cong  L^h_n(\ZH, \alpha, u)
\label{lbundle}
\end{equation}
 given by \cite[12.9]{w2}.
The groups on the right{--}hand side are the algebraic
$L${--}groups of the ``twisted" anti{--}structure defined by choosing
some element  $t \in G-H$ and then setting
$\alpha(x) =w(x)t^{-1}x^{-1}t$ for all $x\in H$, and
$u = w(t)t^{-2}$.
Another choice of $t \in G-H$ gives a scale equivalent
anti{--}structure on $\ZH$.
The same formulas also give a ``twisted" anti{--}structure
 $(\ZG, \alpha, u) $ on $\ZG$, but since the conjugation by $t$
is now an inner automorphism of $G$
 this is  scale equivalent to the standard
structure $(\ZG, w)$.
We can therefore define the  twisted induction map
\[\tilde i_*\colon L^h_n(\ZH, \alpha, u) \to L^h_n(\ZG,w)\]
and the twisted restriction map
\[\tilde\gamma_*\colon L^h_n(\ZG,w) \to L^h_n(\ZH, \alpha, u)\]
as the composites of the ordinary induction or restriction
maps (induced by the inclusion $(\ZH ,w) \to (\ZG,w)$)
with the scale isomorphism.

The  twisted anti{--}structure on $\ZH$
is an example of a ``geometric anti{--}structure"
\cite[p.110]{htw2}:
\[\alpha(g) = w(g)\theta(g^{-1}), \qquad\qquad u = \pm b\]
where $\theta\colon G \to G$ is a group automorphism
with $\theta^2(g) = bgb^{-1}$, $w\circ \theta = w$, $w(b) = 1$
and $\theta(b) = b$.
\begin{example}
For $G$ cyclic, the orientation character restricted to $H$
is trivial,  $\theta(g) = tgt^{-1} = g$ and $u = w(t)t^2$.
Choosing $t \in G$ a generator we get $b=t^{2}$, which is
a generator for $H$.
\end{example}

There is an identification
\cite[Thm. 3]{h1}, \cite[50--53]{htw3}
of the exact sequence (\ref{LNseq})
for the line bundle transfer,
extending the scaling isomorphism
$ L^h_n(\ZG, w)\cong L^h_n(\ZG, \alpha, u)$
and (\ref{lbundle}),
with the long exact sequence of the ``twisted"
inclusion
\[
\dots L^h_n(\ZH, \alpha, u) \RA{\tilde{i}_*}
 L^h_n(\ZG, \alpha, u)
\to L^h_n(\rel{\ZH}{\ZG}, \alpha, u) \to  L^h_{n-1}(\ZH, \alpha, u)
\to\dots
\]
These identifications can then be substituted into the
following ``twisting diagram" in order to compute the various
maps (see \cite[Appendix 2]{htw3} for a complete
tabulation in the case of finite 2{--}groups).

\eqncount
\begin{equation}
\begin{matrix}
\xymatrix@!C@C-6pc{
LN_n(\rel{\ZH}{\ZG},w\phi)     \ar[dr] \ar@/^2pc/[rr]^{\tilde i_*}    &&
L_n(\ZG,w)                     \ar[dr] \ar@/^2pc/[rr]^{\gamma_*}      &&
L_n(\ZH,w)                                                     \\
&  L_{n+1}(\gamma_*)           \ar[dr] \ar[ur]                 &&
L_{n+1}(\rel{\ZH}{\ZG},w\phi)  \ar[dr] \ar[ur]                 \\
L_{n+1}(\ZH,w)                 \ar[ur] \ar@/_2pc/[rr]^{i_*}_{}        &&
L_{n+1}(\ZG,w\phi)             \ar[ur] \ar@/_2pc/[rr]^{\tilde\gamma_*}&&
LN_{n-1}(\rel{\ZH}{\ZG},w\phi)
}\end{matrix}
\end{equation}


The existence of the diagram depends on the identifications
$L_{n+1}(\gamma_*)\cong L_{n}(\tilde\gamma_*)$
and $L_{n+1}(i_*)\cong L_{n}(\tilde i_*)$ obtained geometrically
in \cite{h1} and algebraically in \cite{ra9}.


\section{ Some basic facts  in $K$- and $L$-theory}
\label{calc}
In this section we record various calculational facts
from the literature about
$K${--} and $L${--}theory of cyclic groups.
A general reference for the $K${--}theory
is \cite{o1}, and for $L${--}theory computations is \cite{ht1}.
Recall that $\widetilde K_0(\ca) =K_0(\ca^\wedge)/K_0(\ca)$ for any additive
category $\ca$, and $\wh(\ca)$ is the
quotient of $K_1(\ca)$ by the subgroup defined by the system of stable
isomorphisms.
\begin{theorem}\label{kgroups}
Let $G$ be a cyclic group, $K$ a subgroup.
We then have
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $K_1(\ZG) = (\ZG)^* \subset K_1(\QG)$
Here $(\ZG)^*$ denotes the units of $\ZG$.
\item The torsion in $K_1(\ZG))$ is precisely $\{\pm G\}$,
so $\wh(\ZG)$ is torsion free.
\item The maps $K_1(\ZK) \to K_1(\ZG)$ and
\[\wh(\ZG)/\wh(\ZK) \to \wh(\QG)/\wh(\QK)\]  are injective.
\item $\widetilde K_0(\ZG)$ a torsion group and
the map
$\widetilde K_0(\ZG)\to \widetilde K_0(\bz_{(p)} G$ is the zero map for all
primes $p$.
\item $\Kminus{1}(\ZG)$ is torsion free, and sits in an exact
sequence
\[0 \to K_0(\bz) \to K_0(\zlocal G) \oplus K_0(\QG) \to K_0(\qlocal G)
\to \Kminus{1}(\ZG) \to 0\ .\]
\item $\Kminus{1}(\ZK) \to \Kminus{1}(\ZG)$ is an injection.
\item $\Kminus{j}(\ZG) = 0 $ for $j\geqq 2$.
\end{enumerate}
\end{theorem}
\begin{proof}
The proof mainly consists of references.
See \cite[pp.6,14]{o1} for the first two parts.
Part (iii) follows from (i) and the relation
$(\ZG)^*\cap (\QK)^* = (\ZK)^*$.
Part (iv) is due to Swan \cite{sw1}, and part (vii) is
a result of Bass and Carter \cite{carter1}.
Part (v) gives the arithmetic sequence for computing
$\Kminus{1}(\ZG)$, and the assertion that $\Kminus{1}(\ZG)$ is
torsion  free is easy to deduce (see also \cite{carter1}).
Since $\Res_K\circ \Ind_K$ is multiplication by the index
$[G:K]$, part (vi) follows from (v).
\end{proof}

Tate cohomology of $K_i$-groups play an important role.
The involution on
$K${--}theory is induced by duality on modules.
It is conventionally chosen to
have the boundary map
 \[K_1(\qlocal (G) \to \tilde K_{0}(\ZG)\]
preserve the involution, so to make this happen we choose to have the
involution on $K_0$ be given by sending $[P]$  to $-[P^*]$, and the
involution on $K_1$ is given by sending $\tau$ to $\tau^{\ast}$.
This causes a
shift in dimension in Ranicki-Rothenberg exact sequences
\[\ldots\to H^0(\widetilde K_0(\ca))\to
 L_{2k}^h(\ca)\to L_{2k}^p(\ca)\to H^1(\widetilde K_0(\ca))\to \ldots\]
compared to
\[\ldots\to H^1(\wh(\ca))\to L_{2k}^s(\ca)\to L_{2k}^h(\ca)\to
 H^0(\wh(\ca))\to \ldots\]
and
\[\ldots\to H^1(K_{-1}(\ca))\to L_{2k}^p(\ca)\to \Lminus{1}_{2k}(\ca)\to
 H^0(K_{-1}(\ca))\to \ldots\]
\begin{theorem}\label{Lfacts}
 Let $G$ be a cyclic group, $K$ a subgroup.
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $L_{2k}^s(\ZG)$, $L^p_{2k}(\ZG)$, and
 $\Lminus{1}_{2k}(\ZG)$
 are torsion{--}free  when $k$ is even, and when $k$ is odd the only
torsion is a $\cy 2$-summand
generated by the Arf invariant element.
\item The groups
$L^h_{2k+1}(\ZG)= L^s_{2k+1}(\ZG)=L^p_{2k+1}(\ZG)$
are zero ($k$ even), or $\cy 2$ (if $k$ odd and $|G|$ is even),
detected by projection $G \to \Cy 2$.
\item $\Lminus{1}_{2k+1}(\ZG) = H^1(\Kminus{1}(\ZG))$ ($k$
 even), or
$\cy 2\oplus H^1(\Kminus{1}(\ZG))$ ($k$ odd).
\item The Ranicki-Rothenberg exact sequence gives
\[0\to H^0(\widetilde K_0(\ZG))\to L_{2k}^h(\ZG)\to
L^p_{2k}(\ZG)\to H^1( \widetilde K_0(\ZG))\to 0\]
so $L^h_{2k}(\ZG)$ has the torsion subgroup $H^0(\widetilde K_0(\ZG))$.
\item The double coboundary
$\delta^2\colon  H^0(\widetilde K_0(\ZG)) \to H^0(\wh(\ZG))$
is injective.
\item The maps $L^s_{2k}(\ZK)\to L^s_{2k}(\ZG)$,
$L^p_{2k}(\ZK)\to  L^p_{2k}(\ZG)$ , and
 $\Lminus{1}_{2k}(\ZK)\to
\Lminus{1}_{2k}(\ZG)$ are injective  when $k$
is even or $[G:K]$ is odd.
For  $k$ odd and $[G:K]$ even, the kernel is generated
by the Arf invariant element.
\item In the oriented case,
$\wh(\ZG)$ has trivial involution and
$H^1(\wh(\ZG))=0$.
\end{enumerate}
\end{theorem}
\begin{proof}
See \cite[\S 3, \S 12]{ht1} 
for the proof of part (i) for $L^s$ or $L^p$. 
Part (ii) is  due to Bak for $L^s$ and $L^h$
\cite{bak1}, and is proved in \cite[12.1]{ht1} for $L^p$. 
We can now substitute this information into the Ranicki-Rothenberg sequences above to get part (iv).
Furthermore, we see that the maps
$\Lminus{1}_{n}(\ZG) \to H^n(K_{-1}(\ZG))$ are all surjective,
and the extension giving  $\Lminus{1}_{2k+1}(\ZG)$
actually splits. This gives part (iii).
For part (v) we use the fact that the double
coboundary $\delta^2\colon  H^0(\widetilde K_0(\ZG)) \to H^0(\wh(\ZG))$ can be
identified with the composite
$$H^0(\widetilde K_0(\ZG)) \to L_0^h(\ZG) \to H^0(\wh(\ZG))$$
(see \ptwo, Section \ref{doublecob}).
Part (vii) is due to Wall \cite{o1}.

For $\Lminus{1}_{2k}(\ZG)$ we use the exact sequence
$$0\to \Lminus{1}_{2k}(\ZG)\to L_{2k}^p(\zlocal G)\oplus
L_{2k}^p(\QG) \to  L_{2k}^p(\qlocal G)
$$
 obtained from the braid of exact sequences
given in \cite[3.11]{hm1} by substituting the calculation
$  L_{2k+1}^p(\qlocal G)=0$ from \cite[1.10]{hm3}.
It is also convenient to use the idempotent decomposition
(as in \cite[\S 7]{hm1}) for $G=\Cy{2^r q}$, $q$ odd:
$$\Lminus{1}_{2k}(\ZG) =
\bigoplus_{d\mid q} \Lminus{1}_{2k}(\ZG)(d)$$ 
where the $d$-component, $d\neq q$, is mapped isomorphically
 under restriction  to $L_{2k+1}^p(\ZK, w)(d)$ for $K=\Cy{2^r d}$.
This decomposition extends to a decomposition of the arithmetic
sequence above. The summand corresponding to $d=1$ may be
neglected since $L^p = \Lminus{1}$ for a $2$-group
(since the $K_{-1}$ vanishes in that case).

We now study $L_{2k}^p(\QG)$ by comparing it to
$L_{2k}^p(\qlocal G)\oplus L_{2k}^p(\br G)$ as in
\cite[1.13]{hm3}. 
Let $CL_{n}^p(S) = L^p_n(S \hskip -2.2truept \to
\hskip -2.2truept S_A)$, where  $S$ is a 
factor of $\QG$, and $S_A = \widehat S \oplus (S\otimes\br)$.
If $S$ has type $U$, we obtain  $CL_{2k+1}^p(S) =0$, and we have
an extension $0\to \cy 2 \to CL_{2k}^p(S)
\to H^1(K_0(S_A)/K_0(S))\to 0$. 
We may now assume that $q>1$, implying that all the
factors in the $q$-component of $\QG$ have type $U$.
By induction on $q$ it is enough to
consider the $q$-component of the exact sequence above. 
It can be 
re-written in the form
$$0\to \Lminus{1}_{2k}(\ZG)(q)\to  L_{2k}^p(\zlocal G)(q)\oplus
L_{2k}^p(\br G)(q) \to  CL_{2k}^p(\QG)(q) $$
But $ L_{2k}^p(\zlocal G)(q)\cong H^1(K_0(\zlocal G)(q))$
by \cite[1.11]{hm3}, and the group $H^1(K_0(\zlocal G)(q))$
injects into $CL_{2k}^p(\QG)(q)$.
To see this we use the exact sequence in  Theorem \ref{kgroups} (v),
and the fact that the involution on $K_0(\QG)$ is multiplication by
$-1$.
We conclude that $\Lminus{1}_{2k}(\ZG)(q)$ injects into
$L_{2k}^p(\br G)(q)$ which is torsion-free by \cite[1.9]{hm3}.
 Part (vi) now follows
 from part (i) and the property $\Res_K\circ \Ind_K = [G:K]$.
\end{proof}

\section{ The computation of $L_1^p(\ZG,w)$ }
\label{projective}
Here we correct an error in the statement of \cite[5.1]{hm3}
(Notice however that Table 2
\cite[p.553]{hm3} has the correct answer).
\begin{proposition}
Let $G = \sigma \times \rho$, where $\sigma$ is an abelian
$2${--}group  and $\rho$ has odd order.
Then $L^p_n(\ZG, w) =
L^p_n(\bz\sigma, w) \oplus L^p_{n}(\rel{\bz\sigma}{\ZG}, w)$
where $w \colon G \to \{ \pm 1\}$ is an orientation character.
For $i= 2k$, the second summand is free abelian and
 detected by signatures
at the type $U(\bc)$ representations of $G$
which are non{--}trivial on $\rho$.
For $n = 2k+1$, the second summand  is
a direct sum of $\cy 2$'s, one for each type $U(\GL)$
representation of $G$ which is non{--}trivial on $\rho$.
\end{proposition}
\begin{remark}  Note that type $U(\bc)$ representations of $G$
exist only when $w \equiv 1$, and type $U(\GL)$
representations of $G$ exist only when $w\not\equiv 1$.
In both cases, the second summand is computed by transfer
to cyclic subquotients of order $2^rq$, $q>1$ odd, with
$r \geqq 2$.
\end{remark}
\begin{proof} The given direct sum decomposition follows from the
existence of a retraction of the inclusion $\sigma \to G$
compatible with $w$.
It also follows that
\[ L^{p,h}_{n+1}(\Zreltwo{G}, w) \cong
 L^{p,h}_{n+1}(\Zreltwo{\sigma}, w)\oplus
  L^p_{n}(\rel{\bz\sigma}{\ZG}, w)\]
since the map $ L^h_{n}(\zlocal _2\sigma, w)
 \to L^h_{n}(\Zadictwo{G}, w)$
is an isomorphism.
The computation of the relative groups for $\zreltwo$
can be read off from \cite[Table 2,Remark 2.14]{hm3}:
for each centre field $E$ of a type $U(\GL)$ representation,
the contribution is $H^0(C(E)) \cong \cy 2$ if $i\equiv 1\Mod{2}$.

The detection of $ L^p_{i}(\rel{\bz\sigma}{\ZG}, w)$ by cyclic
subquotients is proved in \cite[1.B.7, 3.A.6, 3.B.2]{htw2}.
\end{proof}
\begin{corollary}\label{galoisaut}
Let $G = \Cy{2^rq}$, for $q > 1$ odd and
$r  \geqq 2$.
Then the group
$$L_{2k+1}^p(\ZG, w) = \bigoplus_{d\mid q} L_{2k+1}^p(\ZG, w)(d)$$
where the $d$-component, $d\neq q$, is mapped isomorphically
under restriction to $L_{2k+1}^p(\ZK, w)(d)$ for $K=\Cy{2^r d}$.
The $q$-component  is given by the formula
 \[L_{2k+1}^p(\ZG, w)(q) = \bigoplus_{i=2}^r CL^K_2(E_i)
\cong(\cy 2)^{r-1}\]
 when $w \not\equiv 1$,
where the summand $CL_2^K(E_i) = H^0(C(E_i))$, $2\leqq i\leqq r$,
corresponds to the type $U(\GL)$ rational representation with
centre field $E_i = \bq(\zeta_{2^iq})$.
\end{corollary}
\begin{remark}\label{cssw-wrong}
 The
calculation of $L^p_1$
 contradicts the assertion in \cite[p.733,l.-8]{cssw1}
 that the projection
map $G \to \Cy{2^r}$ induces an isomorphism on
$L_1^p$ in the non{--}oriented case.
In fact, the projection detects only the $q=1$ component.
This error invalidates the proofs of the main results
of \cite{cssw1} for cyclic groups \emph{not} of $2${--}power order,
so the reader should not  rely on the statements.
In particular, we have already noted that \cite[Thm. 1(i)]{cssw1}
and \cite[Thm. 2]{cssw1} are  incorrect. On the other hand,
the conclusions of  \cite[Thm. 1]{cssw1} are correct for
$6$-dimensional similarities of 
$G=\Cy{2^r}$. We will use \cite[Cor.(iii)]{cssw1}
in Example \ref{inducedsim} and in Section \ref{sE}.
\end{remark}

\begin{remark}
 The $q =1$ component, $L_{2k+1}^p((\ZG, w)(1)$, is isomorphic
to  via the projection 
or restriction map to $L_{2k+1}^p((\bz [C(2^r)],
w)$.  In this case,
the representation with centre field $\bq(i)$ has type $OK(\bc)$ and
contributes $(\cy 2)^2$ to $L_3^p$, hence
$L_1^p(\ZG, w)(1) \cong(\cy 2)^{r-2}$ and 
$L_3^p(\ZG, w)(1) \cong(\cy 2)^{r}$.
\end{remark}

\medskip
We now return to our main calculational
device for determining non{--}linear similarities of cyclic groups,
namely  the ``double coboundary"
 \[\delta^2\colon H^1(\widetilde K_0(\ZG^{-}))\to H^1(\wh(\ZG^{-}))\]
 from the exact sequence
\[0 \to \wh(\ZG) \to \wh(\zlocal G)\oplus K_1(\QG) \to K_1(\qlocal G)
\to \widetilde K_0(\ZG)\to 0\ .
\]
We recall that the discriminant
induces an isomorphism
\[L_1^h(\ZG,w) \cong H^1(\wh(\ZG),w)\]
since $L_i^s(\ZG,w) \cong L'_i(\ZG,w) = 0$
for $i \equiv 0,1 \mod{4}$ by the calculations of 
\cite[3.4.5,5.4]{w2}. 
\begin{proposition}
The kernel of the map $L_1^h(\ZG,w) \to L_1^p(\ZG,w)$
is isomorphic to the image of the double coboundary
$\delta^2\colon H^1(\widetilde K_0(\ZG^{-}))\to H^1(\wh(\ZG^{-}))$
under the
isomorphism $  L_1^h(\ZG,w) \cong H^1(\wh(\ZG^{-}))$
induced by the  discriminant .
\end{proposition}
\begin{proof}
We will use  the commutative braid
\eqncount
\begin{equation} \label{lpbraid}
\begin{matrix}
\xymatrix@!C@C-3pc{
H^1(\widetilde K_0(\ZG^{-}))       \ar[dr] \ar@/^2pc/[rr]^{\delta^2}    &&
{H^1(\wh(\ZG^{-}))}                 \ar[dr] \ar@/^2pc/[rr]               &&
L_0^s(\ZG,w)                                                        \\
&   L_1^h(\ZG, w)                \ar[dr] \ar[ur]                      &&
H^1(\Delta)                    \ar[dr] \ar[ur]                      \\
L_1^s(\ZG,w)                   \ar[ur] \ar@/_2pc/[rr]_{}            &&
L_1^p(\ZG,w)                   \ar[ur] \ar@/_2pc/[rr]               &&
H^0(\widetilde K_0(\ZG^{-}))
}\end{matrix}
\end{equation}

\noindent 
relating the $L^h$ to $L^p$ and the $L^s$ to $L^h$ Rothenberg
sequences.
The term $H^1(\Delta)$ is the
Tate cohomology of the relative group for the double
coboundary defined in \ptwo, Section \ref{doublecob}.
The braid diagram is constructed by diagram chasing using the
interlocking $K$ and $L${--}theory exact sequences, as in
for example \cite[\S 3]{hm1}, \cite[p.560]{hm3}, \cite[p.3]{ra13}
and \cite[6.2]{ra8}. 
We see that the discriminant of an element $\sigma \in
L^h_1(\ZG,w)$ lies in  the image of the double coboundary
if and only if  $\sigma \in \ker (L^h_1(\ZG,w) \to L_1^p(\ZG,w))$.
\end{proof}
The braid diagram in this proof also gives:
\begin{corollary} There is an isomorphism
$L_1^p(\ZG,w) \cong H^1(\Delta)$.
\end{corollary}
\begin{remark} It follows  from Corollary 
\ref{galoisaut} that $H^1(\Delta)$ is fixed
by  the induced maps from group automorphisms of $G$. 
We will generalize this result in the next section.
\end{remark}

\begin{remark}
There is a version of these results for $L_3^p(\ZG,w)$ as well,
on the kernel of the projection map
$ L_3^p(\ZG,w) \to L_3^p(\ZK,w)$,
where $K=\Cy 4$.
The point is that $L_i^s(\ZG,w) \cong L_i^s(\ZK,w)$
is an isomorphism for $i\equiv 2,3\mod{4}$ as well
\cite[3.4.5,5.4]{w2}.
There is also a corresponding braid \ptwo\ (\ref{maindb}) for
$L^s_{2k+1}(\bcatrminus{W})$, $L^h_{2k+1}(\bcatrminus{W})$
and $L^p_{2k+1}(\bcatrminus{W})$ involving the double
coboundary in bounded $K${--}theory.
The cone point inclusion
\[\cc_{pt}(\ZG,w) = \cc_{pt,G^{-}}(\bz) \to \bcatrminus{W}\]
 induces a natural transformation between
the two braid diagrams.
\end{remark}
In Section \ref{A} we will need the following calculation.
We denote by $L_n^{\wh(\ZH)}(\ZG^-)$ the $L${--}group
of $\ZG$ with the non-oriented involution, and Whitehead torsions
allowed in the subgroup $\wh(\ZH) \subset \wh(\ZG)$.
\begin{lemma}\label{vanA}
$L_1^{\wh(\ZH)}(\ZG^-) = 0$, and the 
map $L_0^{\wh(\ZH)}(\ZG^-) \to H^0(\wh(\ZH))$
induced by the discriminant is an injection.
\end{lemma}
\begin{proof}
The Rothenberg sequence gives
\[L_n^s(\ZG^-) \to L_n^{\wh(\ZH)}(\ZG^-) \to
H^n(\wh(\ZH))\ .\]
For $n\equiv1\Mod{4}$ the outside terms are zero, and hence
$L_1^{\wh(\ZH)}(\ZG^-) = 0$.
For $n\equiv 0\Mod{4}$, $L_0^s(\ZG^-)=0$ as noted above
and the injectivity follows.
\end{proof}

In later sections, it will be convenient to stabilize with trivial
representations and use the identification
\[L^p_{n+k}(\bcat{W\times\br^k})
\cong \Lminus{k}_n(\bcat{W}).\]
The composite with the transfer
\[\trf{\br^k}\colon L^p_n(\ZG)
\to L^p_{n+k}(\bcat{W\times\br^k})\]
is just the usual ``change of $K${--}theory" map, which may be
analysed by the Ranicki-Rothenberg sequences
\cite{ra6}.
For $G$ a finite group, $\Kminus{j}(\ZG) = 0$ if $j \geqq 2$ so only
the first stabilization is needed.
\begin{lemma}
For $G = \Cy{2^rq}$ and $w\colon G \to \orient$ non{--}trivial, the
map
 \[L^p_{2k+1}(\ZG,w) \to \Lminus{1}_{2k+1}(\ZG,w)\]
 is injective.
\end{lemma}
\begin{proof}
The group  $\Kminus{1}(\ZG)$
 is a torsion free quotient of $K_0(\qlocal G)$,
and has the involution  induced by $[P]\mapsto -[P^*]$ on
$K_0(\qlocal G)$ \cite[3.6]{hm1}.
This implies first that $H^0(K_0(\qlocal G))=0$,
and so the image of the coboundary
\[H^0(\Kminus{1}(\ZG)) \to H^1(K_0(\zlocal G)) \oplus H^1(K_0(\bq G))\]
consists of the classes $(0,[E])$ where $E$ splits  at
every finite prime dividing $2q$.
We need to compare  the exact sequences in the
following diagram (see \cite{hm3}, \cite{ht1}):
\[
\xymatrix{
&&L^K_{2k+2}(\zlocal G,w)\oplus L^K_{2k+2}(\bq G,w)
\ar[r]\ar[d]&L^K_{2k+2}(\qlocal G,w)\ar[d]\\
0\ar[r]& \Lminus{1}_{2k+2}(\ZG,w)\ar[r]\ar[d] &L^p_{2k+2}(\zlocal G,w)
\oplus  L^p_{2k+2}(\QG,w) \ar[r]\ar[d]& L^p_{2k+2}(\qlocal G,w)\ar[d]\\
0\ar[r]& H^0(\Kminus{1}(\ZG))\ar[r]&H^1(K_0(\zlocal G))
\oplus H^1(K_0( \bq G))\ar[r]&H^1(K_0(\qlocal G))
}
\]
The groups $L^K_{2k+2}(\zlocal G,w)$ reduce to the
$L${--}groups of finite fields, which are zero in type
$U$, and the map $L^p_{2k+2}(\QG,w) \to H^1(K_0( \bq G))$ is surjective.
For each involution invariant field $E$ in the
top component  of $\QG$, the
group  $L^K_{2k+2}(E) = H^0(\Units{E})$ which
maps injectively into
 $L^K_{2k+2}(\hat E) = H^0(\localunits{E}{}{})$, \cite{hm3}.
It follows that the images of
$L^K_{2k+2}(E)$ and $\Lminus{1}_{2k+2}(\ZG,w)$ in $L^p_{2k+2}(E)$
have zero intersection , and so the
composite map $\Lminus{1}_{2k+2}(\ZG,w) \to
H^1(K_0( \QG))$ is an isomorphism onto the classes which
split at all primes dividing $2q$.
Therefore  the map $\Lminus{1}_{2k+2}(\ZG,w) \to H^0(\Kminus{1}(\ZG))$
is surjective, and we conclude that the map
 $L^p_{2k+1}(\ZG,w) \to \Lminus{1}_{2k+1}(\ZG,w)$ is injective.
\end{proof}
\begin{corollary}\label{rstable}
Let $G = \Cy{2^rq}$ and $w\colon G \to \orient$ the non{--}trivial
orientation.
If  $W$ is a $G${--}representation with $W^G = {0}$, then the
map
 \[L^p_{2k+1}(\bcat{W},w) \to
 \Lminus{1}_{2k+1}(\bcat{W},w)\]
 is injective.
\end{corollary}
\begin{proof}
We first note that the cone point maps
$K_0(R G) \to K_0(\ccat{W,G}{R})$ are surjective
for $R = \zlocal, \bq$ or $\qlocal$ since for these
coefficients $RG$ has  vanishing $\Kminus{1}$ groups.
This shows that $\Kminus{1}(\bcat{W})$ is again a
 quotient of $K_{0}(\qlocal G)$.
To see that $\Kminus{1}(\bcat{W})$  is also torsion
free, consider the boundary map
$K_1(\Ccat{W,G}{0}{\qlocal}) \to K_0(\ccat{W}{\qlocal})$
which is just a sum of induction maps
$K_0(\qlocal K) \to K_0(\qlocal G)$ from proper subgroups
of $K\subset G$.
But for $G$ cyclic,  these induction
maps are split injective.
We now complete the argument by comparing the
diagram  above with the
corresponding diagram  for the bounded theory,
concluding that $H^0(\Kminus{1}(\ZG)) \to H^0(\Kminus{1}(\bcat{W}))$
is surjective.
Since $\Lminus{1}_{2k+2}(\ZG,w) \to H^0(\Kminus{1}(\ZG)) $
is also surjective, we are done.
\end{proof}



\section{ The proof of Theorem A}
\label{A}
The condition (i) is equivalent to assuming that  $S(V_1)$
and $S(V_2)$ are freely $G${--}homotopy equivalent.
Condition (ii) is necessary by
Corollary \ref{semifree} which rules out non{--}linear similarities of
semifree representations.
Condition (ii) also implies that $S(V_1)$ is $s${--}normally cobordant
to $S(V_2)$ by \cite[Prop. 2.1]{cs1}, which is another necessary
condition for topological similarity.
Thus under conditions (i) and (ii), there exists a homotopy equivalence
$f\colon S(V_2) \to S(V_1)$, and  an element $\sigma=\sigma(f)
\in L^h_0(\ZG)$ such that
 $\trfrminus(\sigma) = 0 \in L^h_1(\bCatrminus)$ if and only
if $V_1\oplus\rminus\sim_t V_2\oplus\rminus$.

Comparing the $h$- and $s$- surgery exact sequences
 it is easy to see that the
image of $\sigma$ in $ H^0(\wh(\ZG))$ is
 given by the Whitehead torsion
$\{\tau(f)\} = \{\Delta(V_1)/\Delta(V_2)\} \in \wh(\ZG)$ of the
homotopy equivalence
$S(V_2)/G \simeq S(V_1)/G$.

In Section \ref{results}\ we gave the short exact sequence
\[0\to \wh(\ZG)/\wh(\ZH) \to \wh(\rel{\ZH}{\ZG}) \to \ck\to 0\]
where  $K_1(\bCatrminus)/{\scriptstyle \{ \pm G\}})$ is denoted by
$\wh(\bCatrminus) = \wh(\rel{\ZH}{\ZG})$
and $\ck =\ker(\widetilde K_0(\ZH) \to \widetilde K_0(\ZG))$.
We proved in Theorem \ref{ktrans}  that the transfer of the
 torsion element in $\wh(\ZG)$
in $\wh(\rel{\ZH}{\ZG})$ is given by the same element under
the map induced by inclusion
$\wh(\ZG) \to \wh(\rel{\ZH}{\ZG})$.



It follows that the image of $\trf{W}(\sigma)$ in
\[ H^1(\wh(\bCatrminus))= H^1(\wh(\rel{\ZH}{\ZG^{-}}))\]
is given by the image of our well-defined
element
 \[ \{\Delta(V_1)/\Delta(V_2)\} \in  H^1(\wh(\ZG^-)/\wh(\ZH))\]
under the cone point inclusion into
$H^1(\wh(\rel{\ZH}{\ZG^{-}}))$.


 The necessity of the condition is now easy.
To have a non-linear similarity we must have
\[\trfrminus(\sigma) = 0 \in L^h_1(\bCatrminus)\ .\]
Hence
\[\trf{\rminus }(\Delta(V_1)/\Delta(V_2)) = 0
 \in  H^1(\wh(\rel{\ZH}{\ZG^{-}}))\]
must vanish by naturality of the transfer in the Rothenberg sequence.
This element comes from
$H^1(\wh(\ZG^-)/\wh(\ZH))$,
so to vanish in $H^1(\wh(\rel{\ZH}{\ZG^-}))$
it must be in the image from
\[ H^0(\ck) \to H^1(\wh(\ZG^-/\wh(\ZH))\]
under the coboundary.

To prove sufficiency,  we assume that the image of the
transferred element
\[\{\Delta(V_1)/\Delta(V_2)\} \in  H^1(\wh(\rel{\ZH}{\ZG^-}))\] is
zero.
Consider the long exact sequence derived from
the inclusion  of filtered categories
\[\ccat{pt}{\ZG}  \subset \bCatrminus\]
where $\bCatrminus$ has the standard orientation 
\ptwo,  Example \ref{standard},
inducing the  non{--}trivial orientation at the cone point.
The quotient category $\Ccat{\rminus ,G}{0}{\bz}$ of
germs away from $0$ is
canonically isomorphic to $\Ccat{[0,\infty),H}{0}{\bz}$ ,
since by equivariance
what happens on the positive half line has to be copied on the negative
half line, and what happens near $0$ does not matter in the germ category.
Since the action of $H$ on $[0,\infty)$ is trivial, this category is precisely
$\Ccat{[0,\infty)}{0}{\ZH}$ which has the same $K$- and $L$-theory as
$\Ccat{\br}{}{\ZH}$ by the projection map.
By \ptwo, Theorem \ref{ua} we thus get a long exact sequence
\[\ldots \to L^k_n(\ZH) \to L^h_n(\ZG^-)\to
L^h_n(\bCatrminus)\to L^k_{n-1}(\ZH) \to \dots \]
where the map is induced by induction.
Comparing this  sequence with the long
exact sequence for the pair $L^{h,k}_n(\ZH\to\ZG^-)$
(see Theorem \ref{rays}) it follows that
\[L_n^h(\bCatrminus)\cong L_n^{k,h}(\ZH\to\ZG^-)\ .\]
Consider the following diagram with exact rows and columns:
\[
\xymatrix{L_1^{\wh(\ZH)}(\ZG^-) \ar[r]\ar[d]&
L_1^{h,\wh(\ZH)}(\rel{\ZH}{\ZG^-})\ar[r]\ar[d]&
L_0^h(\ZH)\ar[d]\\
L_1^h(\ZG^-)\ar[r]\ar[d]&
L_1^{k,h}(\rel{\ZH}{\ZG^-})\ar[d]\ar[r]&
L_0^k(\ZH)\ar[d]\\
 H^1(\wh(G^-)/\wh(H))\ar[r]&
 H^1(\wh(\rel{\ZH}{\ZG^-}))\ar[r]&
 H^1(\ck)
}
\]
We need to show that
$\trf{\rminus }(\sigma)$ vanishes in order to
produce the non-linear similarity.
We  know that the image of $\Res_H(\sigma)=0$,
and our assumption is that the image
\[\{\Delta(V_1)/\Delta(V_2)\} \in  H^1(\wh(\rel{\ZH}{\ZG^-}))\]
vanishes.
We will finish the argument by showing:
\begin{lemma}\label{lowA}
Suppose that $\sigma \in L_0^h(\ZG)$.
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $ L_1^{h,\wh(\ZH)}(\rel{\ZH}{\ZG^-})$ is torsion{--}free.
\item The torsion subgroup of $L_1^{k,h}(\rel{\ZH}{\ZG^-})$
injects into $H^1(\wh(\rel{\ZH}{\ZG^-}))$, or
equivalently, the torsion subgroup of $L_1^h(\bCatrminus)$
injects into $H^1(\wh(\bCatrminus))$.
\item If $\Res_H(\sigma) = 0 \in L_0^h(\ZH)$, then
$\trfrminus(\sigma)$ is a torsion element.
\end{enumerate}
\end{lemma}
\begin{proof}[The proof of Lemma \ref{lowA}]
For assertion (i), we
consider the diagram
\[
\xymatrix{L_0^s(\ZH)\ar[r]\ar@{  >->}[d]&
L_0^s(\ZG^-)\ar[d]\\
L_0^h(\ZH)\ar[d]\ar[r]&
L_0^{\wh(\ZH)}(\ZG^-)\ar[d]\\
 H^0(\wh(\ZH))\ar@{=}[r]&  H^0(\wh(\ZH))
}
\]
where $L^s_0(\ZG^-) =0$.
Since $L_1^{\wh(\ZH)}(\ZG^-) = 0$ as well (by Lemma \ref{vanA}),
it follows that
\[ L_1^{h,\wh(\ZH)}(\rel{\ZH}{\ZG^-}) =
 \ker(L_0^h(\ZH) \to L_0^{\wh(\ZH)}(\ZG^-)) \cong L_0^s(\ZH)\]
where $L_0^s(\ZH)$ is torsion{--}free.
Part (ii) follows from part (i), since the previous term in
the exact sequence has exponent
 two.
For assertion (iii) we refer to part of the twisting diagram of
Section \ref{rmin},
namely the commutative diagram:
\[
\xymatrix{&L_0^h(\ZG)\ar@{  >->}[d]^{\trfrminus}\ar[dr]^{\Res_H}&\\
L_1^h(\ZG^{-})\ar[r]&L_1^{k,h}(\rel{\ZH}{\ZG^-})\ar[r]
&L_0^k(\ZH)
}
\]
and the fact that $\Res_H(\trfrminus(\sigma)) = 0$ since
it factors through $\Res_H\colon L_0^h(\ZG) \to L_0^h(\ZH)$.
But $L_1^h(\ZG^{-})\cong H^1(\wh(\ZG^{-})$ has exponent $2$.
\end{proof}

\begin{remark}\label{Arem}
Theorem A gives necessary and sufficient conditions for the
existence of $5${--}dimensional similarities.
Consider the situation in dimensions $\leqq 5$.
 By character theory it suffices to consider a cyclic group $G$,
which must be of order divisable by $4$, say $4q$,
with index 2 subgroup $H$.
It suffices by Lemma \ref{LsameW} to consider the following situation
\[V_1\oplus W \topsim V_2 \oplus W\]
where $V_i$ are free homotopy equivalent representations which become
isomorphic once restricted to $H$.
Therefore $S(V_1)$  is normally cobordant to $S(V_2)$.
If $V_i$ are two-dimensional they are determined by one
character, so homotopy equivalence implies isomorphism.
We may thus assume
$V_i$ are at least 4-dimensional, and $\dim W=1$.
 According to
Theorem \ref{maintest}, 
 non-linear similarity is now determined by $\trf{W}(\sigma)$,
where $\sigma$ is an element of infinite order
 in $L^h_0(\ZG)$ hitting the element  in
the structure set determined by the homotopy equivalence
of $S(V_2)/G \simeq S(V_1)/G$.
In case $W$ is the trivial representation we may identify $\trf{W}$
with the map
\[L^h_0(\ZG) \to L^p_0(\ZG)\]
which  we have seen (in Corollary \ref{semifree})
is injective on the elements of infinite order.
Hence we are left with the case where $W$ is
the non-trivial one-dimensional representation $\rminus $,
which is reduced to number theory by Theorem A.
In Corollary \ref{eightteenfive} we work out the number theory
for  $G=\Cy{2^r}$ as an example, showing that $5${--}dimensional
similarities do not exist for these groups. The
general case was done by Cappell and Shaneson 
in 1981, and this preprint has  recently been published \cite{cs3}.
\end{remark}

\section{ The proof of Theorem B}
 \label{B}
The new ingredient in Theorem B is the double coboundary.
As in the last section, we may assume that $f\colon S(V_2) \to
S(V_1)$ is a $G${--}homotopy equivalence which is freely
$G${--}normally cobordant to the identity, giving
an element $\sigma=\sigma(f)\in L_0^h(\ZG)$.
Then $V_1\oplus \rminus\oplus \rplus \sim_t
V_2\oplus \rminus\oplus \rplus$ if and only if
\[\trf{\rminus\oplus\rplus}(\sigma) = 0 \in
L^h_2(\bcat{\rminus\oplus \rplus})\]
by  Theorem \ref{maintest}.
But $L^h_2(\bcat{\rminus\oplus \rplus}) = L_1^p(\bCatrminus)$,
so we may regard the criterion as the vanishing of
$\trfrminus(\sigma) \in L_1^p(\bCatrminus)$ instead.

\medskip
The main commutative diagram is:

\eqncount
\begin{equation} \label{mainda}
\begin{matrix}
\xymatrix@!C@C-5pc{
H^1(\widetilde K_0(\bCatrminus))       \ar[dr] \ar@/^2pc/[rr]^{\delta^2}    &&
H^1(\wh(\bCatrminus))                  \ar[dr] \ar@/^2pc/[rr]               &&
L_0^s(\bCatrminus)                                                          \\
&   L_1^h(\bCatrminus)                 \ar[dr] \ar[ur]                      &&
H^1(\Delta_{\rminus})                  \ar[dr] \ar[ur]                      \\
L_1^s(\bCatrminus)                     \ar[ur] \ar@/_2pc/[rr]_{}            &&
L_1^p(\bCatrminus)                     \ar[ur] \ar@/_2pc/[rr]               &&
H^0(\widetilde K_0(\bCatrminus))
}\end{matrix}
\end{equation}

\noindent where $H^1(\Delta_{\rminus})$ denotes the relative group
of the double coboundary map.
Notice that some of the groups in this diagram
already appeared as relative $L${--}groups in the last section.
We have
\[L_1^h(\bCatrminus) =L_1^{k,h}(\rel{\ZH}{\ZG^{-}})\]
and
\[L_1^s(\bCatrminus) = L_1^{h,\wh(\ZH)}(\rel{\ZH}{\ZG^{-}})\ .\]
By Lemma \ref{lowA} (i) this group is torsion{--}free and
the preceeding term $H^0(\Delta_{\rminus})$ is $2$-torsion.
 Hence $L_1^s(\bCatrminus)$  injects into $L_1^p(\bCatrminus)$, and
we  conclude that the torsion subgroup of 
$L_1^p(\bCatrminus)$ injects
into $H^1(\Delta_{\rminus})$.

However, since $\Res_H(\sigma) = 0 \in L_0^h(\ZH)$,
 Lemma \ref{lowA} (iii) states that
$\trfrminus(\sigma)$ is a torsion element.
Furthermore, its image in $H^1(\Delta_{\rminus})$ is zero
if and only if
\[\{\Delta(V_1)/\Delta(V_2)\} \in  H^1(\wh(\rel{\ZH}{\ZG^-}))\]
lies in the image of the double coboundary,
and this completes the proof of Theorem B.

\bigskip
We conclude this section with an important 
 property of the relative group
$H^1(\Delta_{W\times \rminus})$ of the double
coboundary 
$$\delta^2\colon H^1(\widetilde K_0(\bcatrminus{W})) \to H^1(\wh(\bcatrminus{W}))$$
which we will refer to as \emph{Galois invariance}.
For $G$ a cyclic $2$-group, we establish a similar statement for
the image of  $H^1(\wh(\ZG^-)/\wh(\ZH))$ in the relative group
$H^1(\Delta_{\rminus})$. This sharper version will be
used  in determining the non-linear similarities for cyclic
$2$-groups.
\begin{lemma}\label{galoisinvariance} Let $W$ be a complex
$G${--}representation, with $W^G=0$, containing all the
non-trivial irreducible representations of $G$ with isotropy of 
$2$-power index.
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item 
Automorphisms of $G$ induce the identity on  the
image of  $H^1(\wh(\ZG^-)/\wh(\ZH))$ in the relative group
$H^1(\Delta_{W\times\rminus})$ of the double coboundary.
\item If $G$ is a cyclic $2$-group,
 then automorphisms of $G$ induce the identity on  the
image of  $H^1(\wh(\ZG^-)/\wh(\ZH))$ in the relative group
$H^1(\Delta_{\rminus})$.
\end{enumerate}
\end{lemma}
\begin{proof}
The first step is to prove that the image of
 any  $[u]\in H^1(\wh(\ZG^-)/\wh(\ZH))$
in $H^1(\wh(\rel{\ZH}{\ZG^-}))=H^1(\wh(\bCatrminus))$
equals the discriminant 
$$\trfrminus(\sigma) \in L_1^h(\bCatrminus)$$ for some
$\sigma \in L_0^h(\ZG)$. Consider the diagram
$$\xymatrix{L_0^h(\ZG)\ar[r]\ar[d]_{\trfrminus}&
H^0(\wh(\ZG))\ar[d]^{\trfrminus}\cr
L_{1}^h(\bCatrminus)\ar[r]&H^1(\wh(\rel{\ZH}{\ZG^-}))
}$$
By Theorem \ref{ktrans}, the $K$-theory transfer factors through
the map induced by the cone point inclusion 
$c_*\colon H^1(\wh(\ZG^-)/\wh(\ZH))\to H^1(\wh(\rel{\ZH}{\ZG^-}))$.
Since $L_0^h(\ZG) \to H^0(\wh(\ZG)$ is surjective, there exists
$\sigma \in L_0^h(\ZG)$ with given discriminant $[u]$, and
$\trfrminus(\sigma)$ has discriminant $c_*([u])$. But the natural
map $H^0(\wh(\ZG))\to H^1(\wh(\ZG^-)/\wh(\ZH))$ is also surjective,
so the first step is complete.  By naturality of the transfer
$$\trf{W}\colon L_{1}^h(\bCatrminus) \to L_{2k+1}^h(\bcatrminus{W})$$
 ($\dim W = 2k$), it follows that the image of
$c_*([u])$ in $H^1(\wh(\bcatrminus{W}))$ equals the discriminant
of the element $\trf{W\times\rminus}(\sigma)$.

Our assumption on $W$ implies that $\trf{W\times\rminus}(\sigma)$
is a torsion element (see \ptwo, Section \ref{ratcomp}).
We can therefore apply  \ptwo, Theorem \ref{one}(i): there 
exists a  \emph{torsion} element
$\hat\sigma\in L_{2k +1}^h(\bCatrminus)$, such that
$c_*(\hat\sigma)=\trf{W\times\rminus}(\sigma)$. Now 
from the commutative diagram comparing cone point inclusions:
$$\xymatrix{
L_{2k+1}^h(\bCatrminus)\ar[r]\ar[d]_{c_*}&
H^1(\wh(\rel{\ZH}{\ZG^-}))\ar[d]^{c_*}\cr
L_{2k+1}^h(\bcatrminus{W})\ar[r]&H^1(\wh(\bcatrminus{W})
}
$$
we conclude that
the image of $c_*([u])$ in $H^1(\wh(\bcatrminus{W}))$
equals  the image of the discriminant of $\hat\sigma
\in L_{2k+1}^h(\bCatrminus)$.
The braid diagram used in the proof of Theorem B above
now shows that the image of $c_*([u])$  in 
$H^1(\Delta_{W\times\rminus})$
comes from  the image of
$\hat\sigma\in  L_{2k +1}^p(\bCatrminus)$ in
 $H^1(\Delta_{\rminus})$, via the natural 
map $H^1(\Delta_{\rminus}) \to H^1(\Delta_{W\times\rminus})$.

Finally we consider the cone point inclusion sequence
\[   L_2^h(\brelcat{\rminus}{0})
\to L_1^p(\ZG^-) \to L_1^p(\bCatrminus)  
\to L_1^h(\brelcat{\rminus}{0}) \to  L_0^p(\ZG^-)\]
The $K${--}theory decoration on the relative group is the image
of \[\widetilde K_0(\bCatrminus) \to \widetilde
K_0(\brelcat{\rminus}{0})
\cong K_{-1}(\ZH)\] but since
$K_{-1}(\ZH)\to K_{-1}(\ZG)$ is injective, that image is zero and
we get $L^h$. 
Now $L_2^h(\brelcat{\rminus}{0})\cong L_1^p(\ZH) =0$
and $L_1^h(\brelcat{\rminus}{0}) \cong  L_0^p(\ZH)$ is torsion{--}free.
It follows that the cone point inclusion $L_1^p(\ZG^-) \to
L_1^p(\bCatrminus) $ is an isomorphism onto the torsion subgroup.
But by Corollary \ref{galoisaut} the group $L_1^p(\ZG^-)$
is fixed  by group automorphisms of $G$. This completes
the proof of part (i).

In part (ii) we assume that $G$ is a cyclic $2$-group, so
$K_{-1}(\ZK)=0$ for all subgroups $K\subseteq G$. By
 \ptwo, Corollary \ref{kersimple} we have an isomorphism
$H^1(\Delta_{\rminus})\cong H^1(\Delta_{W\times\rminus})$
and the proof is complete.
\end{proof}


\section{ Cyclic $2${--}Groups: Preliminary Results }
\label{twogroups}
For $G=\Cy{2^r}$ a cyclic $2${--}group, we have stronger
results because $\Kminus{1}(\ZG) =0$.
The results of this section prepare for a complete 
classification of stable and unstable non{--}linear similarities 
for cyclic $2${--}groups, and the computation of
$\rtop{G}$ .

\begin{lemma}\label{vanB} {\rm (R. Oliver, \cite{o2})}
For $G = \Cy{2^r}$, the  cohomology groups $H^*(\ck) =0$, where
$\ck =\ker(\widetilde K_0(\ZH) \to \widetilde K_0(\ZG))$.
\end{lemma}
\begin{proof}
We are indebted to R. Oliver for pointing out that this result follows
from \cite[Thm. 2.6]{o2}, which states (in his notation):
$$D(Z[\Cy{2^{n+3}}) \cong \Image((5-\gamma)\hat\psi_n)$$
where $D(\ZG)$ is the kernel of the
map induced by including $\ZG$ in a maximal order $\cm$ in $\QG$.
By Weber's Theorem, the ideal class groups of 2-power cyclotomic 
fields have odd order, so to prove that  $H^*(\ck) =0$ it is enough
to show that $\ker(D(\ZH) \to D(\ZG)) =0$.

The map $\hat\psi$ is the reduction $\Mod {2^{n+1}}$ of a map
$\psi_n\colon M^n \to M^n$ given by the formula
$$\psi_n(e_i) = \sum_{j=0}^i 2^{i-j}\gamma_{i-j}^{n-j} e_{n-j}$$
for $0\leq i \leq n$, where $\{e_0, \dots, e_n\}$ is a basis
for the direct sum $M^n = \sum_{i=0}^n \zadictwo [\Gamma_i]e_i$
 and
$\Gamma_i$ denotes the cyclic group of automorphisms of
$C(2^{i+2})$ generated by $\gamma(t) = t^5$.

To shorten Oliver's notation, we let
 $D[n+3] := D(Z[C(2^{n+3}))$
so  $D[n+3]$ is identified with the subgroup 
 $\Image((5-\gamma)\hat\psi_n)$ of
$M^n/2^{n+1}M^n$. It is enough to see that the map $$\Ind\colon D[n+3]\to D[n+4]$$
given by the subgroup inclusion induces an injection on this subgroup
$\Image((5-\gamma)\hat\psi_n)$. 

However the map $ \Ind$
corresponds under the identification in \cite[Thm. 2.4]{o2} with the explicit map
$\ind(e_i) = \gamma^{i+1}_i e_{i+1}$ (see last paragraph of \cite[\S 2]{o2}). 
Using this explicit formula, we need to check that $x\in \ker(\hat\psi_{n+1}
\circ \ind)$ implies that $x\in \ker\hat\psi_n$.
 Suppose that $x = \sum_{i=0}^n a_i e_i \in M^n$. Then
 $$\psi_{n+1}(\ind(x)) = 
 \left (\sum_{i=0}^n a_i 2^{i+1}
 \gamma_i^{i+1} \gamma_{i+1}^{n+1}\right )e_{n+1} +
 \sum_{j=0}^{n}\left (\sum_{i=j}^n a_i 2^{i-j} \gamma_i^{i+1}
 \gamma_{i-j}^{n-j}\right ) e_{n-j}  
 $$
 after re-arranging the summations, and 
 $$\psi_n(x) =  \sum_{j=0}^{n}\left (\sum_{i=j}^n a_i 2^{i-j} 
 \gamma_{i-j}^{n-j}\right ) e_{n-j}\ .$$
We can then use the formulas after \cite[Lemma 1.1]{o2} to check that
 $$\gamma_i^{i+1} \gamma_{i-j}^{n-j} = 2 \gamma_{i-j}^{n-j} \in
 \zadictwo [\Gamma_{n-j}]$$
 for $0\leqq j \leqq n$.
 Since $M^{n+1}/2^{n+2}M^{n+1}$ is a direct sum of the group rings
 $\cy{2^{n+2}}[\Gamma_i]$, it follows that 
 $\psi_{n+1}(\ind(x))\equiv 0 \Mod{2^{n+2}}$ implies
 $\psi_n(x)\equiv 0 \Mod{2^{n+1}}$.
\end{proof}
Our results for cyclic $2$-groups can now be improved, starting with
similarities with an $\rminus$ but no $\rplus$ summand.
\begin{theorem}\label{noRplus}
Let  $V_1=t^{a_1} + \dots + t^{a_k} $
and $V_2 = t^{b_1} + \dots + t^{b_k}$ be
 free $G${--}representations, where $G=\Cy{2^r}$.
Let $W$ be a  complex $G${--}representation
with no $\rplus$ or $\rminus$ summands.
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item If $\Res_H (V_1\oplus W) \oplus \rplus\sim_t \Res_H (V_2\oplus W)\oplus \rplus$, then
$V_1\oplus W \oplus \rminus \sim_t V_2\oplus W \oplus \rminus$
if and only if  $S(V_1)$ is $s${--}normally cobordant to $S(V_2)$
and 
$\left\{ \Delta(V_1)/\Delta(V_2) \right\}=0$ in $H^1(\wh(\ZG^{-})/\wh(\ZH))$.
\item If $\Res_H V_1 \cong \Res_H  V_2$, then $V_1\oplus W \oplus \rminus
\sim_t V_2\oplus W\oplus \rminus$ if and only if the class
$\left\{ \Delta(V_1)/\Delta(V_2) \right\}=0$ in $H^1(\wh(\ZG^{-}))$.
\end{enumerate}
\end{theorem}
\begin{proof} Consider first the situation in part (i).
 By \ptwo, Theorem \ref{Cprime},
the image of the final surgery obstruction
$ \trf{W\times \rminus}(\sigma)$ in $L_1^h(\bcatrminus{W}) $
is just the class $\left\{ \Delta(V_1)/\Delta(V_2) \right\}$
considered as an element in $H^1(\wh(\bcatrminus{W})$.
Moreover, the natural map 
 $$H^1(\wh(\ZG^{-})/\wh(\ZH)) \to H^1(\wh(\bcatrminus{W})$$
 factors through
$H^1(\wh(\bCatrminus))$, and  the torsion subgroup of
$L_1^h(\bCatrminus)$ injects into $H^1(\wh(\bCatrminus))$
by Lemma \ref{lowA}(ii).
Therefore, $ \trf{W\times \rminus}(\sigma)=0$
if and only if the Reidemeister torsion invariant vanishes in $H^1(\wh(\bCatrminus))$.
But $H^\ast(\ck) =0$ so the natural map
$$H^1(\wh(\ZG^{-})/\wh(\ZH)) \to
 H^1( \wh(\rel{\ZH}{\ZG^{-}}))\cong H^1(\wh(\bCatrminus))$$ 
induces an isomorphism. This proves part (i).

In part (ii), 
since $\Res_H V_1 \cong \Res_H V_2$, our
Reidemeister torsion quotient represents an element
$\{\Delta(V_1)/\Delta(V_2)\}\in H^1(\wh(\ZG^-))$,
and this group injects into $H^1( \wh(\rel{\ZH}{\ZG^{-}}))$.
The vanishing of the surgery obstruction is now equivalent to 
$\left\{ \Delta(V_1)/\Delta(V_2) \right\}=0$ in $H^1(\wh(\ZG^{-}))$,
by the argument above.
\end{proof}
\begin{corollary}\label{eightteenfive}
The groups $G= \Cy{2^r}$ have no $5${--}dimensional non{--}linear
similarities.
\end{corollary}
\begin{proof}
The Reidemeister torsion quotients for possible
$5$-dimensional similarities are represented by the units
$U_{1,i}$ which form a basis of $H^1(\wh(\ZG^{-}))$ (see
\cite{cs5}, \cite[p.733]{cssw1}).
\end{proof}
Higher{--}dimensional similarities of cyclic $2${--}groups 
were previously studied in the 1980's.
The $6${--}dimensional case was worked out in detail
for cyclic $2${--}groups in \cite{cssw1},
and  general conditions $A${--}$D$ were announced
 in \cite{cssww1} for the classification of non{--}linear similarities
for cyclic $2$-groups in any dimension.
However, in Example \ref{condB} we give a counterexample to the
necessity of \cite[Condition B]{cssww1},
and this invalidates the claimed solution.
Our next result concerns similarities with both $\rminus$ and $\rplus$ summands.
\begin{theorem}\label{eightteenone}
Let
 $V_1=t^{a_1} + \dots + t^{a_k} $
and $V_2 = t^{b_1} + \dots + t^{b_k}$ be
 free $G${--}representations, where $G=\Cy{2^r}$.
Let $W$ be a  complex $G${--}representation
with no $\rplus$ summands.
Then there exists a topological similarity
 $V_1\oplus W \oplus \rminus\oplus \rplus \sim_t
V_2\oplus W \oplus \rminus\oplus \rplus$ if and only if
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $S(V_1)$ is $s${--}normally cobordant to $S(V_2)$,
\item $\Res_H (V_1\oplus W) \oplus \rplus
 \sim_t \Res_H (V_2\oplus W)\oplus \rplus$, and
\item the element
$\left\{ \Delta(V_1)/\Delta(V_2) \right\}$
is in the image of the double coboundary
\[\delta^2\colon
H^1(\widetilde K_0(\rel{\ZH}{\ZG^{-}}))
\to H^1(\wh(\rel{\ZH}{\ZG^{-}}))\ .\]
\end{enumerate}
\end{theorem}
\begin{corollary}\label{eightteentwo}
There is a stable topological  similarity
$V_1\approx_t V_2$ if and only if
$S(V_1)$ is $s${--}normally cobordant to $S(V_2)$
and
$\Res_K \left\{ \Delta(V_1)/\Delta(V_2) \right\}$
is in the image of the double coboundary $\delta^2_K$
for all subgroups $K \subseteq G$.
\end{corollary}
\begin{remark}
The double coboundary $\delta^2_K$ in this statement is the
one for the subgroup $K$ with respect to an index two subgroup
$K_1 < K$.
Corollary \ref{eightteentwo} follows from Theorem \ref{eightteenone}
and \ptwo, Proposition \ref{ninetwo}.
Notice that Theorem \ref{eightteenone} also gives a way to construct
the stable similarity, assuming that the conditions
are satisfied.
In the most complicated case, one would need all proper
subgroups of $G$ appearing as isotropy groups in $W$.
This will be explained  precisely in Section \ref{unstable}.
\end{remark}
\begin{proof}[The proof of Theorem \ref{eightteenone}]
By \ptwo, Corollary \ref{kersimple} the double coboundary
\[\delta_W^2\colon
H^1(\widetilde K_0(\ccat{\Wmax\times \rminus,G}{\bz})) \to
H^1(\wh(\ccat{\Wmax\times \rminus,G}{\bz}))\]
is isomorphic to
\[\delta^2\colon
H^1(\widetilde K_0(\rel{\ZH}{\ZG^{-}}))
\to H^1(\wh(\rel{\ZH}{\ZG^{-}}))\]
under the cone point inclusion.
The result now follows from Theorem C.
\end{proof}
We conclude this section
with two examples showing methods of constructing
non{--}linear similarities. 
\begin{example}\label{condB}
For $G=\Cy{2^r}$, $r\geqq 5$, the element
$(t^9 + t^{1+2^{r-2}} -t -t^{9+2^{r-2}})$ lies in
$\Rtildefree_t(G)$.
To prove this assertion, note that
$(t^9 + t^{1+2^{r-2}} -t -t^{9+2^{r-2}})$ lies in
$\Rtildefree_n(G)$ by applying \cite[Condition A$'$]{cssww1}.
Let $V_1=t^9 + t^{1+2^{r-2}}$, $V_2=t +t^{9+2^{r-2}}$ and
let $W$ denote the complex $2${--}dimensional
representation with isotropy of index $4$.
The surgery obstruction  in $L_3^p(\bcatrminus{W})$ is zero
by a similar argument to that given in \cite[p.734]{cssw1},
based on the fact that $9(1+2^{r-2}) \equiv 1\Mod{8}$
when $r\geqq 5$, and invariance under group automorphisms
(similar to \cite[4.1]{cssw1}).
The Reidemeister torsion invariant
\[ u = \Delta(V_1)/\Delta(V_2) =
 \frac{(t^9-1)(t^{1+2^{r-2}}-1)}{(t-1)(t^{9+2^{r-2}}-1)}\]
can be written in the form $u = \alpha(v)v$ where 
\[ v = \frac{(t^\ell-1)(t^{1+2^{r-2}}-1)}{(t-1)(t^{\ell+2^{r-2}}-1)}\]
 for
$\ell ^2 \equiv 9 \Mod{2^r}$ and $\ell \equiv 1 \Mod{4}$.
The Galois automorphism  is $\alpha(t) = t^\ell$.
Once again we justify these formulas by appeal to a pull{--}back
diagram for the group ring.
Since the surgery obstruction is determined by the image of
$u $ in  $H^1(\Delta_{\rminus})$, which has exponent 2, and group
automorphisms
 of $G$ induce the identity on this group by  Lemma
\ref{galoisinvariance}(ii), the element $(t^9 + t^{1+2^{r-2}} -t
-t^{9+2^{r-2}})$ lies in $\Rtildefree_t(G)$.

We note that this gives a counterexample to the
\emph{necessity} of \cite[Condition B]{cssww1}
for non-linear similarities.
If $f\colon S(V_1)\to
S(V_2)$ is a $G$-homotopy equivalence, then
the Whitehead torsion
$\tau(f) =  \Delta(V_1)/\Delta(V_2)\in \wh(\ZG)$, but 
 its restriction to $\wh(\ZH)$ is given by
 $U_{9,1+2^{r-2}} = U_{1,9}(U_{1,1})^{-1}$, and this is 
\emph{not} a square, since it is non{--}trivial
in $H^0(\wh(\ZH))$ by the results of Cappell{--}Shaneson
on units (see \cite[p.733]{cssw1}).
\end{example}

%
\begin{example}\label{inducedsim}
According to \cite[Thm. 2]{cssww1},
the element $t-t^5$ has order $2^{r-2}$ in $\rtop{G}$
for $G=\Cy{2^r}$, $r>3$, and order $4$ for $G=\Cy{8}$.
We will verify this claim using our methods.
Note that $2^{r-2}(t-t^5) \in \Rtildefree_h(G)$, for
$r\geqq 3$ and
this is the smallest multiple that works.
A short calculation using \cite[Condition A$'$]{cssww1}
(done in \ptwo, Lemma \ref{twentyfour}) also shows that
$2^{r-2}(t-t^5) \in \Rtildefree_n(G)$, and it remains to
consider the surgery obstruction.

For $r=3$ it follows from \cite[Cor.(iii)]{cssw1} that
$2(t-t^5)$ does not give a $6${--}dimensional similarity.
By Theorem \ref{eightteenone}
 it follows that $2(t-t^5)\notin R_t(G)$,
but  is contained in $R_n(G)$.
Since the surgery obstruction has exponent $2$, we get
$4(t-t^5) \approx_t 0$ as claimed.

To handle the general case, 
let  $U_1 = 2^{r-3}t$ and $U_2 = 2^{r-3}t^5$ be the free
representations over $H = \Cy{2^{r-1}}$, and let
$V_1 = 2^{r-3}t$ and $V_2 = 2^{r-3}t^5$ be the
corresponding free
representations over $G = \Cy{2^r}$.
Notice that
$\Ind_H(U_1 - U_2) = (V_1 -V_2) +  (V^\tau_1
-V^\tau_2)$ where $\tau$ is the group automorphism $\tau(t) =
t^{1+2^{r-1}}$. By \cite[Cor(iii)]{cssw1} we have 
$2t\approx_t 2 t^{1+2^{r-1}}$ and $2t^5 \approx_t 2t^{5+2^{r-1}}$,
whenever $r\geqq 4$. Therefore, $2^{r-2}t = V_1\oplus V_1
 \approx_t
V_1 \oplus V^\tau_1$ and  $2^{r-2}t^5 = V_2\oplus V_2
 \approx_t V_2 \oplus V^\tau_2$.

For $r>3$, the assertion is that
$2^{r-2}(t-t^5) \in \Rtildefree_t(G)$ already.
We prove this by induction starting with $2(t-t^5)$ in $R_n(C(8))$.
Suppose that $2^{r-3}(t-t^5)\in R_n(H)$ for $H=\Cy{2^{r-1}}$,
where $r >3$.
Then $\Ind_H(\Delta(U_1)/\Delta(U_2)) =0$ in
$H^1(\wh(\ZG^{-})/\wh(\ZH))$, since
we are dividing out exactly the image of $\Ind_H$.
If $r=4$, it follows from  Theorem \ref{eightteenone} that
$\Ind_H(U_1 - U_2) \in R_t(C(16))$.
For $r\geqq 5$ we conclude by induction that $\Ind_H(U_1 - U_2) \in
R_t(G)$, 
for $G =\Cy{2^r}$ since $R_t(G)$ is closed under induction from
subgroups. 
Now the calculation above for $\Ind_H(U_1 - U_2)$
shows that $2^{r-2}(t-t^5) \in \Rtildefree_t(G)$ for $r\geqq 4$.
\end{example}



\section{The proof of Theorem E}
\label{sE}
%%\section=21%
Our next example is the computation of $\rtop{G}$ for $G=\Cy{2^r}$.
We first choose a nice basis for $\Rtildefree(G)$. Let
\[ a^{(i)}_s(r) = t^{5^i} - t^{5^{2^{r-s-2}+i}}, \quad\text{for\ }
0 \leqq i < 2^{r-s-2} \text{\ and\ } 1\leqq s \leqq r-2\]
and let $\sigma$ denote the automorphism of $G$ given by
$\sigma(t) = t^5$. It is easy to check that 
the $\{a^{(i)}_s(r)\}$ give
an additive basis for $\Rtildefree(G)$.
For later use, we let $\alpha_s(r) = [a^{(0)}_s(r)]$ and
$\beta_s(r) = [a^{(1)}_s(r)]$  denote elements in
  $\rtildetopfree{G}$ for $1\leqq s \leqq r-2$. When the order $2^r$
of $G$ is understood, we will just write $\alpha_s$, $\beta_s$.
These elements admit some stable non{--}linear similarities,
and behave well under induction and restriction.
\begin{lemma}\label{stablerelations}
We have the following relations.
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $\Res_H(a^{(i)}_s(r)) = a^{(i)}_{s-1}(r-1)$  for
 $ 0\leqq i < 2^{r-s-2}$ and
 $s\geqq 2$.
\item $\Res_H(a^{(i)}_1(r)) = 0$.
\item  $a^{(i)}_s(r) \approx_t a^{(i+2)}_s(r)$ 
for  $ 0\leqq i < 2^{r-s-2}-2$,  with 
$1 \leqq s \leqq r-4$ and $r\geqq 5$.
\item $0 \neq \alpha_1(r) + \beta_1(r) \in \Rtildefree_{n,\Top}(G)$
 for $r\geqq 4$, but $2(\alpha_1(r) + \beta_1(r))=0$.
\item $\Ind_H(a^{(i)}_{s-1}(r-1)) = 2a^{(i)}_s(r) -
 a^{(i)}_1(r) + a^{(2^{r-s-2}+i)}_1(r)$ for $2 \leqq s \leqq r-2$.
\item $2^{s-2}(\alpha_s(r) - \beta_s(r)) \notin \Rtildefree_{n, \Top}(G)$
for $s\geqq 2$.
\item $2^{s-1}(\alpha_s(r) + \beta_s(r)) = 0$ for $2\leqq s <r-2$ and 
$r\geqq 5$.
\item $2^s(\alpha_s(r)) =0$ for $1\leqq s \leqq r-2$ and $r\geqq 4$.
\end{enumerate}
\end{lemma}
\begin{proof}
The first two parts are immediate from the definitions.
Part (iii) uses the Cappell{--}Shaneson trick described in Example
\ref{condB}. Let
\[ u(a,b;c,d) = \frac{(t^{5^a}-1)(t^{5^b} -1)}{(t^{5^c}-1)(t^{5^d}-1)}\]
This element represents a unit in $\ZG$ provided that
$a+b\equiv c+d \Mod{2^{r-2}}$. As mentioned in Section \ref{results},
we can calculate in a pull{--}back square for $\ZG$ over the corner where
these elements become cyclotomic units.
Notice that $u(a,b;c,d)^{-1} = u(c,d;a,b)$ and 
$\sigma(u(a,b;c,d) = u(a+1, b+1;c+1, d+1)$.
 Consider the units
$u = u(i, 2^{r-s-2} +i +2; 2^{r-s-2}+i, i+2)$  associated to the
 Reidemeister
torsion quotient $\Delta(V_1)/\Delta(V_2)$ for the element
$a^{(i)}_s(r)-a^{(i+2)}_s(r)$. We can write $u = \sigma(v)v$
where $v = u(i, 2^{r-s-2}+i+1;  i+1, 2^{r-s-2}+i)$  also represents
a unit in $\ZG$.
Since $\dim V_i = 4$ the spheres $S(V_1)$ and $S(V_2)$ are
$G${--}normally cobordant. Then we restrict to
$H$ and use induction on $r$, starting
with $r=4$ where the similarity follows
 from \cite[Cor.(iii),p.719]{cssw1}. Theorem \ref{eightteenone}
and Galois invariance of the surgery obstruction   under  the action of
$\sigma$  (by  Lemma \ref{galoisinvariance}(ii)) completes
the inductive step.

The non{--}existence of a $6${--}dimensional similarity in Part(iv)
follows from the calculation \cite[Cor.(iii),p.719]{cssw1}. 
Theorem \ref{eightteenone} shows that there is no higher
dimensional similarity. 

Part (v) is again immediate, and Part (vi) is an easy calculation
showing that the element $2^{s-2}(\alpha_s(r) - \beta_s(r))
\in \Rtildefree_{h}(G)$ fails the first congruence condition
in \cite[Thm.1.2]{yo1}, which for this case is the same as
the congruence on the sum of the squares of the weights given
in \cite[Condition A$'$]{cssww1}. 
Part (viii) follows
from part (v) and induction on the order of $G$.
In  Example \ref{inducedsim} we did the case $s=r-2$.
For $s\leqq r-3$ we induce up from the similarity
$2\alpha_1(r-s+1) =0$ provided by \cite[p.719]{cssw1},
which applies since we now have  $r-s+1 \geqq 4$.
Part (vii) is proved in a similar way, using part (v), starting
from the element $\alpha_1(r-1) + \beta_1(r-1)$ for $r \geqq 5$.
Inducing this element in $ \Rtildefree_{n,\Top}(\Cy{2^{r-1}})$
gives 
$$2(\alpha_2(r) + \beta_2(r))+ a^{(2^{r-4})}_1(r) - a^{(0)}_1(r)
+ a^{(2^{r-4}+1)}_1(r) - a^{(1)}_1(r)$$
and applying part (iii) now gives $2(\alpha_2(r) + \beta_2(r))=0$.
The required similarities for $s>2$ are obtained by inducing up
from this one, and using the relations $2a^{(i)}_1(r) = 0$ again
to remove the lower terms in the formula from part (v).
\end{proof}
\begin{proof}[The proof of Theorem E]
We already have the generators and 
relations claimed in the statement of
Theorem E, so it remains to eliminate all other possible relations.
In Example \ref{inducedsim} we proved that $\rtildetopfree{C(8)}
=\cy 4$ generated by $\alpha_1 = t-t^5$, where $0 \neq 2\alpha_1
\in \Rtildefree_{n,\Top}(C(8)$.
For $G = C(16)$, we have 
\[ \rtildetopfree{C(16)} = \big\langle \alpha_2, \alpha_1, \beta_1
\big\rangle\]
and we observe that the elements $2\alpha_2 + \alpha_1$
and $2\alpha_2 + \beta_1$ are not Galois invariant, and hence
do not lie in $\Rtildefree_{n,\Top}(G)$ by \ptwo, Theorem \ref{oneD}(iii).
Therefore $\rtildetopfree{C(16)} = \cy 4 \oplus \cy 2\oplus\cy 2$
as claimed in Theorem E. 

We now assume the result for $H=C(2^{r-1})$, with $r\geqq 5$.
By  applying the inductive
assumption, it is not difficult to give generators
for the subgroup $\ker\Res_H \cap \Rtildefree_{h,\Top}(G)$.
Indeed, a generating set consists of the elements (type $I$)
\[ \big\langle 2^{\ell -1}\alpha_\ell + \alpha_1, \alpha_1 + \beta_1
\mid 2\leqq \ell \leqq r-2
\big\rangle\]
together with the elements (type $II$)
\[ \big\langle 2^{\ell -2}(\alpha_\ell -\beta_\ell) 
\mid 3\leqq \ell \leqq r-3
\big\rangle,\]
where the type $II$ elements appear for $r\geqq 6$.
Notice that none of the generators (except $\alpha_1+\beta_1$) 
 are in $\Rtildefree_{n,\Top}(G)$,
and all the generators have exponent 2.

Let $\gamma_\ell = 2^{\ell-1}\alpha_\ell + \alpha_1$ and consider
an linear relation among type $I$ elements of the form
\[\sum n_\ell \gamma_\ell + \epsilon(\alpha_1 +\beta_1) = 0\]
If $\#\{n_\ell \neq 0\}$ is odd, then the left side is not Galois invariant,
so it can't be a relation.  
On the other hand, if $\#\{n_\ell \neq 0\}$ is even, then we can write
the first term as a sum of terms 
$2^{\ell_1 -1}\alpha_{\ell_1}- 2^{\ell_2 -1}\alpha_{\ell_2} \in 
\Rtildefree_{h}(G)$ with each
$\ell_i \geqq 2$.  Suppose
first that $\epsilon =1$. If the first sum on the lefthand side was in
$\Rtildefree_{n,\Top}(G)$, then its  surgery obstruction would be
a square  and hence trivial by Theorem \ref{eightteenone}.
This contradicts the fact that $\alpha_1 +\beta_1\neq 0$.

We are left with the possibility that $\#\{n_\ell \neq 0\}$ is even
and $\epsilon=0$. Suppose that $\ell_0\leqq r-3$ 
is the minimal index such
that $n_{\ell_0} =1$, and write the left side as
\[ 2^{\ell_0 -1}
\big [ \sum_{\ell >\ell_0} n_\ell 2^{\ell-\ell_0}\alpha_\ell
 +\alpha_{\ell_0}\big ] = 0\ .\] 
However, if we call  the term in brackets $\omega$ and 
restrict it $(\ell_0 -1)$ steps to
the  subgroup of index $2^{\ell_0-1}$ in $G$, we
get
\[ \sum_{\ell >\ell_0} n_\ell 2^{\ell-\ell_0}\alpha_{\ell-\ell_0+1}
 +\alpha_1\]
and this is not Galois invariant. We can arrange the signs of the
coefficients $n_\ell$ so that $\omega \in \Rtildefree_{h,\Top}(G)$.
Since twice this element is trivial,
its normal invariant order equals 2. Now by \ptwo, Theorem \ref{oneD}(iv)
we conclude that the normal invariant order (over $G$) of the
bracketed term $\omega$ must be $2^{\ell_0}$. Hence there is no
relation of this form.

Next we let $\xi_\ell = 2^{\ell -2}(\alpha_\ell -\beta_\ell)$,
and suppose that we have a relation of the form
\[ \sum_{\ell\geq\ell_0} n_\ell\xi_\ell =0\]
where $\ell_0$ is the minimal non-zero coefficient index as before.
If $\#\{n_\ell \neq 0\}$ is odd, then the left-hand side fails the first
congruence test for the normal invariant and so it can't be a relation.
If $\#\{n_\ell \neq 0\}$ is even we
 write the left side
as  $2^{\ell_0 -2}\omega$, with $\omega \in \Rtildefree_{h, \Top}(G)$, and
restrict $\omega$ down to a subgroup $K$ of index $2^{\ell_0 -2}\leqq 2^{r-5}$. 
 The
restriction  has the form
\[  \Res_K(\omega) =\sum_{\ell >\ell_0} n_\ell 2^{\ell-\ell_0}
(\alpha_{\ell-\ell_0+2} -\beta_{\ell-\ell_0+2})
+ (\alpha_2 + \beta_2) \]
Since $\#\{n_\ell \neq 0\}$ is even, this can be re-written 
as a sum $\Res_K(\omega) = \theta - \sigma(\theta)$ with
$\theta \in \Rtildefree_{h, \Top}(K)$. If
$\Res_K(\omega) \in \Rtildefree_{n,\Top}(K)$, this would imply
that its surgery obstruction was trivial, hence $\Res_K(\omega)=0$.
But restriction one more step down gives $\alpha_1 + \beta_1 \neq 0$
and we have a contradiction. It follows that the normal invariant
order of $\Res_K(\omega)$ equals 2, and that of $\omega$
is  $2^{\ell_0 -1}$ so the original relation was trivial.

We are left with the possibility of further relations among the type $I$
and type $II$ elements of the form
\[ \sum n_\ell \gamma_\ell + \sum m_\ell \xi_\ell  
+\epsilon (\alpha_1 + \beta_1)=0.\]
It is easy to reduce to a relation of the form
\[ \sum n_\ell 2^{\ell-1}\alpha_\ell +
\sum m_\ell2^{\ell-2}(\alpha_\ell - \beta_\ell) = 0\]
where $\#\{n_\ell \neq 0\} = 2\cdot\nu$ is even and
$\#\{m_\ell \neq 0\} + \nu$ is even. This follows from the Galois 
invariance and the fact that the $p_1${--}obstruction
for normal cobordism is non-trivial
 for the order 2 elements
of the form
$(2^{\ell_1-1}\alpha_{\ell_1}  - 2^{\ell_2-1}\alpha_{\ell_2})$.

Now if $\ell_0$ denotes the minimal index such that $n_\ell$
or $m_\ell$ is non{--}zero, we have two cases. First, if
$m_{\ell_0} \neq 0$, we factor out $2^{\ell_0-2}$ and
 restrict our relation to the subgroup $K$ of
index $2^{\ell_0-2}$.
 We obtain
\[ 2^{\ell_0-2}\big [
\sum_{\ell\geqq \ell_0}n_\ell 2^{\ell-\ell_0+1}\alpha_{\ell-\ell_0+2}
+ \sum_{\ell >\ell_0} m_\ell 2^{\ell-\ell_0}
(\alpha_{\ell-\ell_0+2} -\beta_{\ell-\ell_0+2})
+ (\alpha_2 - \beta_2) \big ] \]
If the normal invariant obstruction for the term in brackets
vanishes, then its surgery obstruction
is zero: when $\nu$ is even we 
can rewrite the sum of type $II$ elements as above to get a
Reidemeister torsion obstruction of the form $\theta - \sigma(\theta)$, 
with $\theta \in \Rtildefree_{h, \Top}(K)$.
The remaining type $I$ terms are collected in 
pairs $(2^{\ell_1+1}\alpha_{\ell_1+2}  - 2^{\ell_2+1}\alpha_{\ell_2+2})$
whose surgery obstructions are squares.
When $\nu$ is odd, we replace $\alpha_2 - \beta_2$ by the expression 
$(\alpha_2 - 2^t\alpha_{t+2}) - (\beta_2 -2^t\beta_{t+2}) + 2^t(\alpha_{t+2} - \beta_{t+2})$
whose torsion has the form $\theta - \sigma(\theta)$ plus a square, and continue as for
$\nu$ even.
The vanishing of this surgery obstruction 
contradicts
$ (\alpha_1 + \beta_1) \neq 0$ on restricting one step further down.

In the remaining case, if $n_{\ell_0} \neq 0$ and $m_{\ell_0} = 0$, we 
factor out $2^{\ell_0-1}$ and  restrict to
the subgroup of index $2^{\ell_0-1}$.
As above, the other factor is  not Galois invariant and we get a
contradiction to the existence of a  normal invariant. 
This completes the
proof.
\end{proof}

\section{Non{--}Linear Similarity for Cyclic $2$-Groups}
\label{unstable}
%%\section=22% 
In this final section we will apply our previous results to give
explicit necessary and sufficient 
conditions for the existence of a  non-linear similarity
$V_1\oplus W \sim_t V_2 \oplus W$ for representations of finite
cyclic $2$-groups. The main result is Theorem \ref{unstablefour}.

\begin{lemma}\label{unstableone} Let $G=\Cy{2^r}$ be a cyclic
$2$-group.
A basis for $\Rtildefree_t(C(8))$ is given by $4(t-t^5)$.
 A basis for $\Rtildefree_t(G)$, $r\geqq 4$,  is given by the elements
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
 \item $2^s \alpha_s(r) $ for $1\leqq s \leqq r-2$, and
\item $2(\alpha_1(r) + \beta_1(r))$, 
\end{enumerate}
together with  (provided $r\geqq 5$) the elements
\begin{enumerate}
\item[(iii)] $2^{s-1}(\alpha_s(r) + \beta_s(r)) $ for $2\leqq s
\leqq r-3$,  and 
\item[(iv)] $\gamma^{(i)}_s(r)=(a^{(i)}_s(r)- a^{(i+2)}_s(r))$ 
for  $ 0\leqq i < 2^{r-s-2}-2$ and $1\leqq s \leqq r-4$.
\end{enumerate}
\end{lemma}
\begin{proof}
This an immediate consequence of  Theorem E.
\end{proof}
The next step is to 
rewrite the basis is a more convenient form.
It will be useful to introduce some notation for
certain subsets of $\Rtildefree(G)$. Let
$$\ca(r) = \{ 2\alpha_1(r), 2(\alpha_1(r) + \beta_1(r))\}$$
for $r\geqq 4$ and set $\ca(3) = \{ 4\alpha_1(3)\}$.
Next, let
$$\widehat \cb(r) = \{ 2\alpha_1(r), 
\alpha_1(r)+\beta_1(r)\}$$ for $r \geqq 4$, and let
$\widehat \cb(3) =
\{ 2\alpha_1(3)\}$. Finally, let
$$\cc(r) = \{ \gamma^{(i)}_s(r)\Vertical 
0\leqq i < 2^{r-s-2}-2 \text{\ and \ } 1\leqq s \leqq r-4\}$$
when $r\geqq 5$, and otherwise $\cc(r)=\emptyset$.

Recall the notation $G_k$ for the subgroup of index $2^k$ in $G$.
We let $\Ind_k$ denote induction of
representations from
$G_k$ to $G$, and $\Res_k$ denote restriction of representations
from $G_k$ to $G_{k+1}$.
Define 
$$\cb(r) = \{ \Ind_k(\chi) \Vertical \chi \in
\widehat \cb(r-k)\}$$ for $r\geqq 4$ and $0 < k < r-2$.
Note that $x \in \cb(r)$ implies  that
$x = \Ind_k(\chi)$ and $\Res_k(\chi) =0$.


\begin{lemma}\label{unstabletwo}
 The  free abelian group $\Rtildefree_t(G)$
 has an integral basis given by the elements in the set
$\ca(r)\cup \cb(r) \cup \cc(r)$.
\end{lemma}
\begin{proof}
This follows easily from the relations in Lemma 
\ref{stablerelations}, particularly the induction formula
$$\Ind_H(a^{(i)}_{s-1}(r-1)) = 2a^{(i)}_s(r) -
 (a^{(i)}_1(r) - a^{(2^{r-s-2}+i)}_1(r))$$
valid for $2 \leqq s \leqq r-2$. Notice that the second
term on the right-hand side is just $\alpha_1(r) - \beta_1(r)$ if
$s=r-2$, and otherwise it is a linear combination of the
basis elements $\gamma^{(j)}_1(r)$. 
The result is easy for 
$r=3$ or $r=4$, and the inductively we assume it for $r-1$.
Then $\Ind_H(2^{s-1}\alpha_{s-1}(r-1)) = 2^s\alpha_s(r)$,
plus terms in $\ker\Res_H$ which are all contained in the span
of $\ca(r)$ and $\cc(r)$. Similarly, 
$$\Ind_H(2^{s-2}(\alpha_{s-1}(r-1) + \beta_{s-1}(r-1))) =
2^{s-1}(\alpha_{s}(r) + \beta_{s}(r)),$$
 plus terms in the span of
$\ca(r)$ and $\cc(r)$. This shows that integral linear combinations of
the set $\ca(r)\cup\cb(r)\cup\cc(r)$ span $\Rtildefree_t(G)$. Since
$\Res_H(\gamma^{(i)}_s(r)) = \gamma^{(i)}_{s-1}(r-1)$, 
$\Res_H\circ\Ind_H = 2$, and $\ca(r) \subset \ker\Res_H$, we
conclude by induction that there are no non-trivial integral
relations  among the elements of $\ca(r)\cup\cb(r)\cup\cc(r)$.
\end{proof}

Using the basis given in Lemma \ref{unstabletwo},
we now define the set of {\it weights\/} 
$\theta(x) = \{i_1, i_2, \dots, i_\ell\}$ 
of an element $x \in \Rtildefree_t(G)$. 
This will be a subset of $\{1, 2, \dots, r-2\}$ arranged in
strictly ascending order.
 It will be used to identify the  minimal set
of isotropy subgroups needed for the construction of a non-linear
similarity for  $x \in \Rtildefree_t(G)$. 
\begin{definition}\label{weights}
The weights for $x \in \ca(r)\cup\cb(r)\cup\cc(r)$ are given as follows:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
 \item If $x \in \ker\Res_H$ then $\theta(x) = \{ 1\}$.
\item If $x = \Ind_{k}(2\alpha_1(r-k))$, for $r>r-k\geqq 4$, 
then $\theta(x) =\{ k+1\}$.
\item If $x = \Ind_k(\alpha_1(r-k) + \beta_1(r-k))$, for $r>r-k\geq 4$,
 or $x=\Ind_k(2\alpha_1(3))$, 
then  $\theta(x) = \{k, k+1\}$.
\item If $x = \gamma^{(i)}_s(r)$, then  $\theta(x) = \{ 1,2,\dots, s\}$.
\end{enumerate}
 If $x = \sum n_\ell $ is an integral linear combination of
elements
 $\omega_\ell \in \ca(r)\cup\cb(r)\cup\cc(r)$, then
$\theta(x) = \bigcup \{\theta(\omega_\ell)\Vertical n_\ell \not=0\}$.
\end{definition}
 In other words, after collecting the indices of the subgroups
involved in the unique linear combination of basis elements for $x$,
we arrange them in ascending order ignoring repetitions to produce
$\theta(x)$.
\begin{definition}\label{evenodd}
We say that an element $x\in \Rtildefree_t(G)$ is {\it even\/}
if $x = \sum n_\ell \omega_\ell$,  $\omega_\ell \in \ca(r)\cup\cb(r)\cup\cc(r)$,
 has
$n_\ell \equiv 0 \Mod{2}$ whenever one of the following holds:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item  $\omega_\ell = 2\alpha_1(r) \in \ca(r)$, or
\item $\omega_\ell \in \cc(r)$.
\end{enumerate}
Otherwise, we say the element $x$ is {\it odd\/}.
An element $x$ has {\it mixed\/} type if $x$ is even, but $n_\ell\not\equiv 0\Mod{2}$
for some $\omega_\ell=\Ind(2\alpha_1(r-k))$ with $r>r-k\geqq 4$.
Such an element $x$ has {\it depth\/} equal to the minimum $k$ for which $x$
contains  a constituent $\Ind(2\alpha_1(r-k))$ with odd multiplicity.
\end{definition}

\begin{lemma}\label{unstablethree}
An element $x=[V_1 - V_2] \in \Rtildefree_t(G)$ is even if and only if
its Reidemeister torsion invariant $\{\Delta(V_1)/\Delta(V_2)\} = 0$ in
$H^1(\wh(\ZG^-)/\wh(\ZH))$.
\end{lemma}
\begin{proof}
 The Reidemeister torsion invariants for elements of $\ker\Res_H$
lie in   $H^1(\wh(\ZG^{-}))$, which has a basis of units 
$U_{1,i}$ for $1\leqq i < 2^{r-1}$ and $i \equiv 1 \Mod 4$ 
 (see \cite[\S 5]{cs5}). In particular, 
 $\{\Delta (2\alpha_1(r))\}= U_{1,1} \neq 0$, and
 $\{\Delta (\alpha_1(r)+\beta_1(r))\}= U_{1,5} \neq 0$,
  but $\{\Delta(2(\alpha_1(r) + \beta_1(r)))\} = U_{1,5}^2 = 0$ in
$H^1(\wh(\ZG^{-}))$. Note that  $H^1(\wh(\ZG^{-}))$ injects into
$H^1(\wh(\ZG^-)/\wh(\ZH))$.  
 
 Suppose first that
 $x=[V_1 - V_2] \in \Rtildefree_t(G)$ is even. It follows that   its Reidemeister torsion invariant
is either a square or induced up from $H$, so $\{\Delta(V_1)/\Delta(V_2)\} = 0$
 in $H^1(\wh(\ZG^-)/\wh(\ZH))$.
 Conversely, suppose that $x \in \Rtildefree_t(G)$ has an odd coefficient $n_\ell$
for   $\omega_\ell = 2\alpha_1(r)$ or $\omega_\ell = 
 \gamma^{(i)}_1(r)$ in $ \ker\Res_H $. In these cases, 
 $\{\Delta(V_1)/\Delta(V_2)\} \not= 0$
 in $H^1(\wh(\ZG^-)/\wh(\ZH))$.
 
Finally, suppose that $n_\ell$ is odd for some 
$\omega_\ell =\gamma^{(i)}_s(r) \in \cc(r)$ with $s\geqq 2$.
If the Reidemeister torsion invariant for $\omega_\ell$ were trivial
in  $H^1(\wh(\ZG^-)/\wh(\ZH))$, then its image under the twisted restriction
map 
would also be trivial. This is the map defined by composing
the twisting isomorphism 
$$H^1(\wh(\ZG^-)/\wh(\ZH)) \cong H^0(\wh(\ZG)/\wh(\ZH))$$
with $\Res_H$, followed by another twisting isomorphism
at the index two level.
But $$\Res_H(\gamma^{(i)}_s(r)) = \gamma^{(i)}_{s-1}(r-1)\ ,$$
so after restricting $s -1$ steps we arrive at  $\gamma^{(i)}_{1}(r-s +1)
\in \ker\Res_{s-1}$, and a contradiction as in the previous case.
\end{proof}

For $2\leqq i \leqq r-2$, let $W_i$ denote any irreducible
$2$-dimensional real representation of $G$ with isotropy 
group $G_{i}$, let $W_1=\rminus$, and let $W_0 = \rplus$.
The terms defined in Definitions  \ref{weights} 
and  \ref{evenodd}
will  be used in
the statement of our classification result.
\begin{theorem}\label{unstablefour}
Let $G=\Cy{2^r}$ and $V_1$, $V_2$ be free $G$-representations.
Suppose that $x = [V_1-V_2]\in \Rtildefree_t(G)$.
Then $V_1 \oplus W \sim_t V_2 \oplus W$ for a given
$G$-representation $W$ if and only if
the representation $W$ contains:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item  a summand $W_{k}$ for
each $k \in \theta(x)$,
\item a summand $W_t$ for some $0 \leqq t \leqq depth(x)$ when 
$x$ has mixed type, and
\item a summand $\rplus$ when $x$ is odd.
\end{enumerate}
\end{theorem}
\begin{proof}
The sufficiency of the given conditions  follows immediately
from  Theorem \ref{eightteenone}, the basic list of $6$-dimensional similarities
in \cite[Thm.1 (iii)]{cssw1},  and the
following commutative diagram
\[
\xymatrix{ L^h_{2k}(\ZH)\ar[r]^{i_*}
\ar[d]_{\trf{\Res_H W}}&L^h_{2k}(\ZG)\ar[d]_{\trf{W}}\\
 L^h_{2k+m}(\mathcal C_{\Res W, H}(\bz))\ar[r]^{i_*}
&L^h_{2k+m}(\mathcal C_{W,G}(\bz))
}\]
or the corresponding $L^p$ version if 
 $W$ contains an  $\rplus$ sub-representation.
 The summand $\rplus$ will be unnecessary exactly when 
$\{\Delta(V_1)/\Delta(V_2)\} = 0$ in $H^1(\wh(\ZG^-)/\wh(\ZH))$,
by Theorem \ref{noRplus}. 
By Lemma \ref{unstablethree} this happens precisely when $x $ is even.
 
 For all  elements  $[V_1 - V_2] \in \Rtildefree_t(H)$ we have
 $\dim V_i =2k \equiv 0 \Mod 4$.
To handle the surgery obstructions of induced representations, note that
$$\left\{ \Delta(\Ind V_1)/\Delta(\Ind V_2) \right\}
= i_*\left\{ \Delta(V_1)/\Delta(V_2) \right\}\in H^0(\wh(\ZG))$$
for any  $x=[V_1 - V_2] \in \Rtildefree_t(H)$. For the surgery
obstruction of an $H$-homotopy equivalence $f\colon S(V_2) \to S(V_1)$ we have the relation
$$\sigma(\Ind (f)) = i_*(\sigma(f)) + i_*(\beta) \in L_0^h(\ZG)$$ 
for some  $\beta \in L_0^s(\ZH)$. It follows that 
 $$i_* (\trf{\Res W}(\sigma(f))) = \trf{W}(\sigma(\Ind(f))) 
 \in L^h_{m}(\mathcal C_{W,G}(\bz))$$
 provided that $\Res W $ contains an $\rminus$ summand.
 This formula gives the existence of unstable similarities for elements
$\Ind_k(\chi) \in \cb(r)$ under the given conditions on $W$.

The necessity of  condition (i) follows immediately from
\cite[Thm. 5.1]{hp2}, once we relate the odd $p$-local components used there
to our setting. The surgery obstruction group $L^h_{2k}(\ZG) $ has a natural splitting,
after localizing at any odd prime, indexed by the divisors of $|G| = 2^r$.
An element $\sigma \in  L^h_{2k}(\ZG)$ of infinite order
will have a non-trivial projection in the ``top"  $2^r$-component
provided that $\Res_H(\sigma) =0$. 
This applies to our basis elements $2\alpha_1(r)$, $2(\alpha_1(r) + \beta_1(r))$,
$4\alpha_1(3)$,
and $\gamma^{(i)}_1(r)$. If any of these occurs as a constituent in $x$, a summand
$\rminus$ is necessary to produce the non-linear similarity.

More generally, $\sigma$ will have a 
non-trivial projection in the 
$2^{r-k}$-component for $k >0$ provided that
 $\Res_{G_k}(\sigma)$ is non-zero but
 $\Res_{G_{k+1}}(\sigma)=0$.
In this case
the representation $W$ must contain at least one summand  $W_{k+1}$
 which restricts to $\rminus$ for the
subgroup $G_k$. To see this
we restrict the surgery obstruction
for our element $x = [V_1-V_2]$ to each $G_k$ for $k+1 \in
\theta(x)$ and apply \cite[Thm. 5.1]{hp2}.
This establishes part of condition (i) for $x$ containing any one of the basis
elements $\Ind_k(\chi) \in \cb(r)$.
We will deal below with the necessity
of the additional summand
$W_k$  for the basis elements $\Ind_k(\alpha_1(r-k) + \beta_1(r-k))$,
or $\Ind_k(2\alpha_1(3))$,
having $\theta=\{k, k+1\}$.
 

For the elements  $\gamma^{(i)}_s(r)$ with $s \geqq 2$, we use the relation
$(2x - \Ind_H(\Res_H x)) \in \ker \Res_H$
and the induction formula in Lemma \ref{stablerelations} again. This shows that the
elements $\gamma^{(i)}_s(r)$ have a non-trivial projection into each 
component of index $\leqq 2^s$, and we again apply \cite[Thm. 5.1]{hp2}.
Condition (i) is now established for any $x$ containing
some $\gamma^{(i)}_s(r)$ as a constituent. 

Condition (ii) applies only to elements $x$ of mixed type. These contain a constituent
$\Ind_k(2\alpha_1(r-k))$, for $r-k\geqq 4$, with odd multiplicity.
It says that $W$ must contain a summand which restricts to a sum of $\rplus$ 
representations over the subgroup $G_k$. 
Suppose if possible
that $W=W_{k+1}\oplus U$ where $U$ has isotropy groups contained in $G_{k+1}$.
Then $K_1(\bcat{W_{k+1}}) \cong K_1(\bcat{W}) $ by \ptwo, Lemma \ref{orb},
and $$\wh(\bcat{W_{k+1}}) \cong \wh(\ZG)/\wh(\bz G_{k+1})\ .$$
However, the results of
\cite[Prop. 1.2]{cs5} show that the inclusion induces an injection
$$\Coh{0}{\wh(\bz G_k)/\wh(\bz G_{k+1}) }\to 
\Coh{0}{\wh(\ZG)/\wh(\bz G_{k+1}) }\ .$$
It follows that the surgery obstruction $\trf{W}(\sigma)$ is non-zero,
contradicting the existence of such a
non-linear similarity. 

To see that a summand $W_k$ is also necessary when $x$ has a constituent
of the form $\Ind_k(\alpha_1(r-k) + \beta_1(r-k))$, $r\geqq 4$, 
or $\Ind_k(2\alpha_1(3))$, we start with the argument
of the last paragraph again.
 It shows that $W$ must contain some $W_t$
for $t \leqq k$. However the representations $W_t$ with $t<k$ all restrict to
$\rplus$ or 
$\Rplus{2}$ over $G_k$, so it is enough to eliminate similarities of the form
$W =W_2 \oplus \rplus$ for $G$ with $k=1$. 
Restriction from $G$ to $G_{k-1}$ then gives the general case.
However, a non-linear similarity of this form is ruled out by considering
the surgery obstruction $\trf{W}(\Ind_H(\sigma)) \in \Lgroup{p}{2}{\bcat{W}}$. 
Over the subgroup
$H$ the representation $\Res_H W_2 =\Rminus{2}$, and the surgery obstruction
$\trf{\Res_H W}(\sigma)$ is computed from the twisting diagrams tabulated in 
\cite[Appendix 2]{htw3}. In particular,
 it is non-zero in
$ \Lgroup{p}{2}{\ZH}/\Lgroup{p}{2}{\ZK}$ 
 (see  \cite[Table 2: $U\to U$, p.123]{htw3}),
 which injects into
 $\Lgroup{p}{2}{\ccat{\Res_H W,H}{\bz}}$, 
where $K$ denotes the subgroup of index $4$ in $G$.
Since
 $$0 \to \Lgroup{p}{2}{\ZG}/\Lgroup{p}{2}{\ZK}
\to \Lgroup{p}{2}{\bcat{W}} \to \Lgroup{p}{0}{\ZK} \to 0$$ 
we check 
that the inclusion $H\subset G$ induces an injection 
$$\Lgroup{p}{2}{\ccat{\Res_H W,H}{\bz}} \to 
\Lgroup{p}{2}{\bcat{W}}$$
and hence the surgery obstruction $\trf{W}(\sigma)\neq 0$ from 
the commutative diagram  above.
\end{proof}
%\bibliographystyle{amsplain}
%\bibliography{ekpmain}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
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%\bigskip
%\noindent
%\leftline{\textsc{The second part of this paper is}:}
%\vskip .3cm
%\vbox{\hbox{I. Hambleton and E. K. Pedersen,
%{\it Topological equivalence of linear representations  for}}
%\hbox{{\it cyclic groups}: \parttwo, Forum Math. (to appear).}}
%\bigskip
\end{document}

sim2_final.tex
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\begin{document}
  \title[Similarities of cyclic groups]{Topological equivalence of linear representations\\
 for cyclic groups: \parttwo}
\author{Ian Hambleton}
\address{Department of Mathematics \& Statistics,
McMaster University\newline
\indent Hamilton, Ont.,  Canada,  L8S 4K1}
\email{ ian@math.mcmaster.ca}

\author{Erik K. Pedersen}
\address{Department\ of\ Mathematical Sciences\\
SUNY at Binghamton\newline
\indent Binghamton,  NY, 13901}
\email{ erik@math.binghamton.edu}
\thanks{Partially supported by NSERC grant A4000
and  NSF grant  DMS 9104026. The authors also
wish to thank the Max Planck Institut f\"ur Mathematik, Bonn, for
its hospitality and support.}
\begin{abstract} In  the two parts of this paper we prove that the Reidemeister torsion invariants
determine  topological equivalence of $G${--}representations,
for $G$ a finite cyclic group.
\end{abstract}
\date{Feb. 19. 2004}
\maketitle
\section{ Introduction\label{intro}}
Let $G$ be a finite group and $V$, $V'$ finite dimensional
real orthogonal representations of $G$.
Then $V $ is said to be
\emph{topologically equivalent} to $V'$
(denoted $V \sim_t V'$) if there exists a homeomorphism $h\colon
V \to V'$ which is $G$-equivariant.
If $V$, $V'$ are topologically equivalent, but not linearly isomorphic,
then such a homeomorphism is called a non-linear similarity.
These notions were introduced and studied by de Rham \cite{drh2},
\cite{drh1}, and developed extensively in
 \cite{cs1}, \cite{cs2}, \cite{hsp1},  \cite{mr1}, and
\cite{cssw1}.
In the two parts of this paper, referred to as \pone\
and \ptwo, we complete de Rham's program by showing that
Reidemeister torsion invariants and number theory determine
non-linear similarity for finite cyclic groups.

A  $G$-representation is called \emph{free} if
each element $1\neq g \in G$ fixes only the zero vector.
Every representation of a finite cyclic group has a
unique maximal free subrepresentation.

\begin{thmmain} Let $G$ be a finite cyclic group
and $V_1$, $V_2$ be free
$G${--}representations. For any $G${--}representation $W$, the
existence of a non-linear similarity
$V_1\oplus W \sim_t V_2\oplus W$ is entirely 
determined by explicit
congruences in the weights of the free summands $V_1$, $V_2$,
 and the ratio
$\Delta(V_1)/\Delta(V_2)$ of their Reidemeister torsions, 
up to an algebraically described indeterminacy.
\end{thmmain}
\noindent
The notation and the  indeterminacy are given in Section
\ref{results} and a detailed statement of results in
Theorems A{--}E. 
This part of the paper contains the foundational results
and calculations in 
bounded algebraic $K$- and $L$-theory needed to prove
the main results on non-linear similarity. 
The study of non-linear similarities $V_1\oplus W \sim_t V_2\oplus W$ increases in difficulty with the number of isotropy types in
$W$. We introduce a new method using excision in bounded surgery theory, based on the \emph{orbit type filtration}, to organize and deal with these difficulties.
We expect that this technique will be useful for other applications.
Our most general results about non-linear similarity for arbitrary cyclic groups are Theorem C and its extensions (see
 Sections \ref{C} and \ref{rminustwo}).

In Sections \ref{splitting} and \ref{sD} we study the group
$\rtop{G}$ of $G${--}representations modulo \emph{stable} topological equivalences  (see \cite{cs2} where $\rtop{G}\otimes \bq$ is computed).
As an application of our general results,  we determine the
structure of the torsion in $\rtop{G}$, for $G$ any cyclic group
(see Theorem \ref{oneD}), and in Theorem D we give the  calculation of $\rtop{G}$
for $G = \Cy{4q}$, for $q$ odd, correcting \cite[Thm. 2]{cssw1}.
One interesting feature is that Corollary \ref{sixdim}
and  Theorem D indicate a connection between
 the orders of the \emph{ideal class groups} for
cyclotomic fields and topological equivalence of linear
representations.  

\tableofcontents
\section{ Statement of Results\label{results}}
For the reader's convenience, we  recall some notation from Part \partone, and then give the main results of both parts.
Theorems A and B  are proved in Part \partone\ and Theorems
C and D are proved in Part \parttwo. The proof of Theorem E
is divided between the two parts.

Let $G = \Cy{4q}$, where $q >1$, and let $H = \Cy{2q}$ denote
the subgroup of index 2 in $G$.
The maximal odd order subgroup of $G$ is denoted $\Godd$.
We fix a generator $G=\langle t \rangle$ and a primitive $4q^{th}$-root of
unity $\zeta = \exp{2\pi i/{4q}}$.
The group $G$ has both a trivial $1$-dimensional real representation,
denoted $\rplus$, and a non-trivial $1$-dimensional
real representation, denoted $\rminus$.

A \emph{free} $G${--}representation is a sum of faithful
1-dimensional complex representations.
Let $t^a$, $a \in \mathbf Z$, denote the complex numbers
 $\bc$ with action
$t\cdot z = \zeta^az$ for all $z \in \bc$.
This representation is free if and only if $(a, 4q) = 1$, and
 the coefficient $a$ is well{--}defined only modulo $4q$.
Since $t^a \cong t^{-a}$ as real $G${--}representations,  we
can  always choose the weights $a \equiv 1 \Mod{4}$.
This will be assumed unless otherwise mentioned.

Now suppose that $V_1 = t^{a_1} + \dots + t^{a_k}$
is a free $G${--}representation.
The Reidemeister torsion invariant of $V_1$ is defined
as
\[\Delta(V_1) = \prod_{i=1}^{k} (t^{a_i} -1) \in \mathbf Z [t]/\{\pm t^m\} \ .\]
Let $V_2 = t^{b_1} + \dots + t^{b_k}$ be another free representation,
such that $S(V_1)$ and $S(V_2)$ are $G$-homotopy equivalent. 
This just means that the products of the weights $\prod a_i \equiv \prod b_i
\Mod{4q}$.
Then the Whitehead torsion of any $G$-homotopy equivalence  is
determined by the element
\[\Delta(V_1)/\Delta(V_2) = \frac{\prod (t^{a_i}-1)}{\prod(t^{b_i}-1)}
\]
since $\wh(\ZG)\to \wh(\QG)$ is monic
 \cite[p.14]{o1}.  

Let $W$ be a finite-dimensional  $G$-representation. 
 A necessary condition for a non-linear similarity $V_1\oplus W \sim_t V_2\oplus W$ 
is the existence of a  $G$-homotopy equivalence
 $f\colon S(V_2) \to S(V_1)$ such that
$f \ast id\colon S(V_2\oplus  U)\to S(V_1\oplus  U)$ is freely 
$G$-normally cobordant to the identity map on $S(V_1\oplus  U)$,
for \emph{all} free $G${--}representations $U$
(see \pone, Section \ref{basics}).
If $V_1$ and $V_2$ satisfy this
condition, we say that $S(V_1)$ and $S(V_2)$ are \emph{$s${--}normally cobordant}. 
This  condition for non-linear similarity can
be decided by explicit congruences in the weights of $V_1$
and $V_2$ (see \cite[Thm. 1.2]{yo1}).

This quantity, $\Delta(V_1)/\Delta(V_2)$ is the basic invariant
determining non-linear similarity. It represents a unit in the group
ring $\ZG$, explicitly described for $G = \Cy{2^r}$ by Cappell and
Shaneson in
\cite[\S 1]{cs5} using a pull-back square of rings. 
To state concrete results we need to evaluate this invariant
modulo suitable indeterminacy. 

The involution $t \mapsto t^{-1}$ induces the identity on $\wh(\ZG)$,
so we get an element
\[\left\{ \Delta(V_1)/\Delta(V_2) \right\}
\in H^0(\wh(\ZG))\]
where we use $H^i(A)$ to denote the Tate cohomology
$H^i(\cy2;A)$ of $\cy2$ with coefficients in $A$.

Let $\wh(\ZG^-)$ denote the Whitehead group $\wh(\ZG)$
together with the involution induced by $t \mapsto -t^{-1}$.
Then for  $\tau(t) = \frac{\prod(t^{a_i}-1)}{\prod(t^{b_i}-1)}$,
we compute
\[\tau(t)\tau(-t)=
\frac{\prod(t^{a_i}-1)\prod((-t)^{a_i}-1)}
{\prod(t^{b_i}-1)\prod((-t)^{b_i}-1)}
= \prod\frac{(t^2)^{a_i}-1}{((t^2)^{b_i}-1)}\]
which is clearly induced from $\wh(\ZH)$.
Hence we also get a well defined
element
 \[\left\{ \Delta(V_1)/\Delta(V_2) \right\}
\in H^1(\wh(\ZG^-)/\wh(\ZH))\ .\]
This calculation takes place over the ring
 $\Lambda_{2q} =\bz [t]/(1 + t^2 + \dots + t^{4q-2})$,
 but the result holds
over
$\ZG$ via the involution{--}invariant pull-back square
\[
 \begin{matrix} \ZG  & \to &\Lambda_{2q}\cr
\downarrow &&\downarrow\cr
\bz[\cy 2]& \to & \cy{2q}[\cy 2]
\end{matrix}
\]


Consider the exact sequence of modules with involution:
\eqncount
\begin{equation}\label{okseq}
K_1(\ZH) \to K_1(\ZG)\to
 K_1(\rel{\ZH}{\ZG}) \to \widetilde K_0(\ZH)\to \widetilde K_0(\ZG)
\end{equation}
and define
$\wh(\rel{\ZH}{\ZG}) =K_1(\rel{\ZH}{\ZG})/{\scriptstyle \{\pm
G\}}$ . We then have a short exact sequence
\[0\to \wh(\ZG)/\wh(\ZH) \to \wh(\rel{\ZH}{\ZG}) \to \ck\to 0\]
where $\ck =\ker(\widetilde K_0(\ZH) \to \widetilde K_0(\ZG))$.
Such an exact sequence of $\cy 2$-modules induces a long exact
sequence in Tate cohomology.
In particular, we have a coboundary map
\[\delta\colon H^0(\ck) \to H^1(\wh(\ZG^-)/\wh(\ZH))\ .\]
Our first result deals with isotropy groups of index 2, as
is the case  for all the non{--}linear similarities
constructed in \cite{cs1}. 
\begin{thma}
Let $V_1=t^{a_1} + \dots + t^{a_k} $
and $V_2 = t^{b_1} + \dots + t^{b_k}$ be free 
$G${--}representations,
with $a_i \equiv b_i\equiv 1 \Mod{4}$.
There exists a topological similarity $V_1\oplus \rminus \sim_t
V_2\oplus \rminus$ if and only if
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $\prod a_i \equiv \prod b_i \Mod{4q}$,
\item $\Res_H V_1 \cong \Res_H V_2$, and
\item the element
$\left\{ \Delta(V_1)/\Delta(V_2) \right\}
\in H^1(\wh(\ZG^-)/\wh(\ZH))$
is in the image of the coboundary
$\delta\colon H^0(\ck) \to H^1(\wh(\ZG^-)/\wh(\ZH))$.
\end{enumerate}
\end{thma}
\begin{remark}
The proof of this result is in Part \partone, but note that
Condition (iii) simplifies for $G$ a cyclic $2$-group
since $H^0(\ck) = 0$ in that case (see \pone, Lemma \ref{vanB}).
Theorem A  should be compared with \cite[Cor.1]{cs1},
where more explicit conditions are given for ``first-time" similarities
of this kind under the assumption
that $q$ is odd, or a $2$-power, or $4q$ is a ``tempered" number.
See also Theorem \ref{Cprime} for a more general result concerning
similarities without $\rplus$ summands.
The case $\dim V_1 = \dim V_2 = 4$ gives a reduction to
number theory for the existence of $5$-dimensional
similarities (see \pone, Remark \ref{Arem}).
\end{remark}

Our next result uses a more elaborate setting for the invariant.
Let
\[
\Phi = \begin{pmatrix} \ZH  & \to &\Zadictwo{H}\cr
\downarrow &&\downarrow\cr
\ZG & \to & \Zadictwo{G}
\end{pmatrix}
\]
and consider the exact sequence
\eqncount
\begin{equation}0 \to K_1(\rel{\ZH}{\ZG}) \to K_1(\rel{\Zadictwo{H}}{\Zadictwo{G}})
\to K_1(\Phi) \to \widetilde K_0(\rel{\ZH}{\ZG}) \to 0\ .
\end{equation}
Again we can define the Whitehead group versions by dividing
out trivial units $\{\pm G\}$, and get a double coboundary
\[\delta^2\colon H^1(\widetilde K_0(\rel{\ZH}{\ZG^{-}}))
\to H^1( \wh(\rel{\ZH}{\ZG^{-}}))\ .\]
There is a natural map
$H^1(\wh(\ZG^-)/\wh(\ZH))\to H^1( \wh(\rel{\ZH}{\ZG^{-}}))$.
We will use the same notation 
$$\{ \Delta(V_1)/\Delta(V_2) \} \in H^1( \wh(\rel{\ZH}{\ZG^{-}}))$$
 to denote the image of our Reidemeister torsion invariant.
The non-linear similarities handled by the next result have
isotropy of index $\leq 2$.
\begin{thmb}
Let $V_1=t^{a_1} + \dots + t^{a_k} $
and $V_2 = t^{b_1} + \dots + t^{b_k}$ be free $G${--}representations.
There exists a topological similarity
 $V_1\oplus \rminus\oplus \rplus \sim_t
V_2\oplus \rminus\oplus \rplus$ if and only if
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $\prod a_i \equiv \prod b_i \Mod{4q}$,
\item $\Res_H V_1 \cong \Res_H V_2$, and
\item the element
$\left\{ \Delta(V_1)/\Delta(V_2) \right\}$
is in the image of the double coboundary
\[\delta^2\colon H^1(\widetilde K_0(\rel{\ZH}{\ZG^{-}}))
\to H^1(\wh(\rel{\ZH}{\ZG^{-}}))\ .\]
\end{enumerate}
\end{thmb}
This result can be applied to $6$-dimensional similarities.
\begin{corollary}\label{sixdim}
Let $G=\Cy{4q}$, with $q$ odd, and suppose that the
fields $\bq(\zeta_{d})$ have odd class number for
all $d \mid 4q$.
Then $G$ has no $6$-dimensional non-linear similarities.
\end{corollary}
\begin{remark} For example, the class number condition is satisfied for $q\leq 11$, but not for $q=29$.
The proof of the Corollary \ref{sixdim} is given in Section \ref{sixdiml} assuming
Theorem B, which is proved in Part \partone.
This result corrects \cite[Thm.1(i)]{cssw1}, and shows that the
computations of $\rtop{G}$ given in \cite[Thm. 2]{cssw1}
are incorrect.
\end{remark}
Our final example of the computation of  bounded transfers is suitable
for determining stable non{--}linear similarities inductively,
with only a minor assumption on the isotropy subgroups.
To state the algebraic conditions, we must again generalize the
indeterminacy for the Reidemeister torsion invariant
to include bounded $K${--}groups (see Section \ref{excision}).
In this setting $\widetilde K_0(\rel{\ZH}{\ZG}) = \widetilde K_0(\ccat{\rminus,G}{\bz})$
and $\wh(\rel{\ZH}{\ZG})=\wh(\ccat{\rminus,G}{\bz})$.
We  consider the analogous double coboundary
\[\delta^2\colon
H^1(\widetilde K_0(\ccat{W\times \rminus,G}{\bz})) \to
H^1(\wh(\ccat{W\times \rminus,G}{\bz}))\]
and note that there is a map
$\wh(\ccat{\rminus,G}{\bz})\to
\wh(\ccat{W\times \rminus,G}{\bz})$
induced by the inclusion on the control spaces. 
We will again use the same notation
$$\{\Delta(V_1)/\Delta(V_2)\}\in H^1(\wh(\ccat{W\times \rminus,G}{\bz}))$$
 for the image of the Reidemeister torsion invariant in this new domain.

\begin{thmc}
Let $V_1=t^{a_1} + \dots + t^{a_k} $
and $V_2 = t^{b_1} + \dots + t^{b_k}$ be
 free $G${--}representations.
Let $W$ be a  complex $G${--}representation
with no $\rplus$ summands.
Then there exists a topological similarity
 $V_1\oplus W \oplus \rminus\oplus \rplus \sim_t
V_2\oplus W \oplus \rminus\oplus \rplus$ if and only if
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $S(V_1)$ is $s$-normally cobordant
 to $S(V_2)$,
\item $\Res_H (V_1\oplus W) \oplus \rplus
 \sim_t \Res_H (V_2\oplus W)\oplus \rplus$, and
\item the element
$\left\{ \Delta(V_1)/\Delta(V_2) \right\}$
is in the image of the double coboundary
\[\delta^2\colon
H^1(\widetilde K_0(\ccat{\Wmax\times \rminus,G}{\bz})) \to
H^1(\wh(\ccat{\Wmax\times \rminus,G}{\bz}))\ , \]
where $0\subseteq \Wmax \subseteq W$ is a
complex subrepresentation of real dimension $\leq 2$, with  maximal
isotropy group among the isotropy groups
of $W$ with 2-power index.
\end{enumerate}
\end{thmc}
\begin{remark}\label{unstablenormal}
 The existence of a
 similarity $V_1\oplus W\sim_t V_2\oplus W$ implies that
 $S(V_1)$ and $S(V_2)$ are  $s$-normally cobordant.
In particular, $S(V_1)$ must be freely $G$-normally cobordant to
$S(V_2)$ and this (unstable) normal invariant condition is enough
to give us a surgery problem. Crossing with $W$ defines the bounded transfer map
$$\trf{W}\colon L^h_{2k}(\ZG) \to L^h_{2k + \dim W}(\bcat{W})$$
introduced in \cite{hp1}. The vanishing of the surgery obstruction
is equivalent to the existence of a similarity (see \pone, Theorem \ref{maintest}). The computation of
the bounded transfer in  $L${-}theory leads  to condition (iii), and an
expression of the obstruction purely in terms of bounded $K$-theory. To carry
out this computation we may need  to stabilize in the free part,
and this uses the $s$-normal cobordism condition.
\end{remark}
\begin{remark} Note that $\Wmax=0$ in condition 
(iii) if $W$ has no isotropy
subgroups of $2$-power index.
Theorem C suffices to handle stable topological similarities,
but leaves out cases where $W$ has an
odd number of $\rminus$ summands (handled in  Theorem
\ref{Cprime} and the results of Section
\ref{rminustwo}). 
Simpler conditions can be given when $G = \Cy{2^r}$
 (see \pone, Section \ref{twogroups}).

The double coboundary in (iii) can also be expressed in more
``classical" terms by using the short exact sequence
\eqncount
\begin{equation} 0 \to \wh(\bCatrminus) \to 
\wh(\bcatrminus{\Wmax}) \to
K_1(\brelCatrminus{\Wmax}) \to 0       \label{ses}
\end{equation}
derived in Corollary \ref{kersimple}.
We have $K_1(\brelCatrminus{\Wmax}) = \Kminus{1}(\ZK)$, where
$K$ is the isotropy group of $\Wmax$, and
$\wh(\bCatrminus)=\wh(\rel{\ZH}{\ZG})$.
The indeterminacy in Theorem C
is then generated by the double coboundary
\[\delta^2\colon H^1(\widetilde K_0(\rel{\ZH}{\ZG^{-}}))
\to H^1(\wh(\rel{\ZH}{\ZG^{-}}))\]
used in Theorem B and the coboundary
\[\delta\colon \Coh{0}{\Kminus{1}(\ZK)} \to \Coh{1}{\wh(\rel{\ZH}{\ZG^{-}})}\]
from the Tate cohomology sequence of (\ref{ses}).

\end{remark}

Finally, we   apply these results to $\rtop{G}$. Since its rank is
known  (see  \cite{cs2} or Section \ref{ratcomp}), it remains to determine its torsion subgroup.
In Section \ref{splitting}, we will define a filtration
\eqncount
\begin{equation}R_t(G) \subseteq R_n(G) \subseteq R_h(G) \subseteq R(G)
\label{tnh}
\end{equation}
on the real representation ring
$R(G)$, inducing a filtration on
 $\rtop{G} =  R(G)/R_t(G)$.
Here the subgroup
$$R_t(G) = \{  (V_1 - V_2) \mid V_1\oplus W
\sim_t V_2\oplus W \text{ for\ some\ }W\}$$
is generated by stable topological similarity.
Note that $R(G)$ has the following nice basis:
$\{ t^i, \delta, \epsilon\,\mid\, 1\leqq i \leqq 2q-1\}$, 
where $\delta=[\rminus]$ and $\epsilon=[\rplus]$
(although  we do not have $i\equiv 1 \Mod{4}$ for all  the weights).


Let $\rfree{G} = \{ t^a \mid (a, 4q) = 1\} \subset R(G)$
 be the subgroup
generated by the free representations.
To complete the definition, we let $\rfree{\Cy 2} = \{\rminus\}$
and $\rfree{e} = \{\rplus\}$.
Then inflation and fixed sets of representations defines an isomorphism
\[R(G) = \bigoplus\limits_{K \subseteq G} \rfree{G/K}\]
 and this
direct sum splitting can be intersected with $R_t(G)$
 to define 
$\Rfree_t(G)$.
We let $\rtopfree{G} = \rfree{G}/\Rfree_t(G)$.
Since  inflation and fixed sets preserve topological similarities, we obtain an induced splitting
\[\rtopfree{G} = \bigoplus\limits_{K \subseteq G} \rtopfree{G/K}\ .\]
By induction on the order of $G$, we see that it suffices to
study the  summand $\rtopfree{G}$.

Let $\rtildefree{G} = \ker(\Res\colon\rfree{G} \to \rfree{\Godd})$,
 and then project into $\rtop{G}$ to define
\[\rtildetopfree{G} = \rtildefree{G}/\Rfree_t(G)\ .\]
In Section \ref{ratcomp} we prove that $\rtildetopfree{G}$ 
is \emph{precisely}
the torsion subgroup of $\rtopfree{G}$.
Here is a specific computation (correcting \cite[Thm. 2]{cssw1}).
\begin{thmd}
Let $G=\Cy{4q}$, with $q>1$ odd, and  suppose that the
fields $\bq(\zeta_{d})$ have odd class number for
all $d \mid 4q$.
Then $\rtildetopfree{G} = \cy{4}$ generated by
$(t-t^{1+2q})$.
\end{thmd}
For any cyclic group $G$, we use normal cobordism and homotopy equivalence to define a filtration
$$\Rfree_t(G) \subseteq	 \Rfree_n(G) \subseteq\Rfree_h(G) \subseteq\rfree{G} $$
leading by direct sum to the filtration of $R(G)$ mentioned above.
Both $\rtildefree{G}/\Rtildefree_h(G)$ and
$\Rtildefree_h(G)/\Rtildefree_n(G)$
are torsion groups  which can be explicitly
determined by congruences in the weights
 (see  Section \ref{normalinvariant} and \cite[Thm.1.2]{yo1}).
The subquotient $\Rtildefree_n(G)/\Rfree_t(G)$ always has exponent 
two (see Section \ref{sD}).


We conclude this list of sample results with a calculation of
$\rtop{G}$ for cyclic $2$-groups (see \pone\ for the proof).

\begin{thme}
Let $G=\Cy{2^r}$, with $r\geqq 4$.
Then
\[\rtildetopfree{G} = \big\langle \alpha_1, \alpha_2, \dots, 
\alpha_{r-2}, \beta_1, \beta_2, \dots,  \beta_{r-3} 
\big\rangle\]
subject to the relations $2^s \alpha_s = 0$ for $1\leqq s \leqq r-2$,
and $2^{s-1}(\alpha_s + \beta_s) =0$ for $2\leqq s \leqq r-3$, 
together with $2 (\alpha_1+\beta_1) =0$.
\end{thme}
\noindent
The generators for $r\geqq 4$ are given by the elements
\[\begin{matrix} \alpha_s = t - t^{5^{2^{r-s-2}}} & \text{and}&
\beta_s =t^5 - t^{5^{2^{r-s-2}+1}}.\end{matrix} \]
We remark that  $\rtildetopfree{\Cy{8}} = \cy 4$ generated by
$t-t^5$. 
In \pone, Theorem \ref{unstablefour}
we use this information to give a complete
topological classification of linear representations for cyclic
$2$-groups. 

\begin{acknowledgement} The authors would like to express their
appreciation to the referee for many constructive comments and suggestions. In
particular, the referee pointed out an improvement
to the statement of Theorem E. If we let $\psi_i = \alpha_{i+1}
+\beta_{i+1}$ for $1\leq i\leq r-4$, then $\rtildetopfree{\Cy{2^r}}$
is a direct sum of cyclic groups generated by $\alpha_1,\dots,\alpha_{r-2},
\beta_1$ and $\psi_1,\dots, \psi_{r-4}$
where the order of the cyclic group generated by a basis element
with subscript $i$ is $2^i$. This basis displays the group structure
explicitly.
\end{acknowledgement}


\section{ A splitting of $R_{\Top}(G)$}\label{splitting}
In this section we point out an elementary
splitting of $\rtop{G}$, and some useful filtrations.
For $G$ any finite group, we denote by $R(G)$ the \emph{real}
representation ring of $G$.
Elements in $R(G)$ can be given as formal differences
$(V_1 - V_2)$ of $G${--}representations, and
$(V_1 - V_2) \sim 0$ if and only if there exists a representation
$W$ such that $V_1\oplus W \cong V_2 \oplus W$.

Notice that for $K$ any normal subgroup of $G$,
 taking fixed sets gives a retraction of
the inflation map
\[\Inf_K\colon R(G/K) \to R(G)\]
defined by pulling back a $G/K$ representation
using the  composition with the quotient map $G \to G/K$.
More explicitly,
\[\Fix_K\colon R(G) \to R(G/K)\]
is defined by $\Fix_K (V_1 - V_2) = (V_1^K -V_2^K)$ for
each normal subgroup $K \triangleleft\, G$.
Then $\Fix_K\circ \Inf_K= id\colon R(G/K) \to R(G/K)$.
\begin{definition} A $G${--}representation $V$ is
\emph{free} if $V^K = \{ 0\}$ for all non{--}trivial normal
 subgroups $1\neq K \, \triangleleft\, G$.
\end{definition}

This is the same as the usual definition
(no non{--}identity element of $G$
fixes any non{--}zero vector) for cyclic groups.
We let
\eqncount
\begin{equation}\Rfree(G) =
\bigcap \{ \ker\Fix_K \mid 1\neq K\triangleleft\, G\}
\end{equation}
denote the subgroup of $R(G)$ such that
there is a representative
of the stable equivalence class $(V_1 - V_2)$ with
$V_1, V_2$ free representations.
\begin{proposition}There is a direct sum splitting
\[R(G) = \bigoplus\limits_{K\triangleleft\, G} \Rfree (G/K)\]
indexed by the normal subgroups $K$ in $G$.
\end{proposition}
\begin{proof}
Let $V(K)$ denote the $G${--}invariant subspace given by
the sum of all
the irreducible sub{--}representations of $V$ with kernel exactly
$K$. This is a free $G/K$ representation. 
The decomposition above is given
by mapping $(V_1-V_2)$ to the elements $(V_1(K) - V_2(K))$.
\end{proof}

Inside $R(G)$ we have the subgroup of stably topologically
similar representations
\eqncount
\begin{equation}
R_t(G) = \{  (V_1 - V_2) \mid V_1\oplus W
\sim_t V_2\oplus W \text{ for\ some\ }W\}
\end{equation}
and the quotient group is $\rtop{G}$ by definition.
We define $\Rfree_t(G) =\Rfree(G)\cap R_t(G)$.
Since $R_t(G)$ is preserved by inflation and
taking fixed sets, we obtain
\begin{corollary} There is a direct sum decomposition
\[\rtop{G} = \bigoplus\limits_{K\triangleleft\, G} \Rtopfree(G/K)\]
where the summands are the quotients
$\Rfree (G/K)\big / \Rfree_t(G/K)$.
\end{corollary}
We will also need a certain filtration of $R(G)$.
First we define
\eqncount
\begin{equation}\Rfree_h(G) = \{  (V_1 - V_2) \mid S(V_1)
\simeq_G S(V_2) \text{ \ for\ $V_1$\ and $V_2$\ free}\}
\end{equation}
where $\simeq_G$ denotes  $G${--}homotopy equivalence.
This is a subgroup of $\Rfree(G)$, in fact a sub{--}Mackey functor
since it has induction and restriction for subgroups of $G$.
We define 
\eqncount
\begin{equation}
R_h(G) = \bigoplus\limits_{K\triangleleft\, G} \Rfree_h(G/K)
\end{equation}
If there exists a $G${--}homotopy equivalence
 $f\colon S(V_1) \to S(V_2) $ such that
\[S(V_1 \oplus U)=S(V_1){\ast} S( U)
\RA{\ f{\ast} 1\ } S(V_2){\ast} S( U) = S(V_2\oplus U)
\]
is freely $G${--}normally cobordant to the identity for all free
$G${--}representations $U$, then we say that
$S(V_1)$ and $S(V_2)$ are $s$-normally cobordant, and we
write $S(V_1) \bumpeq_G S(V_2)$. Define
\eqncount
\begin{equation}
\Rfree_n(G) = \{ \alpha\in \Rfree_h(G)  \mid \exists V_1, V_2 \text{\ with\ }\alpha=(V_1-V_2)
 \text{ \ and\ } S(V_1)
\bumpeq_G S(V_2)\}
\end{equation}
and note that $\Rfree_n(G)$ is also a subgroup of $\Rfree(G)$.
Indeed, if $(V_1 - V_2)$ and $(V'_1 - V'_2)$
are in $\Rfree_n(G)$, there exist $G$-homotopy equivalences
 $f\colon S(V_1) \to S(V_2) $
and $f'\colon S(V'_1) \to S(V'_2) $
with $f{\ast} 1$ and $f'{\ast} 1$
normally cobordant to the identity
under any stabilization.
But $f{\ast} f' \simeq_G (1{\ast} f')\circ (f{\ast} 1)$, so we
just glue together the normal cobordisms
for $f{\ast} 1$ (after stabilizing by $U=V'_1$) and for
$1{\ast} f'$ (after stabilizing by $U=V_2$) along
the common boundary $id\colon S(V_2 \oplus V'_1)$.
 As above, we define
 \eqncount
 \begin{equation}
R_n(G) = \bigoplus\limits_{K\triangleleft\, G} \Rfree_n(G/K)
\end{equation}
  Since $\Rfree_t(G) \subseteq \Rfree_n(G)$, we have defined a filtration
\eqncount
\begin{equation}
R_t(G) \subseteq R_n(G) \subseteq R_h(G) \subseteq R(G)\label{filt}
\end{equation}
of $R(G)$, natural with respect to  
restriction of representations. All the terms except possibly
$R_n(G)$ are also natural with respect to induction
of representations.

\begin{remark} It follows from the proof of \cite[3.1]{yo1} that 
$S(V_1\oplus U_0)$ is $s$-normally cobordant to
$S(V_2\oplus U_0)$, for some free $G$-representation $U_0$, 
if and only if $S(V_1)$ is $s$-normally cobordant to
$S(V_2)$. It follows that we could have used the latter condition
to define $\Rfree_n(G)$.
\end{remark}

\section{ A rational computation}
\label{ratcomp}
In this section we use \pone,  Theorem \ref{maintest} and 
the splitting of the last
section to describe the torsion subgroup of $\rtopfree{G}$.
We also give a new proof of Cappell and Shaneson's result
computing $\rtop{G}\otimes \bq$  for all finite groups $G$.

First we consider cyclic groups.
 Let $G=\Cy{2^rq}$ be a cyclic group, where $q\geqq 1$ is odd.
The Odd Order Theorem \cite{hsp1},
\cite{mr1} gives $R(G) = \rtop{G}$ if $r \leqq 1$, and we
recall the definition
\[\Rtildetopfree(G) = \Rtildefree(G)/\Rfree_t(G)\]
where
$\Rtildefree(G) = \ker\{ \Res\colon \rfree{G} \to
\rfree{\Godd}\}$.
Here $\rtildefree{\Cy{2q}} = \rtildefree{\Cy{q}} = 0$,
for $q\geqq 1$ odd.

\begin{theorem}\label{threeone}
For $G$ cyclic,
the kernel and cokernel of
\[\Res\colon \rtopfree{G} \to \rtopfree{\Godd}=\rfree{\Godd}\]
are $2${--}primary torsion groups.
\end{theorem}
\begin{corollary}\label{cyclicranks}
Let $G=\Cy{2^rq}$, $q$ odd, be a finite cyclic group.
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item The torsion subgroup of $\rtopfree{G}$
is $\rtildetopfree{G}$.
\item The rank of $\rtopfree{G}$ is $\varphi(q)/2$
(resp. $1$) if $q>1$ (resp.  $q=1$).
\item We have the formula
\[\text{rank} (\rtop{G}) =
\begin{cases} (r+1)\bigl\{
 \sum_{1 \not=d \mid q} \varphi(d)/2 + 1\bigr\},& \text{for $q>1$}\cr
(r+1)& \text{for $q=1$}.
\end{cases}
\]
where $\varphi(d)$ is the Euler function.
\end{enumerate}
\end{corollary}
\begin{proof}
The first part follows directly from
Theorem \ref{threeone}. For parts (ii) and (iii)
 count the free representations of $\Godd$.
\end{proof}
Now let $\Kodd$ be the composite of all subfields of $\br$
of the form $\bq (\zeta+\zeta^{-1})$ where
$\zeta \in \bc$ is an odd root of unity.
\begin{lemma}\label{threetwo}
For $G$ cyclic,
the composition of the natural map $R_\Kodd(G) \to R(G)$ and
$\Res\colon R(G) \to R(\Godd)$ induces
a $p${--}local isomorphism $\Rfree_\Kodd(G) \to \Rfree(\Godd)$,
for any odd prime $p$.
\end{lemma}
\begin{proof}
According to a result of Brauer \cite[Thm.24]{se1}
 $R_\Kodd(\Godd) = R(\Godd)$, and any representation
of $G$ can be realized over the field $\bq (\zeta_{|G|})$.
In addition,  the restriction map
 \[\Res\colon R_\Kodd(G) \to R_\Kodd(\Godd)\]
is a $p${--}local surjection
(since $\Res_{\Godd}\circ \Ind_{\Godd}$ is just multiplication by $2^r$).
But the rank of $R_\Kodd(G)$  given in \cite[12.4]{se1} equals the rank of
$R(\Godd)$, so we are done.
\end{proof}
\begin{corollary}\label{threethree}
For $G$ cyclic and any odd prime $p$,
the natural map  $R_\Kodd(G) \to \rtop{G}$ induces
a $p${--}local isomorphism.
\end{corollary}
\begin{proof}
This follows from the Lemma \ref{threetwo} and
Theorem \ref{threeone}.
\end{proof}
In \cite{cs2}, Cappell and Shaneson obtained
the following result by a different argument. It computes the rank of
$\rtop{G}\otimes \bq$ for any finite group $G$.
\begin{theorem}
Let $\Kodd$ be the composite of all subfields of $\bc$
of the form $\bq (\zeta+\zeta^{-1})$ where
$\zeta \in \bc$ is an odd root of unity.
Then for any finite group $G$,  the natural map $R_\Kodd(G) \to R(G)$ induces
an isomorphism $R_\Kodd(G)\otimes \bz _{(p)}
\cong \rtop{G}\otimes \bz _{(p)}$ for any
odd prime $p$.
\end{theorem}

\begin{proof}
The result holds for cyclic groups $G$ by Corollary \ref{threethree},
and we apply induction theory to handle general finite groups.

First we observe that the Mackey functors
$R(G)\otimes \bz _{(p)} $ and $R_\Kodd(G)\otimes \bz _{(p)}$
are  generated by induction from
$p${--}elementary subgroups (see
\cite[Thm.27]{se1}  and note that
$\Gamma_\Kodd = \{\pm 1\}$).
Therefore,  by \cite[11.2]{o1}
they are $p${--}elementary computable in the sense
of Dress induction theory \cite{dre1}.
We may therefore assume that $G$ is $p${--}elementary.


Now if $G$ is $p${--}elementary, it is a product of a
$p${--}group and a cyclic group prime to $p$.
Any irreducible complex representation of $G$ is then
induced from a linear representation on a subgroup
 \cite[Thm.16]{se1}.

It follows that $\Rtop(G)$ is generated by  generalized induction
(i.e. inflation followed by induction) from cyclic subquotients.
Consider the following commutative diagram
\[
\xymatrix{R_\Kodd(G) \ar[r]\ar@<1ex>@{  >->}[d]^{\Res}&
R(G)\ar[r]\ar@<1ex>[d]&
\rtop{G}\ar@<1ex>[d]\\
{\bigoplus\limits_{C}}
 R_\Kodd(C) \ar@<1ex>[u]^{\Ind}\ar@<1ex>[r]&
{\bigoplus\limits_{C}}
R(C)\ar@<1ex>[u] \ar@<1ex>[r]&
{\bigoplus\limits_{C}}
 \rtop{C}\ar@<1ex>@{>>}[u]
}
\]
It follows from
 Corollary \ref{threethree}  that the top composite
 map $R_\Kodd(G)\otimes \bz _{(p)} \to \rtop{G})\otimes \bz _{(p)}$
is surjective.

The sum of the (ordinary) restriction maps
to cyclic subgroups induces a rational injection on $R_\Kodd(G)$
(see \cite[2.5,2.10]{swe1}).
Since $R_\Kodd(G)$ is torsion{--}free, it follows again from
Corollary \ref{threethree} that the map
 $R_\Kodd(G)\otimes \bz _{(p)} \to \rtop{G})\otimes \bz _{(p)}$
is injective.
\end{proof}


\begin{proof}[The proof of Theorem \ref{threeone}]
Since an $\Kodd${--}representation is free if and only if it
is the sum of Galois conjugates of free $G${--}representations,
we can decompose $R_\Kodd(G)$ as in Section \ref{splitting}, and
conclude that $\Rfree_\Kodd(\Godd) = \Rfree(\Godd)$.
 It remains to show that $\Rtildetopfree(G) $ is
a torsion group with 2{--}primary exponent.
For this we use the filtration of \S \ref{splitting}.

For $\Rtildefree(G)/\Rtildefree_h(G) $ this is easy
since the $k${--}invariant gives a homomorphism (via joins
of free  $G${--}spheres) to $\Units{(\cy{2^r})}$ and this is
a 2{--}group.
The next quotient is also 2{--}primary torsion, by results
of \cite{yo1}: a sufficiently large 2{--}power join of a $G${--}homotopy
equivalence between two  free $G${--}spheres,
which are linearly equivalent over $\Godd$, becomes $s${--}normally
cobordant to the identity.
The point is that the normal invariant is detected by a
finite number of 2{--}power
congruences
conditions among the Hirzebruch $L${--}classes of the tangent bundles
of the lens spaces, and this can be satisfied after sufficiently
many joins.

Finally, the last quotient $\Rtildefree_n(G)/ \Rtildefree_t(G) $
is shown to be $2${--}primary torsion in the next proposition.
\end{proof}
\begin{proposition}
Let $G=\Cy{2^rq}$, $q$ odd,  and assume
\[\sigma \in \ker(\Res \colon  L^h_{2k}(\ZG) \to L^h_{2k}(\bz \Godd)).\]
Then there exists a complex representation
$W$ with $W^G =0$ such  that $\trf{W}(2^r\sigma)=0$.
\end{proposition}
\begin{proof}
We will take $W = \rminus \oplus \rminus \oplus W_0$,
where $W_0$ is the sum of all the irreducible $2${--}dimensional
representations of $G$ with isotropy of $2${--}power index.
Note that the $\rminus$-transfer is just the compact
$I_{-}$ transfer of one-sided codimension 1 surgery followed by
adding rays to infinity, so whenever the $I_{-}$transfer is $0$, the
$\rminus$-transfer will have to be $0$.
This was discussed in more detail
in \pone, Section \ref{rmin}.

\smallskip

\noindent\textbf {Step 1:}
If $G$ has odd order, there is nothing to prove.
Otherwise, let $H\subset G$ be of index 2.
If $\Res_H(\sigma)=0$, then
$\trfrminus(\sigma) \in
\Image(L_1^h(\ZG^-) \to L_1^h(\rel{\ZH}{\ZG^-}))$.
But  $L_1^h(\ZG^-)$ has exponent 2
\cite[12.3]{ht1}, so $\trfrminus(2\sigma) =0$.
Then take $W=\rminus\oplus\rminus$.
Note that this case applies to $G=\Cy{2q}$,
so we can always get started.

\smallskip

\noindent\textbf{Step 2:}
We may assume that $r\geqq 2$.
If $\Res_H(\sigma)\ne 0$ note that
\[\Res_H(2\sigma -\Ind_H\Res_H(\sigma))
= 2\Res_H(\sigma) - 2  \Res_H(\sigma) = 0.\]
By induction  $\Res_H W_0$
works for $\Res_H(\sigma)$: say
\[\trf{\Res_H W_0}(2^{r-1}\Res_H(\sigma))=0\]
 and $W_0^H=0$.
 Let $\dim W_0=m$ and consider the commutative diagram
\[
\xymatrix{ L_{2k}(\ZG) \ar[r]^
{\Res} \ar[d]_{\trf{W_0}}&L_{2k}(\ZH)\ar[r]^{\Ind}
\ar[d]_{\trf{\Res_H W_0}}&L_{2k}(\ZG)\ar[d]_{\trf{W_0}}\\
L_{2k+m}(\mathcal C_{W_0,G}(\bz))\ar[r]^{\Res}& L_{2k+m}(\mathcal
C_{\Res W_0,H}(\bz))\ar[r]^{\Ind}&L_{2k+m}(\mathcal C_{W_0,G}(\bz)).
}\]
 From this we get
\[2^{r-1}\cdot\trf{W_0}(\Ind_H\Res_H(\sigma)) = 0\ .\]
The first step implies that $\trf{\Rminus{2}}(2\sigma_1) = 0$,
where $\sigma_1 = 2\sigma - \Ind_H\Res_H(\sigma)$.
Let $W= \rminus\oplus\rminus\oplus W_0$, so that
we have $W$ complex and $W^G=0$.
Note that $\trf{W}= \trf{\Rminus{2}}\circ\trf{W_0}=
\trf{W_0}\circ\trf{\Rminus{2}}$.
But
\[2^r\cdot \trf{W}(\sigma)
= 2^{r-1}\trf{W}(2\sigma - \Ind_H\Res_H(\sigma)) +
2^{r-1}\trf{W}(\Ind_H\Res_H(\sigma))\]
and both terms vanish
(because $r\geqq 2$ and by the property of $W_0$
respectively).
\end{proof}
A similar argument to that in Step 2 above gives:
\begin{proposition}\label{trftorsion}
If $\Res_H(\trf{W}(x))=0$ for $x \in L_0^h(\ZG)$,
then $4\cdot\trf{W\times\rminus}(x) =0$.
\end{proposition}
\begin{proof}
Since
\[4\cdot \trf{W\times\rminus}(\sigma)
= 2\cdot\trf{W}\trfrminus(2\sigma - \Ind_H\Res_H(\sigma)) +
2\cdot\trfrminus\trf{W}(\Ind_H\Res_H(\sigma))\]
we conclude as above that both terms vanish.
\end{proof}

\section{ Excision in bounded surgery theory}
\label{excision}

A small additive category with involution $\ca$ is a small additive
category together with a contravariant endofunctor $*$ such that
$*^2= 1_\ca$.
 Ranicki defines algebraic $L${--}theory $L^h_*(\ca)$ for such
categories and corresponding spectra
$\BL^h(\ca)$ with $L_*^h(\ca)=\pi_*(\BL^h(\ca))$ \cite{ra5}.
The obstruction
groups for bounded surgery are obtained this way for appropriately
chosen additive categories.
We shall also need a simple version of such
groups.
 For this, the additive category must come equipped with a
system of stable isomorphisms and a subgroup $s\subset K_1(\ca)$,
such that any composition resulting in an automorphism defines an
element in $s$.
 The point here is that whenever two objects are stably
isomorphic, there is a canonically chosen stable isomorphism,
canonical up to automorphisms defining elements of $s$.
 In this situation
Ranicki refines the definition of $\BL^h(\ca)$ to give the simple
 $L${--}theory
spectrum $\BL^s(\ca)$, by requiring appropriate isomorphisms to give
elements of $K_1(\ca)$ belonging to the subgroup $s$.
More generally, we also get the
$\BL^ {a}(\ca)$-spectra for any involution invariant subgroup
$a$ with $s \subset a \subset K_1(\ca)$, coinciding
with $\BL^ h(\ca)$ when $a = K_1(\ca)$.
\begin{example} Let $\ca$ be the category of free $\ZG$-modules
with a
$G$-invariant $\bz$-basis, and $\ZG$-module morphisms.
Two objects are stably isomorphic if and only if they have the same rank.
The preferred isomorphisms are chosen to be the ones sending a
$\bz$-basis to a
$\bz$-basis, so automorphisms define elements of $\{\pm G\}\subset
K_1(\ZG)$.
In this situation one obtains Wall's  $L^s$-groups.
\end{example}
The theory of projective $L${--}groups fits into the scheme
as follows: one defines  $\BL^p(\ca) = \BL^h(\ca^{\wedge})$, where
$\ca^\wedge$ is the idempotent completion of $\ca$.
The objects of $\ca^\wedge$ are pairs $(A,p)$ with $A$ an object of
$\ca$ and $p^2=p$.
The morphisms $\phi\colon(A,p) \to (B,q)$ are the  $\ca$-morphisms
$\phi\colon A\to B$ with $q\phi p = \phi$.
Again it is possible to ``partially'' complete $\ca$.
If $K_0(\ca)\subset \ck\subset K_0(\ca^\wedge)$ is an
involution invariant subgroup , we define $\ca^{\wedge k}$ to be the full
subcategory of $\ca^\wedge$ with objects defining elements of
$\ck\subset K_0(\ca^\wedge)$.
This way we may define $\BL^k(\ca) =
\BL(\ca^{\wedge k})$. Similarly to the above, for $\ck=K_0(\ca)\subset
K_0(\ca^\wedge)$, $L_*^k(\ca)$ is naturally isomorphic to $L^h_*(\ca)$.
The quotient $\widetilde K_0(\ca) = K_0(\ca^\wedge)/K_0(\ca)$
is called the reduced projective class group of $\ca$.
\begin{example} If $\ca$ is the category of free $\ZG$-modules
then
$\ca^\wedge$ is isomorphic to the category of projective
$\ZG$-modules and the $L^p_*(\ca)$ are Novikov's original $L^p$-groups.
\end{example}
Suppose $M$ is a metric space with the finite group $G$
acting by isometries, $R$ a ring with involution. In \cite[3.4]{hp1}
we defined an additive
category ${\mathcal G}_{M,G}\,(R)$
 with involution as follows:
\begin{definition} An object A is a  left $R(G)$-module together
with a map $f\colon A \longrightarrow F(M)$, where $F(M)$ is the set of finite
subsets of $M$, satisfying
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $f$ is
$G$-equivariant.
\item $A_x = \{ a \in A \mid f(a) \subseteq \{ x \} \}$ is a finitely generated
free sub $R$-module
\item As an $R$-module $A = \bigoplus_{x \in M} A_x$
\item $f(a+b) \subseteq f(a) \cup f(b)$
\item For each ball $B \subset M$, $\{x \in B\mid A_x \ne 0\}$ is finite.
\end{enumerate} A morphism $\phi\colon A \longrightarrow B$ is a morphism of
$RG$-modules, satisfying the following condition: there exists $k$ so
that the components $\phi^m_n \colon A_m \longrightarrow B_n$ (which are
$R$-module morphisms) are zero when $d(m,n) > k$. 
 Then ${\mathcal G}_{M,G}\,(R)$
 is an additive category in an obvious way.
 The full subcategory of ${\mathcal G}_{M,G}\,(R)$, for which all the
 object modules are required to be \emph{free} left $RG$-modules, is denoted
 $\ccat{M,G}{R}$.

Given an object $A$, an $R$-module homomorphism $\phi \colon A
\longrightarrow R$ is said to be locally finite if the set of $x \in M$ for
which $\phi(A_x) \neq 0$ is finite. We define $A^* = \text{Hom}_R^{lf}(A,R)$,
the set of locally finite $R$-homomorphisms.
We want to make $*$ a functor from ${\mathcal G}_{M,G}\,(R)$
  to itself to make ${\mathcal G}_{M,G}\,(R)$ a category
with involution.
 We define $f^* \colon A^* \longrightarrow FM$ by $f^*(\phi) =
\{x\mid\phi(A_x) \neq 0\}$ which is finite by assumption. $A^*$ has an
obvious right action of $G$ turning it into a right $RG$ module given by
$\phi g(a) = \phi(ga)$, and $f^*$ is equivariant with respect to the right action on
$M$ given by $xg = g^{-1}x$.
To make $*$ an endofunctor of ${\mathcal G}_{M,G}\,(R)$
we need to replace the right action by a left action.
We do this by the standard way in surgery theory by  letting $g$ act on the
left by letting $g^{-1}$ act on the right.
In the unoriented case, given a
homomorphism $w\colon   G \longrightarrow
\{\pm 1\}$, we let $g$ act on the left of $A^*$ by $w(g)\cdot g^{-1}$ on
the right. The involution $*$ induces a functor on the subcategory  
$\ccat{M,G}{R}$, so that $\ccat{M,G}{R}$
 is also a category with involution.
\end{definition}
\begin{example}\label{standard}Let $\rho_W\colon G \to O(W)$ be an orthogonal
action of $G$ on a finite dimensional real vector space $W$.
We take $M=W$ with the action through $\rho_W$, and orientation
character $\det(\rho_W)$.
This will be called the \emph{standard orientation} on $\bcat{W}$.
\end{example}

\begin{remark} We will need to find a system of stable
isomorphisms for the category $\ccat{M,G}{R}$ to be able to do
simple $L${--}theory.
To do this we choose a point $x$ in each $G$-orbit,
and an $RG_x$-basis for $A_x$, where $G_x$ is the isotropy subgroup
of $x$.
We then extend that by equivariance to an $R$-basis of the
module.
Having a basis allows defining an isomorphism in the usual
fashion.
In each case we need to describe the indeterminacy in the
choices coming from the choice of $R$-basis and points in the orbit.
For our particular choices of $M$ it will be easy to determine the subgroup $s$,
so we will not formulate a general statement.
\end{remark}

We will study the $L${--}theory of the categories $\ccat{M,G}{R}$ using
excision.
Let $N$ be a $G$-invariant metric subspace of $M$.
 Denoting
$\ccat{M,G}{R}$ by $\cu$, let $\ca$ be the full subcategory on modules
$A$ so that $A_x=0$ except for $x$ in some bounded neighborhood of
$N$.
The category $\ca$ is isomorphic to $\ccat{N,G}{R}$ in an obvious
fashion.
The quotient category $\cu/\ca$, which we shall denote by
$\Ccat{M,G}{N}{R}$, has the same objects as $\cu$ but two morphisms
are identified if the difference factors through $\ca$, or
 in other words, if they  differ in a bounded neighborhood of $N$.
This is
a typical example of an additive category $\cu$ which is $\ca$-filtered
in the sense of Karoubi.
 We recall the definition.
\begin{definition} Let $\ca$ be a full subcategory of an additive
category $\cu$.
Denote objects of $\ca$ by the letters A through F and
objects of $\cu$ by the letters U through W.
We say that $\cu$ is
$\ca$-filtered, if every object $U$ has a family of decompositions
$\{U=E_\alpha\oplus U_\alpha\}$, so that
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item For each $U$, the decompositions form a filtered poset under the
partial order that $E_\alpha\oplus U_\alpha \le E_\beta\oplus U_\beta$,
whenever $U_\beta \subseteq U_\alpha$ and $E_\alpha \subseteq
E_\beta$.
\item Every map $A \to U$, factors $A \to E_\alpha \to E_\alpha\oplus
U_\alpha = U$ for some $\alpha$.
\item Every map $U\to A$ factors $U=E_\alpha \oplus U_\alpha
\to E_\alpha
\to A$ for some $\alpha$.
\item For each $U,V$ the filtration on $U\oplus V$ is equivalent to the
sum of filtrations $\{U=E_\alpha \oplus U_\alpha\}$ and $\{ V = F_\beta
\oplus V_\beta\}$ i.e. to $U\oplus V = (E_\alpha\oplus F_\beta) \oplus
(U_\alpha\oplus V_\beta)$
\end{enumerate}
\end{definition}
The main excision results were proved in \cite{pw1},
\cite{cap1}, \cite{cp1}, \cite{ra7}.
 We give a slight generalization of the $L${--}theory results.
Let $\BK $ denote  the Quillen $K${--}theory spectrum, and
$\BK ^{-\infty}$   its non-connective delooping
(with the $\Kminus{i}$-groups as homotopy groups).
\begin{theorem}\label{ua}
Let $\cu$ be an $\ca$-filtered additive category with
involution.
Consider the map
$i\colon  K_0(\ca^\wedge) \to K_0(\cu^\wedge)$
induced by inclusion, and
 let $k= i^{-1}(K_0(\cu))$.
There are fibrations of spectra
\[\BK (\ca^{\wedge k})\to \BK (\cu) \to \BK (\cu/\ca)\]
and
\[\BK ^{-\infty}(\ca)\to \BK ^{-\infty}(\cu)\to \BK ^{-\infty}(\cu/\ca)\]
If $\cu$ and $\ca$ admit compatible involutions there is a
fibration of spectra
 \[\BL^k(\ca)\to \BL^h(\cu)\to \BL^h(\cu/\ca)\ .\]
More generally, if
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $a\subset K_i(\ca)$,$b\subset K_i(\cu^\wedge)$, and
$c\subset K_i((\cu/\ca)^\wedge)$, for $i\leqq 1$,
\item  $a= i^{-1}(b)$ and $b\to c$ is onto,
\item  $a$, $b$, and $c$ contain $K_0(\ca)$, $K_0(\cu)$ and
$K_0(\cu/\ca)$ respectively, if $i=0$, and
\item  $a$, $b$ and $c$
contain the indeterminacy subgroup given by the system of stable
isomorphisms in the case $i=1$,
\end{enumerate}
then we have a fibration of spectra
\[\BL^a(\ca)\to \BL^b(\cu) \to \BL^c(\cu/\ca)\]
\end{theorem}
\begin{proof} The $K${--}theory statements are implicitly contained in
 \cite{pw1}.
 A simpler, more modern proof and explicit statements are given in
\cite{cap1}.
The first $L${--}theory statement was proved in \cite{cp1},
and the other $L${--}theory statements follow by the following argument:
we have an exact sequence
\[K_0(\ca^{\wedge k})\to K_0(\cu)\to K_0(\cu/\ca) \to 0,\]
where  the map from $K_0(\cu)\to K_0(\cu/\ca)$ is
onto because the categories have the same objects.
Letting $I$ denote
the image of $K_1(\cu/\ca)$ in $K_0(\ca^{\wedge k})$,
we consider the diagram of short exact sequences:
\[
\xymatrix{ 0\ar[r]& I \ar@{=}[d]\ar[r]& K_0(\ca^{\wedge
k})\ar[d]\ar[r]& K_0(\cu)\ar[r]\ar[d]& K_0(\cu/\ca)\ar[r]\ar@{=}[d]& 0\\
0\ar[r]& I\ar[r]\ar@{=}[d]& a\ar[r]\ar@{=}[d]& b'\ar[d]\ar[r]&
K_0(\cu/\ca)\ar[d]\ar[r]& 0\\ 0\ar[r]& I\ar[r]\ar@{=}[d]& a\ar[d]\ar[r]&
b\ar[r]\ar[d]& c\ar[r]\ar[d]& 0\\ 0\ar[r]& I\ar[r]&
K_0(\ca^\wedge)\ar[r]& K_0(\cu^\wedge)\ar[r]& K_0((\cu/\ca)^\wedge) }
\]
 The vertical arrows are either equalities or inclusions.
We define $b'$ simultaneously as the pullback of $a/I\to b\to c$
and as the pushout of
$0\to K_0(\ca^{\wedge k})/I\to K_0(\cu)\to K_0(\cu/\ca)\to 0$.
We have
\[a/K_0(\ca^{\wedge k}) \cong  b'/K_0(\cu),\]
so using the Ranicki{--}Rothenberg
 fibrations of spectra \cite{ra7}
\[
\xymatrix{
\BL^k(\ca)\ar[r]\ar[d]&
 \BL^a(\ca)\ar[r]\ar[d]&
  {\mathbb H}(a/K_0(\ca^{\wedge k}))\ar[d]\\
\BL^h(\cu)\ar[r]& \BL^{b'}(\cu) \ar[r]& {\mathbb H} (b'/K_0(\cu))
}
\]
we get a fibration
\[\BL^a(\ca)\to \BL^{b'}(\cu)\to \BL^h(\cu/\ca).\]
We now repeat this
argument using the isomorphisms $b/b'\cong c/K_0(\cu/\ca)$
 to obtain the desired fibration of spectra.
 Since $L^h$-groups may be understood as simple
$L$-groups with all of $K_1$ as allowed torsions,
the above  bootstrapping argument
extends to fibrations of the $L$-spectra stated, using the isomorphism
\[K_1(\cu/\ca)/\ker(\bd) \cong \text{image}(\bd)\]
 where
$\bd$  is the boundary map $\bd \colon   K_1(\cu/\ca) \to
K_0(\ca^\wedge)$.
\end{proof}


In Section \ref{rminustwo} we need to use bounded surgery groups with
geometric anti{--}structure generalizing
the definition of \pone, Section \ref{rmin} (see \cite{hp1}).
The new ingredient is
 a counterpart  to the automorphism $\theta\colon H \to H$
at the metric space level.


Let  $\theta_H \colon H\longrightarrow H$ be a group automorphism
so that the data $(H, \theta_H, w, b)$ gives a geometric
anti{--}structure on $RH$.
 Let  $\theta_M \colon M \longrightarrow M$ be an isometry
 with the properties
 $\theta_M(g \cdot m) = \theta_H(g) \cdot \theta_M(m)$,
$\theta_M^2(m) = bm$, and  $\theta^2_H(g) = bgb^{-1}$.

Given an object $A \in {\mathcal G}_{M,H}\,(R)$,  we have
the functor  $*$  from
 ${\mathcal G}_{M,H}\,(R)$ to itself so that ${\mathcal G}_{M,H}\,(R)$
 is a category with involution.
We may then twist the involution $*$ by composing with the
functor sending $(A,f)$ to
$(A^{\theta},f^{\theta})$ where $A^{\theta}$ is the
 same $R$-module, but $g$ acts on the left by multiplication by
$\theta(g)$ and $f^{\theta} =\theta^{-1}_M \cdot f$.
 This defines the bounded anti{--}structure on
 ${\mathcal G}_{M,H}\,(R)$ and on the subcategory
$\ccat{M,H}{R}$ of free $RH$ modules.

\begin{example}\label{bdLNseq}
Bounded geometric anti{--}structures arise geometrically
as above.
The bounded $\rminus$ transfer sits in the long
 exact sequence
 \begin{multline}
 LN_n(\bcatrminus{W},w\phi) \to
L^h_n(\bcat{W},w) \to
L^h_{n+1}(\bcatrminus{W},w\phi)\\
 \to LN_{n-1}(\bcatrminus{W},w\phi)\to
 L^h_{n-1}(\bcat{W},w) \to\dots
\end{multline}
where $w=\det(\rho_W)$ is the standard orientation
(see Example \ref{standard}).
The bounded $LN${--}group
\[LN_n(\bcatrminus{W},w\phi)
\cong L_{n}(\ccat{W,H}{\bz},\alpha,u)\]
where $\theta_W(x) = t\cdot x$ and $\theta_H(h) = tht^{-1}$
for a fixed $t \in G-H$.

Conversely, given a bounded geometric antistructure
$(\theta_H,\theta_M,b,w)$,
we can define $G=\langle H,t\mid t^{-1}ht=\theta_H(h),t^2=b\rangle$
and $t\cdot m = \theta_M(m)$.
Then $\ccat{M,G}{R}$ induces $(\theta_H,\theta_M,b,w)$
as above, showing that all geometric antistructures
arise by twisting and restricting to an index two subgroup.
\end{example}
The $L$-theory of these bounded geometric anti{--}structures
also has a useful vanishing property which we now wish to
formulate.
We first give a basic construction.
\begin{definition}
If $\ca$ is an additive category,
then the \emph{opposite category}  $\ca^{op}$
is the category with the same objects as $\ca$
 but $\hom_{\ca^{op}}(A,B)= \hom_{\ca}(B,A)$.
The product category $\ca\times\ca^{op}$
is an additive category with involution given by
 $\ast\colon (A,B)=(B,A)$ on objects and
$\ast\colon (\alpha,\beta)=(\beta,\alpha)$
on morphisms.
\end{definition}

Clearly $K_i(\ca^{op})=K_i(\ca)$, so we can identify
$K_i(\ca \times \ca^{op}) = K_i(\ca) \times K_i(\ca)$.

\begin{lemma}\label{Rop}
 Let $b \subseteq K_i(\ca)$  for some $i\leqq 1$, and
$q=b\times b\subseteq K_i(\ca\times\ca^{op})$.
 Then  $L_{n}^q(\ca\times\ca^{op})=0$ for all $n$.
\end{lemma}
\begin{proof}
Let ${\mathcal P}(\ca)$ denote the category with the
same objects as $\ca$, but
with morphisms given by $\ca${--}isomorphisms.
Then it suffices to prove that the quadratic category
${\mathcal Q}(\ca \times \ca^{op}) \simeq {\mathcal P}(\ca)$
via the hyperbolic map (see \cite[p.122]{w8}).
This shows that $L^h_*(\ca \times \ca^{op}) =0$ and
other decorations follow trivially from the
Ranicki Rothenberg exact sequences
(note that the Tate cohomology $H^*(q)=0$).
The result for
 lower $L${--}groups follows by replacing
 $\ca$ by $\cc_{\br}(\ca)$.

Suppose $(\nu_1,\nu_2)\colon (A,B)\to (B,A)$ is a non{--}singular
quadratic form representing an element in
 ${\mathcal Q}(\ca \times \ca^{op})$.
 This means that the bilinearization $\nu_1 +\nu_2$
is an isomorphism, and we are allowed to change
$(\nu_1,\nu_2)$ by terms of the
form $(\alpha,\beta)-(\beta,\alpha)$.
We have
\[
(\nu_1,\nu_2) + (\nu_2,0) -(0,\nu_2) = (\nu_1 +\nu_2,0)
\]
and the right hand side is a hyperbolic form.
\end{proof}
We encounter the $\ca\times \ca^{op}$ situation in the
following setting:
\begin{example}
Let $M = M_1\cup M_2$ be a metric space
given as the union  of two sub{--}metric spaces $M_1$
and $M_2$, where we denote $M_1\cap M_2$ by $N$.
Suppose that
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item  $G$ acts by isometries on $M$, such that
each $g \in G$ preserves or switches $M_1$ and
$M_2$ in this decomposition,
\item  $H=\{ g\in G\mid g(M_1)=M_1\}$
is an index two subgroup of $G$,
\item for every $k>0$
 there exists an $l>0$ such that, if $x \in M_1$ (resp. $x \in M_2$)
with $ d(x,N) > l$, then
$d(x,M_2) > k$ (resp. $d(x,M_1) > k$).
\end{enumerate}
The category $\cc_{M,H}(R)$ has a
bounded geometric antistructure $(\alpha,u)$ given by
$(\theta_H,\theta_M,b,w)$ as in Example \ref{bdLNseq}, with
$\theta_M(m) = t\cdot m$ and $\theta_H(h)=tht^{-1}$
 for a fixed $t \in G -H$.
Next, observe that the category
 \[\cc_{M,H}^{>N}(R) = \cc_{M_1,H}^{>N}(R) \times \cc_{M_2,H}^{>N}(R)\]
because of our separation condition (iii).
Moreover, the functor
 $T\colon \cc_{M_1,H}^{>N}(R) \to \cc_{M_2,H}^{>N}(R)^{op}$
defined by $T(A,f) = (A^*, \theta^{-1}_M\circ f^*)$ on objects
and $T(\phi) = \phi^*$ on morphisms is an equivalence of categories.
We are thus in the $\ca\times \ca^{op}$ situation described above and
$L^h_{\ast}(\cc_{M,H}^{>N}(R),\alpha,u)=0$ by Lemma \ref{Rop}.
\end{example}


For any bounded geometric antistructure,
 notice that
the action of $\theta_M$ on $M$ takes $H${--}orbits
to $H${--}orbits since $\theta_M(g\cdot m) = \theta_H(g)\cdot
\theta_M(m)$.
Let $M_{(H,\theta)}$ denote the subset consisting of $H${--}orbits
in $M$ which are \emph{fixed} by the $\theta${--}action.
Then  $M_{(H,\theta)} = \{ m\in M \Vertical \theta_M(m) \in H\cdot m\}$.
Note that $M_{(H,\theta)}$  is a $H${--}invariant subspace of $M$.
\begin{theorem}\label{optrick}
Suppose that $(\ccat{M,H}{R},\alpha,u)$ has a bounded geometric
antistructure $(\alpha,u)$
given by $(\theta_H, \theta_M, b, w)$, such that:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $M = O(K)$, where  $K$ is a finite $H${--}CW complex
and $M$ has the cone of the given $H${--}action on $K$,
\item $\theta_M$ is induced by a simplicial map on $K$,
\item $M_{(H,\theta)} \subseteq O(L):=N$ for some
$H${--}invariant subcomplex $L \subset K$, and
\item for some $i \leqq 1$, $I \subset K_i(\bCat{M}{H}{N}(R))$
 is a subgroup with $H^*(I)=0$.
\end{enumerate}
Then
$L_n^I(\bCat{M}{H}{N}(R),\alpha,u)=0$
for all $n$.
\end{theorem}
\begin{corollary}
Let $G=\Cy{2^rq}$, $q$ odd, be a cyclic group and
 $H \subset G$ the subgroup of index 2.
 Let $W$ be
a $G${--}representation, and
$N=\bigcup \{W^K \mid [G:K]\ \text{ is\ odd}\}$.
Then $L_n^I(\bCat{W}{H}{N}(\bz),\alpha,u)(q)=0$
on the  top component, where $(\alpha, u)$ is
the antistructure given above.
\end{corollary}
\begin{proof}[The proof of Theorem \ref{optrick}]
We extend the given $H${--}action on $M$ to a
simplicial action
of $G=\langle H,t\mid t^{-1}ht=\theta_H(h), t^2=b\rangle$ as described
above.
The proof is by induction on cells,
so suppose that $K$ is obtained
from $L$ by attaching exactly one $G${--}equivariant $k${--}cell
$D^k \times G/G_0$.
Since $M_{(H,\theta)}\subset O(L)=N$, it follows that
$G_0 \subset H$ and we may write $G/G_0 = H/G_0\,\sqcup \, tH/G_0$.
Now we define $M_1=O(L \cup (D^k\times H/G_0))$ and consider
the category $\ca = \Ccat{M_1,H}{O(L)}{R}$.
By construction, we have
\[\Ccat{M,H}{N}{R} = \ca \times  \ca^{op}\]
which has trivial $L${--}theory by Lemma \ref{Rop}.
Since the Tate cohomology of the $K_1$
decoration $I$ vanishes, we get
$L_n^I(\bCat{M}{H}{N}(R),\alpha,u)=0$.
\end{proof}

\section{ Calculations in bounded $K$-theory}
\label{boundedK}
We begin to compute the bounded transfers $\trf{W}$ by considering
the bounded $K${--}theory analogue.
In this section, $G = \Cy{2^rq}$ is cyclic of
order $2^rq$, with $r\geqq 2$ and $q\geqq 1$ odd.
By  \pone, Theorem \ref{twopoweriso} we can restrict our attention to those $W$
where the isotropy subgroups have $2${--}power index.
Let $G_i\subseteq G$ denote the subgroup
of index $[G:G_i]=2^{i}$ for $i=0, 1,\dots, r$.
As above, we reserve the notation $H < G$ for
the subgroup of index 2.

Any real, orthogonal $G${--}representation $W$
can be decomposed uniquely into isotypical
direct summands indexed by the subgroups
$K\subseteq G$, where in each summand $G$
operates  with isotropy group $K$ away from
the origin.
Since we assume that $W$ has isotropy of $2${--}power
index, we can write
\[W = W[0]\oplus W[1]\oplus \dots \oplus W[r]\]
where $W[i]$ is isotypic with isotropy group
$G_i$.
Thus $W[0]=\br^k$ is a trivial $G${--}representation,
and $W[1]$ is a sum of $\rminus$ factors.
We say that $W$ is \emph{complex} if $\dim W[0]$ and 
$\dim W[1]$ are even (in this case, $W$ is the underlying real
representation of a complex representation).
If $W$ is complex, then
 $\Wmax \subseteq W$  denotes a 
complex sub-representation of real $\dim\leq 2$
with maximal proper isotropy subgroup. If $W=W[0]$ then 
$\Wmax=0$.
Then $\Wmax$ is either irreducible
or $\Wmax = \rminus\oplus\rminus$.

We study bounded $K${--}theory by means of equivariant
filtrations of the control space.
The basic sequence is (see \cite{hp1}):
\[ \dots \to K_{i+1}(\brelcat{V}{U}) \RA{\bd_{i+1}} K_i(\bcat{U}) \to
K_i(\bcat{V}) \to K_i(\brelcat{V}{U}) \to \dots\]
valid for $U\subseteq V$ a closed $G${--}invariant subspace.
If $W_1$ is a complex representation with
 $\dim W_1=2$ and   isotropy group $K\neq G$,
let $U =\Uell$ be the union of $[G:K]$ rays from the origin
in $W_1$, which are freely permuted by $G/K$.
Then $W_1\setminus \Uell$ is a disjoint union of open fundamental
domains for the free $G/K${--}action. 
If $W = W_1\oplus W_2$, we call
$W_2 \subset \Uell\times W_2 \subset W$
the \emph{orbit type} filtration of $W$.

Recall that $t$ denotes a generator of $G$, and thus acts as an
isometry on the control spaces $M$ we use in the bounded categories
$\bcat{M}$.
Let $t_{\ast}$ denote the  action
 of $t $ on bounded $K${--}theory  induced by
 its action on the control space.

\begin{lemma}\label{orb}
Let $W = W_1\oplus W_2$, where $W_1$ is
a complex $2${--}dimensional
 sub-representation of $W$ with
minimal isotropy subgroup $ K \neq G$.
Then
 \begin{align*}
&K_{i+1}(\brelcat{W}{\Uell\times W_2})
\cong K_{i-1-k}(\ZK)\\
&K_{i}(\brelcat{\Uell\times W_2}{W_2})
\cong K_{i-1-k}(\ZK),
\end{align*}
where $k=\dim W_2$.
The boundary map $\bd_{i+1} = 1 - t_{\ast}$ in
the long exact sequence
of the orbit type filtration for $W$.
\end{lemma}
\begin{proof}
The bounded category $\brelcat{\Uell\times W_2}{W_2}$
of germs away from $W_2$  has effective fundamental
group $K$, as defined in \cite[3.13]{hp1}.
It therefore has the same $K${--}theory as
$\cc_{\br^{k+1}}(\cc_{pt}(\ZK))$.
The other case is similar.
The identification of $\bd_{i+1}$ with $1-t_{\ast}$ is
discussed in detail in the proof of Proposition \ref{boundary}.
\end{proof}
Since $\Kminus{j}(\ZK) = 0$ for $j\geqq 2$ by \cite{carter1},
this Lemma gives vanishing results for bounded $K${--}theory
as well.
\begin{lemma}\label{five}
Suppose that $W$ is complex, and $W^G=0$.
Then the inclusion map induces an isomorphism
\[K_i(\cc_{\Wmax\times \rminus,G}(\bz)) \to
K_i(\cc_{W\times \rminus,G}(\bz))\]
for $i \leqq 1$.
\end{lemma}
\begin{proof}
This is an argument using the orbit type filtration.
Let $W = \Wmax\oplus W_2$, and suppose that $W_2\neq 0$
or equivalently $\dim W_2\geqq 2$, since $W$ is complex.
Write $W = W' \oplus W''$ with $W'\subseteq W_2$,
 $\dim W' = 2$ and $\Iso(W') = K$ minimal.
We  choose $W''$ containing $\Wmax$,
and by induction we assume the result holds for $W''$.

Then applying the first part of Lemma \ref{orb}, we get the calculations
\eqncount
\begin{equation}K_i(\brelcatrminus{\Uell\times W''}{W''})
=K_{i-2-|W''|}(\ZK)\label{filta}
\end{equation}
and
\eqncount
\begin{equation}K_i(\brelcatrminus{W}{\Uell\times W''})
=K_{i-3-|W''|}(\ZK)\ .\label{filtb}
\end{equation}
Since  $\dim W'' \geqq 2$, we get the vanishing results
$K_i(\brelcatrminus{\Uell\times W''}{W''}) = 0$
for $i \leqq 2$ by \cite{carter1},
and
$K_i(\brelcatrminus{W}{\Uell\times W''}) =0$ for  $i\leqq 3$.
From the filtration sequence, it follows that
$K_i(\brelcat{W\times\rminus}{W''\times \rminus}) =0$
for $i\leqq 2$, and therefore
\[K_i(\bcat{W''\times\rminus})
\RA{\approx} K_i(\bcat{W\times\rminus})\]
for $i\leqq 1$.
We are done, by induction.
\end{proof}
\begin{corollary}\label{maxk}
$K_i(\brelcat{W\times\rminus}{\Wmax\times\rminus}) =0$
for $i \leqq 2$.
\end{corollary}
\begin{proof}
We continue the notation from above, and look at part of the
filtration sequence
\[  K_i(\brelcat{W''\times\rminus}{\Wmax\times\rminus})
 \to K_i(\brelcat{W\times\rminus}{\Wmax\times\rminus})
\to  K_i(\brelcat{W\times\rminus}{\Wmax\times\rminus})\ .\]
The first term is zero for $i\leqq 2$  by induction on dimension, and the
third term is zero for $i\leqq 2$ as above.
\end{proof}
We can obtain a little sharper result with some additional work.
First a useful observation:
\begin{lemma}\label{flipsign} Let $\ca$ be an additive category (with involution).
Then the  map $\br\to\br$ sending $x$ to $-x$ induces minus the identity on
$K$-theory (and $L$-theory) of $\cc_{\br}(\ca)$.
\end{lemma}
\begin{proof}
The category $\cc_{\br}(\ca)$ is filtered by the full subcategory
whose objects have support in a bounded neighborhood of $0$.
This subcategory is
equivalent to $\ca$ and the quotient category may be identified with
$\Ccat{[0,\infty)}{0}{\ca}\times \Ccat{(-\infty,0]}{0}{\ca}$
via the projection maps in an obvious way.

Consider the diagram
\[
\xymatrix{\ca\ar@{=}[d]\ar[r]&\cc_{\br}(\ca)\ar[d]\ar[r]&
\cc_{[0,\infty)}^{>0}(\ca)\times \cc _{(-\infty,0]}^{>0}(\ca)\ar[d]\\
\ca\ar[r]&\cc_{[0,\infty)}(\ca)\ar[r]&\cc_{[0,\infty)}^{>0}
}
\]
where the vertical map is induced by $x\mapsto |x|$.
 In the lower horizontal row, $K$ and $L${--}theory of the
 middle term is trivial, so the
boundary map will be an isomorphism.
The lower row splits off the upper row in
two different ways, one induced by including $[0,\infty)\subset \br$ and the
other by sending $x\in[0,\infty)$ to $-x\in\br$.
Under these two splittings we
may identify $K$ or $L${--}theory of the quotient
$\cc_{[0,\infty)}^{>0}(\ca)\times \cc_{(-\infty,0]}^{>0}(\ca)$ with
$\cc_{[0,\infty)}^{>0}\times \cc_{[0,\infty)}^{>0}$ and under this
identification, the flip map of $\br$ corresponds to interchanging the two
factors.
On $K${--}theory (or $L${--}theory)
we conclude that the exact sequence is
of the form
\[ 0\to A_*\to A_*\times A_*\RA{+} A_*\to 0\ .\]
The flip action on the last term is trivial, and on the middle
term it interchanges the two factors, so the inclusion must send $a$ to
$(a,-a)$.
Hence the flip action on the first term must be $a\mapsto -a$.
\end{proof}

As above, $t_{\ast}$ denotes the induced action
 of a generator $t \in G$
on $K${--}theory, and $\varepsilon\colon G \to \orient$ the non-trivial  action
of $G$ on $\rminus$.

\begin{proposition}\label{boundary}
Let $W$ be a
complex $2$-dimensional $G${--}representation with $W^G=0$,
and isotropy subgroup $K$.
Then under the isomorphisms of (\ref{filta}) and (\ref{filtb})
the complex
\[ K_{i+1}(\brelcatrminus{W}{\Uell})
\RA{\bd_a} K_{i}(\brelCatrminus{\Uell}) \RA{\bd_b}
K_{i-1}(\brelcat{\rminus}{0})\]
with $\bd_b\circ\bd_a =0$ is isomorphic to
\[K_{i-2}(\ZK) \RA{\bd'_a} K_{i-2}(\ZK) \RA{\bd'_b} K_{i-2}(\ZH)\]
where
$\bd'_a = 1-t_{\ast} =2$ and
$\bd'_b = \Ind_H\circ (1-t_{\ast}) = 0$.
\end{proposition}
\begin{proof}
The orbit type filtration is based on a $G${--}equivariant
simplicial model for $W$, where $G$ acts through the
projection to $G/K$.
The third term in the complex is $K_{i-1}(\brelcat{\rminus}{0})
=K_{i-2}(\ZH)$ and the identification of the  boundary maps
follows from the definition of the germ categories.

To compute $\bd_a$, we use the isomorphisms between the domain of
$\bd_a$  and $K_{i-2}(\bz K)$  obtained by noticing that every element is induced
from an  element  of
\[K_{i+1}(\Ccat{C\times \br,K}{\bd C\times \br}{\bz)},\]
where $C$ is the region between two adjacent half{--}lines of $\Uell$.
This follows since the regions in the complement of $\Uell\times\rminus$
 are disjoint and the boundedness condition ensures there is no
interference.
 Similarly the isomorphism of the range of $\bd_a$  with
$K_{i-2}(\bz)$ is obtained by noticing that every element is induced from
$\cc_{h\times \br,K}^{>\br}$ where $h$
 is just one half{--}line in $\Uell$, and we can think of
$\bd C=h\cup th$.
To compute the boundary we first take the standard
boundary to $\cc_{\bd C\times \br,K}(\bz)$ which is an isomorphism, and then
map away from $0\times \br$.
 It follows from the proof of Lemma \ref{flipsign} above
that this map is of the form $a \mapsto (a,-a)$ in $K$ or $L${--}theory.
In this picture, the support of one of the boundary components
is along $h$ and the other along $th$.
We need to use the group action to associate both elements to
the same ray.
Since $t$ flips the $\br${--}factor, we get a change of sign before
adding, so $t_\ast = \varepsilon(t) = -1$ and  $\bd_a$ sends $a$ to $2a$.

To compute $\bd_b$, we start with an element in the source of $\bd_b$ which
as above is identified via induction with
$K_i(\brelcat{h\times \br,K}{\br})$.
The boundary first sends this isomorphically to $\cc_{\br,K}(\bz)$,
 then by induction to $\bCatrminus$,  and
then via the natural map to the range of $\bd_b$, which is
$\cc_{\rminus,G}^{>0}$.
We have a commutative square
\[
\xymatrix{K_1(\Ccat{h \times \br,K}{\br}{\bz})
\ar[r]^{\Ind} \ar[d]_{\bd}
&K_1(\brelCatrminus{\Uell})\ar[d]^{\bd_b}\\
K_0(\Ccat{\br,K}{0}{\bz}) \ar[r]^{\Ind}&K_0(\Ccat{\rminus,G}{0}{\bz})
}
\]
where $\Ccat{\br,K}{0}{\bz} =\cc_{\br,K}(\bz)\times
\cc_{\br,K}(\bz)$.
Under this identification, the natural map to the germ category
$K_0(\cc_{\br,K}(\bz)) \to K_0(\Ccat{\br,K}{0}{\bz})$
is just $a \mapsto (a,-a)$, and
the induction map
\[\Ind\colon K_0(\Ccat{\br,K}{0}{\bz}) \to K_0(\Ccat{\rminus,G}{0}{\bz})\]
is  given by $(a,b)\mapsto \Ind_H(a+t_{\ast} b)$.
In this case the action of $t$ on
$\cc_{\rminus,G}(\bz)$ is  the identity since any element is
invariant under the action of $G$, hence under the action of $t$.
It follows that $\Ind(a,-a) = \Ind_H(a-t_{\ast}a) =0$ as required.
\end{proof}


\begin{lemma}\label{boundtwo}
Let $W$ be a complex $2$-dimensional $G${--}representation
with proper isotropy group $K$.
The boundary map
\[K_{i+1}(\brelCatrminus{\Uell}) \to K_i(\bCatrminus)\]
is zero for $i\leqq 1$.
\end{lemma}
\begin{proof}
If $i \leqq -1$ the domain of this boundary map is zero, so the result is trivial.
For $i=1$ we use the injection $\wh(\bCatrminus) \to \wh(\ccat{\rminus,G}{\zadictwo})$,
which follows from the vanishing of $SK_1(\ZG)$. But
$\wh(\ccat{\rminus,G}{\zadictwo}) = \wh( \Zadictwo{G})/\wh(\Zadictwo{H})$ and 
the group $ \wh( \Zadictwo{G}) = \prod_{d\mid q} \zadictwo [\zeta_d] G_2$,
where $G_2 \subset G$ is the $2$-Sylow subgroup and $q$ is the odd part of the order
of $G$. Now Oliver \cite[Thm. 6.6]{o1}  constructed a short exact sequence
$$1 \to \wh(\zadictwo [\zeta_d] G_2) \to 
\zadictwo [\zeta_d] G_2 \to \langle -1\rangle \times G_2 \to 1$$
by means of the integral $2$-adic logarithm. This sequence is natural with respect to
inclusion of subgroups, so we may use it to compare $ \wh( \Zadictwo{G})$ and
$ \wh( \Zadictwo{H})$. Since each corresponding term injects, and the middle quotient is 
$\bz$-torsion free, we conclude that $\wh(\ccat{\rminus,G}{\zadictwo})$ is
 also $\bz$-torsion free. Since $K_{2}(\brelCatrminus{\Uell} = K_{0}(\ZK)$ is
torsion (except for $\bz = K_0(\bz)$ which is detected by projection to the
trivial group),
the given boundary map is zero.


For $i=0$, we use the surjection $K_0(\qlocal K) \to \Kminus{1}(\ZK)$,
and compute  with $\qlocal$
coefficients and $i=1$.
We will list the steps, and leave the details to the reader.
First, compute that $K_1(\ccat{\rminus,G}{\qlocal})=K_1(\qlocal G)/
K_1(\qlocal H)$ surjects onto $K_1(\ccat{\Uell\times\rminus,G}{\qlocal})$,
by means of a braid containing the cone point inclusions
into $\ccat{\rminus,G}{\qlocal}$ and $\ccat{\Uell\times\rminus,G}{\qlocal}$.
Second, prove that $K_1(\ccat{\Uell\times\rminus,G}{\qlocal})$
fits into a short exact sequence
\[0 \to K_1(\ccat{\Uell,H}{\qlocal}) \to K_1(\ccat{\Uell,G}{\qlocal})
\to  K_1(\ccat{\Uell\times\rminus,G}{\qlocal}) \to 0\]
by means of a braid containing the inclusion
$\Uell \subseteq \Uell\times\rminus$.
Finally, compute the first two terms
$K_1(\ccat{\Uell,G}{\qlocal})=K_1(\qlocal G)/K_1(\qlocal K)$,
and $K_1(\ccat{\Uell,H}{\qlocal})=K_1(\qlocal H)/K_1(\qlocal K)$
by comparing the groups under the inclusion $H < G$.
We conclude that
\[K_1(\ccat{\rminus,G}{\qlocal}) \cong
K_1(\ccat{\Uell\times \rminus,G}{\qlocal})\] under the inclusion map,
and hence $\bd =0$.
\end{proof}


\begin{corollary}\label{kersimple}
Let $W$ be a complex $G${--}representation with $W^G=0$.
Then the inclusion induces  an isomorphism
$K_i(\bCatrminus) \to K_i(\bcatrminus{W})$
for $i\leqq 0$, and an injection for $i=1$.
If $\Kminus{1}(\ZK)=0$ for the maximal proper isotropy
 groups $K$ of $W$,
then $K_1(\bCatrminus) \cong K_1(\bcatrminus{W})$.
\end{corollary}
\begin{proof}
We may assume that $\dim W=2$, and apply the filtering
argument again.
By (\ref{filta}) and (\ref{filtb}) we get
$K_i(\brelCatrminus{W}) =0$ for $i\leqq 0$, and
$K_1(\brelCatrminus{W})$ is a quotient of
$K_1(\brelCatrminus{\Uell})=\Kminus{1}(\ZK)$.
Since the composition
\[K_{1}(\brelCatrminus{\Uell})
\to K_{1}(\brelCatrminus{W}) \to K_0(\bCatrminus)\]
is zero  by Lemma \ref{boundtwo}, the result follows for $i\leqq 0$.

Similarly, $K_2(\brelCatrminus{W})$
is a quotient of $K_2(\brelCatrminus{\Uell}) = K_0(\ZK)$,
because the boundary map $K_2(\brelcatrminus{W}{\Uell})=\Kminus{1}(\ZK)$
to $K_1(\brelCatrminus{\Uell})=\Kminus{1}(\ZK)$ is multiplication
by $2$, and hence injective.
Then we make the same argument, using Lemma \ref{boundtwo}. 

If we also assume $\Kminus{1}(\ZK)=0$, then
$ K_1(\brelCatrminus{W}) =0$ so we get the isomorphism
$K_1(\bCatrminus) \RA{\approx} K_1(\bcatrminus{W})$.
\end{proof}

\section{ The double coboundary}
\label{doublecob}
The composite $\Delta$
of maps from the $L^h-L^p$ and $L^s -L^h$ Rothenberg
sequences
\[
\xymatrix{
H^{i}(\widetilde K_0(\ZG))\ar[r]\ar[dr]^{\Delta} & \Lgroup{h}{i}{\ZG}\ar[d]\\
&  H^i(\wh(\ZG))}
\]
(see \cite{ht1}) has an algebraic description
by means of a ``double coboundary" homomorphism
\[\delta^2\colon H^{i}(\widetilde K_0(\ZG))\to H^i(\wh(\ZG))\]
In this section, we will give a brief description due to
Ranicki \cite{ra13} of this homomorphism
(see also \cite[\S 6.2]{ra8} for related material on
``interlocking" exact sequences in $K$ and $L${--}theory).

Let $X$ be a space with a $\cy 2$ action $T\colon X \to X$,
and define homomorphisms
\[\Delta\colon H^i(\pi_n(X)) \to H^i(\pi_{n+1}(X))\]
by sending $g\colon S^n \to X$ to
\[h\cup (-1)^iTh\colon S^{n+1} = D^{n+1}_{+} \cup_{S^n} D^{n+1}_{-}
\to X\]
for any null-homotopy $h\colon D^{n+1} \to X$ of the map
$g + (-1)^{i+1}Tg\colon S^n \to X$.

The maps $\Delta$ lead to a universal description of double
coboundary maps, as follows.
Let $f\colon X \to Y$ be a $\cy 2${--}equivariant map of
spaces with $\cy 2$ action, and consider the long exact
sequence
\[\dots \to \pi_n(X) \RA{f} \pi_n(Y) \to \pi_n(f) \to \pi_{n-1}(X)
\to \pi_{n-1}(Y) \to \dots\]
We define
$I_n=\ker(f\colon \pi_n(X) \to \pi_n(Y))$ and
$J_n=\Image(\pi_n(Y) \to \pi_{n}(f))$, and get an exact
sequence
\[0 \to \pi_n(X)/I_n \to \pi_n(Y) \to \pi_n(f) \to I_{n-1} \to 0\]
which can be spliced together from the short exact sequences
\eqncount
\begin{equation}
\begin{split} \label{tateseq}
&0 \to  \pi_n(X)/I_n \to \pi_n(Y) \to J_n \to 0\\
&0 \to J_n \to \pi_n(f) \to I_{n-1} \to 0\ .
\end{split}
\end{equation}

Then it follows directly from the definitions that the double
coboundary
\[\delta^2\colon H^i(I_{n-1}) \RA{\delta} H^{i+1}(J_n) \RA{\delta}
H^i(\pi_n(X)/I_n)\]
from the Tate cohomology sequences induced
by (\ref{tateseq}) is given by the composite
\[\delta^2\colon H^i(I_{n-1}) \RA{inc_{\ast}} H^i(\pi_{n-1}(X))
\RA{\Delta} H^i(\pi_n(X)) \RA{proj_{\ast}} H^i(\pi_n(X)/I_n)\ .\]
If we can pick the map $f\colon X \to Y$ appropriately,
say with $I_n=0$ and $I_{n-1} = \pi_{n-1}(X)$, this gives
an algebraic description of $\Delta$.

In later sections we will use the relative Tate cohomology groups
$H^i(\Delta)$, which are just (by definition) the relative Tate
cohomology groups \cite[p.166]{ra8} of the map $\pi_n(Y) \to \pi_n(f)$ in
the long exact sequence above.
These groups fit into the commutative braids given in \cite{ra13}
which will be used in the proofs of Theorems A-C.


We now give some examples, with $G$ denoting a finite cyclic
group as usual. These arise from homotopy groups of certain fibrations
of algebraic $K$-theory spectra.
\begin{example}
There is an exact sequence \cite{ht1}
\[0 \to \wh(\ZG) \to \wh(\Zadictwo{G}) \to  \wh(\Zreltwo{G})
\to \widetilde K_0(\ZG) \to 0\]
of $\cy 2$ modules and the associated double coboundary in Tate
cohomology equals
\[\Delta\colon H^{i}(\widetilde K_0(\ZG)) \to H^i(\wh(\ZG))\ .\]
The point here is that $\ker(\wh(\ZG) \to \wh(\Zadictwo{G}))=0$
\cite{o1}, and the map $\widetilde K_0(\ZG) \to
\widetilde K_0(\Zadictwo{G})$ is zero by a result of Swan
\cite{sw1}.
We could also use the exact sequence
\[0 \to  \wh(\ZG) \to \wh(\zlocal G)\oplus \wh(\bq G)\to
 \wh(\qlocal G)) \to \widetilde K_0(\ZG) \to 0\]
to compute the same map $\Delta$.
\end{example}
\begin{example}
There is an exact sequence
\begin{multline*}
0 \to \wh(\bCatrminus) \to \wh(\ccat{\rminus, G}{\zlocal\oplus \bq})
\to \\
\wh(\ccat{\rminus,G}{\qlocal})  \to \widetilde K_0(\bCatrminus) \to 0
\end{multline*}
where $\wh(\bCatrminus) = \wh(\rel{\ZH}{\ZG})$ and
$\widetilde K_0(\bCatrminus) = \widetilde K_0(\rel{\ZH}{\ZG})$.
The injectivity on the left follows because
 $K_2(\ccat{\rminus,G}{\qlocal})$ is a quotient of $K_2(\qlocal G)$,
mapping trivially through $K_1(\ZG)$ into $\wh(\bCatrminus)$
(since $SK_1(\ZG)=0$ \cite{o1}).
We therefore get an algebraic description of $\delta^2\colon
H^i(\widetilde K_0(\rel{\ZH}{\ZG^{-}})) \to H^i(\wh(\rel{\ZH}{\ZG^{-}}))$
as used in the statement of Theorem B.
\end{example}

\begin{example}
There is an exact sequence
\begin{multline}
0 \to \wh(\bcatrminus{W}) \to
 \wh(\ccat{W\times \rminus,G}{\zlocal \oplus \bq})
\to\\
 \wh(\ccat{W\times \rminus,G}{\qlocal})
 \to \widetilde K_0(\bcatrminus{W}) \to 0
\end{multline}
for any complex $G${--}representation $W$ with $W^G=0$.
We therefore get an algebraic description of $\delta^2\colon
H^i(\widetilde K_0(\bcatrminus{W})) \to H^i(\wh(\bcatrminus{W}))$
as used in the statement of Theorem C.
\end{example}
\begin{lemma}
For complex
 $G${--}representations $W_1\subseteq W$ with $W^G=0$,
there is a commutative diagram
\[
\xymatrix{H^i(\widetilde K_0(\bcatrminus{W_1}))
\ar[d]_{c_{\ast}}\ar[r]^{\delta^2}
& H^i(\wh(\bcatrminus{W_1}))\ar[d]^{c_{\ast}}\\
H^i(\widetilde K_0(\bcatrminus{W})) \ar[r]^{\delta^2}
& H^i(\wh(\bcatrminus{W}))
}
\]
where the vertical maps are induced by the inclusion $W_1\subseteq W$.
\end{lemma}
For our applications, the main point of the double coboundary
description is that it permits these maps induced by
cone point inclusions to be computed using bounded $K${--}theory,
instead of bounded $L${--}theory.

The double coboundary maps also commute with restriction
to  subgroups of $G$.
\begin{proposition}\label{ninetwo}
Let $G_1<G$ be a subgroup of odd index, and $H_1<G_1$ have index $2$,
then there are twisted restriction maps
$$H^i(\widetilde K_0(\rel{\ZH}{\ZG^-})) \xrightarrow{\Res}
 H^i(\widetilde K_0(\rel{\ZH_1}{\ZG_1^-}))
$$
 and
 $$H^i(\wh(\rel{\ZH}{\ZG^-})) \xrightarrow{\Res}
H^i(\wh(\rel{\ZH_1}{\ZG_1^-}))$$
such that the diagram
\[
\xymatrix{H^i(\widetilde K_0(\rel{\ZH}{\ZG^-}))
\ar[r]^{\delta^2}\ar[d]_{\Res}&
H^i(\wh(\rel{\ZH}{\ZG^-}))\ar[d]^{\Res}\\
H^i(\widetilde K_0(\rel{\ZH_1}{\ZG_1^-}))\ar[r]^{\delta_1^2}&
  H^i(\wh(\rel{\ZH_1}{\ZG_1^-}))
}
\]
commutes.
\end{proposition}
\begin{proof}
The vertical maps are twisted restriction maps given by
composing
the  twisting isomorphisms 
$H^i(\widetilde K_0(\rel{\ZH}{\ZG^-})) \cong
 H^{i+1}(\widetilde K_0(\rel{\ZH}{\ZG}))$ 
and
$H^i(\wh(\rel{\ZH}{\ZG^-}) \cong H^{i+1}(\wh(\rel{\ZH}{\ZG})$,
discussed in \pone, Section \ref{rmin},
with the  restriction maps induced by the inclusion
$\ZG_1\to \ZG$ of rings with involution.
Since $G_1 <G$ has odd index, $H_1 = H\cap G_1$ and the composition
$\Res_{G_1}\circ \Ind_{H}$ lands in the image of $\Ind_{H_1}$
by the double coset formula.
\end{proof}
This can be generalized to the double coboundary maps
used in the statement of Theorem C, under certain
conditions.
\begin{proposition}\label{ninethree}
Let $G_1<G$ be an odd index subgroup, and $H_1<G_1$ have index $2$.
Suppose that
$W$ only has proper isotropy subgroups  of $2${--}power index.
Then there are a twisted restriction maps
$H^i(\widetilde K_0(\bcatrminus{W}))\to
H^i(\widetilde K_0(\ccat{\Res W\times\rminus,G_1}{\bz}))$
and
$H^i(\wh(\bcatrminus{W})) \to
 H^i(\wh(\ccat{\Res W\times\rminus,G_1}{\bz}))$
which commute with the corresponding
 double coboundary maps $\delta^2_W$ and $\delta^2_{\Res W}$.
\end{proposition}

\section{ Calculations in bounded $L${--}theory}
 \label{boundedL}

Suppose that $\sigma=\sigma(f) \in L^h_0(\ZG)$
is the surgery obstruction arising from a normal cobordism
between $S(V_1)$ and $S(V_2)$, as in the statement of
\pone,  Theorem \ref{maintest}.
In this section, we establish two important properties of
$\trf{W}(\sigma)$ in preparation for the proof of Theorem C.
Unless otherwise mentioned, all bounded categories will
have the standard orientation (see Example \ref{standard}).

For a complex $G${--}representation $W$ the standard
orientation is trivial, and
the cone point inclusion  $0 \in W$ induces the map
\[c_*\colon  L_n^h(\ZG) \to L_{n}^h(\bcat{W})\]
Note that the presence
of an $\rminus$ factor introduces a non-trivial orientation
at the cone point
\[c_*\colon  L_{n+1}^h(\ZG, w) \to  L_{n+1}^h(\bcatrminus{W})\]
where $w\colon G \to \orient$ is the non-trivial
projection.
The properties are:
\begin{theorem}\label{one}
Suppose  $W$ is a complex $G${--}representation with
no  $\rplus$ summands.
If $\trf{W\times \rminus}(\sigma)
\in L^h_{2k+1}(\bcatrminus{W})$
is a torsion element, 
then
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item there exists a torsion element $\hat\sigma\in 
L_{2k+1}^h(\bCatrminus)$ such that 
\[c_*(\hat\sigma) = \trf{W\times \rminus}(\sigma)
\in  L^h_{2k+1}(\bcatrminus{W})\]
\item there exists a torsion
 element $\hat\sigma \in L^p_{2k+1}(\ZG, w)$
such that
\[c_*(\hat\sigma) = \trf{W\times \rminus}(\sigma)
\in  L^p_{2k+1}(\bcatrminus{W})\]
where $\dim W = 2k$.
\end{enumerate}
\end{theorem}
We remark that the condition ``$\trf{W\times \rminus}(\sigma)$
is a torsion element"  follows from  the assumption
$\Res_H (V_1\oplus W) \oplus \rplus
 \sim_t \Res_H (V_2\oplus W)\oplus \rplus$ in 
Theorem C,
as an immediate consequence of Proposition \ref{trftorsion}.
Before giving the proof,  we need some preliminary
results.
When the $L${--}theory decoration is not explicitly given,
we mean $\Linfty$.

\begin{lemma}
Let $W$ be a complex $G${--}representation.
Then
\[ L_{2k+1}(\bcat{W})\otimes \bq = 0\]  for $k \geqq 0$
\end{lemma}
\begin{proof}
We argue by induction on $\dim W$, starting with
\[L_{2k+1}(\ZG) = \cy 2\oplus H^1(\Kminus{1}(\ZG))\]
 which is all $2${--}torsion.
It is enough to prove the result for the top component
$ L_{2k+1}(\bcat{W})(q)$, and therefore by \pone, 
Theorem \ref{twopoweriso} we may
assume that the isotropy groups of $W$ all have
$2${--}power index.
Since we are working with $\Linfty$, we may ignore
$\rplus$ summands of $W$.

Let $W = W'\oplus W''$ where $\dim W'=2$ and
$W'$  has minimal isotropy group $K$.
 We assume the result for $W''$,
and let $\Uell \subset W'$ be a $G$-invariant
set of rays from the origin, dividing $W'$ into fundamental
domains for the free $G/K$-action.


Then
\[L_{n}(\bCat{\Uell \times W''}{G}{W''}(\bz))
=L_{n-1}(\bcat{W''})
=L_{n-1-|W''|}(\ZK),\]
which is torsion for $n$ even, and
\[L_{n}(\bCat{W}{G}{\Uell \times W''}(\bz))
=L_{n-2}(\bcat{W''})
=L_{n-2-|W''|}(\ZK),
\]
which is torsion for $n$ odd.
Moreover, we have a long exact sequence
\[
\dots\to L_{n}(\bCat{\Uell \times W''}{G}{W''}(\bz)) \to
L_{n}(\bCat{W}{G}{W''}(\bz)) \to
L_{n}(\bCat{W}{G}{\Uell \times W''}(\bz)) \to \dots
\]
We claim that  the first map in this sequence is
rationally  injective.
The previous map in the long exact sequence
is
\[ L_{n+1}(\bCat{W}{G}{\Uell \times W''}(\bz))
\longrightarrow{\bd}
 L_{n}(\bCat{\Uell \times W''}{G}{W''}(\bz)),\]
which may be identified (using the isomorphisms above
and Proposition \ref{boundary})
with a geometrically induced map
\[L_{n-1-|W''|}(\ZK) \RA{1-u} L_{n-1-|W''|}(\ZK)\]
``multiplication by $1-w(t)$", where $\bar t = w(t) t^{-1}$
from the action of $t$ in the antistructure used to define the
$L$-group.

In the oriented case, $w(t)= +1$ and
this boundary map is zero.
Therefore, we conclude that
\[\xymatrix{L_{2k+1}(\bCat{\Uell \times W''}{G}{W''}(\bz))
 \ar[r]^{\approx}&
L_{2k+1}(\bCat{W}{G}{W''}(\bz))} \]
is a rational isomorphism
and so
\[L_{2k+1}(\bCat{W}{G}{W''}(\bz)) = L_{2k-|W''|}(\ZK)\ .\]
Finally, we will substitute this computation and our
inductive assumption into the long exact sequence
\begin{multline*}
 \dots \to L_{2k+1}(\bcat{W''}) \to
 L_{2k+1}(\bcat{W}) \to\\
L_{2k+1}(\bCat{W}{G}{W''}(\bz)) \to
L_{2k}(\bcat{W''})  \to
\end{multline*}
and obtain an exact sequence
\[0 \to L_{2k+1}(\bcat{W}) \to
L_{2k+1}(\bCat{W}{G}{W''}(\bz)) \to L_{2k}(\bcat{W''}).\]
However, the second map in this sequence can be identified
with the inclusion map
\[
\Ind_K\colon  L_{2k}(\cc_{W'', K}(\bz))\to  L_{2k}(\bcat{W''})\]
and the composition $\Res _K \circ  \Ind_K$ is multiplication
by $[G:K]$, which is a rational isomorphism.
Therefore $\Ind_K$ is injective
and $ L_{2k+1}(\bcat{W}) =0$.
\end{proof}

Another computation we will need is
\begin{lemma}\label{fourtors} $L_{n}(\brelCatrminus{W})\otimes \bq = 0$
for W a complex G-representation.
\end{lemma}
\begin{proof}
We may assume that $W^G=0$  and argue by
 induction on the dimension of W.
We write $W = W' \oplus W''$,
where $\Iso(W' ) = K$ is minimal ($2${--}power index isotropy subgroups
may be assumed as usual).
We have 2 long exact sequences (all $L${--}groups
are  tensored  with $\bq$):

(i)  from the inclusion $W''\oplus \rminus \subset W \oplus \rminus$.
For short, let $A = \brelCatrminus{W}$,
$B = \brelCatrminus{W''}$, and then
$A/B = \brelcatrminus{W}{W''}$.
We need the
piece of the $L$-group sequence:
\eqncount
\begin{equation}
 ... \to L_{n}(B) \to  L_{n}(A) \to  L_{n}(A/B) ... \label{foura}
\end{equation}
and note that $L_{n}(B) = 0$ by induction.

(ii) we study $L_{n}(A/B)$ by looking at the usual rays
 $\Uell \subset W'$ which divide the 2-dim representation
$W'$ into G/K chambers.
 Let
    \[D_0 =\brelcatrminus{\Uell\times W''}{W''}\]
and
 \[   D_1 =\brelcatrminus{W}{\Uell\times W''}\]
Then we need the $L${--}group sequence
\eqncount
\begin{equation} \to L_{n}(D_0) \to  L_{n}(A/B) \to  L_{n}(D_1) \to L_{n-1}(D_0)
\label{fourb}
\end{equation}
But the groups $L_{n}(D_i)$ are the ones we have been computing
 by using the chamber structure.
 In particular,
\[L_{n}(D_0) = L_{n-2}(\cc_{W'',K}(\bz))\]
and
\[L_{n}(D_1) = L_{n-3}(\cc_{W'',K}(\bz)).\]
But since $K$ is the minimal isotropy group in $W$,
it acts trivially on $W''$
and these $L$-groups are just
\[
        L_{n}(D_0) = L_{n-2-|W''|}(\ZK) \]
and
\[L_{n}(D_1) = L_{n-3-|W''|}(\ZK).\]
Now the boundary map
\eqncount
\begin{equation}
        L_{n}(D_1) \to L_{n-1}(D_0)
\end{equation}
in the sequence is just multiplication by
 $1-w(t)=2$ ($w(t)=-1$ since $\rminus$
is non-oriented), and
this is a rational isomorphism.
Therefore, $L_{n}(A/B) = 0$ by (\ref{fourb}) and
substituting back into (\ref{foura}), the  $L_n(A)=0$.

\end{proof}
Now a more precise result in a special case:
\begin{lemma}\label{four}
Let W be a complex $2${--}dimensional G-representation
with $W^G=0$.
Then $L^s_{3}(\brelCatrminus{W}) = 0$
\end{lemma}
\begin{proof}
We have an exact sequence
\begin{multline*}
\dots \to L^s_3(\brelCatrminus{\Uell})
\to L^s_3(\brelCatrminus{W})\to\\
 L^I_3(\brelcatrminus{W}{\Uell})
\to L^s_2(\brelCatrminus{\Uell})
\to \dots
\end{multline*}

arising from the orbit type filtration.
Here
\begin{align*}
I &= \Image(K_2(\brelCatrminus{W}) \to K_2(\brelcatrminus{W}{\Uell}))\\
&=\ker(K_2(\brelcatrminus{W}{\Uell}) \to K_1(\brelCatrminus{\Uell}))\\
&=\ker(\Kminus{1}(\ZK) \RA{2} \Kminus{1}(\ZK))
\end{align*}
by Proposition \ref{boundary}, where $K$ is the isotropy subgroup of $W$.
But  $\Kminus{1}(\ZK)$ is torsion{--}free,  so $I = \{ 0 \} \subseteq K_2$.
Now substitute the computations
 \begin{align*} &L^s_3(\brelCatrminus{\Uell}) = L_1^p(\ZK) =0\\
&L^I_3(\brelcatrminus{W}{\Uell}) = L^p_0(\ZK)\\
&L^s_2(\brelCatrminus{\Uell}) = L^p_0(\ZK)
\end{align*}

into  the exact sequence.
The boundary map
\[L^I_3(\brelcatrminus{W}{\Uell})
\to  L^s_2(\brelCatrminus{\Uell})\]
 is multiplication by $2$ so
$ L^s_3(\brelCatrminus{W})=0$.
\end{proof}
\begin{remark}\label{fourex}
 The same method shows that
$ L^s_1(\brelCatrminus{W})=L_3^p(\ZK)=\cy 2$
for $\dim W=2$ as above, assuming that $K\neq 1$.
\end{remark}

Our final preliminary result is a Mayer{--}Vietoris sequence:
\begin{lemma}\label{two} Let $W$ be a complex $G${--}representation
with $W^G=0$.
Let $W = W_1 \oplus W_2$ be a direct
sum decomposition, where $W_1=\Wmax$.
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item
There is a long exact sequence
\begin{multline*}
\dots \to L^I_{n+1}(\brelcat{W\times \rminus}
{W_1\times \rminus\cup W_2\times \rminus})
\RA{\bd_{n+1}}
L^s_{n}(\brelCatrminus{W})
 \to \\
L^s_{n}(\brelcatrminus{W}{W_1})\oplus
 L^s_{n}(\brelcatrminus{W}{W_2})
 \to
L^I_{n}(\brelcat{W\times \rminus}
{W'\times \rminus\cup W''\times \rminus})
\end{multline*}

of bounded $L$-groups, where
\[I = \Image\left [ K_2(\brelcatrminus{W}{W_2}) \to
K_2(\brelcat{W\times \rminus}
{W'\times \rminus\cup W''\times \rminus})\right ]\]
is the decoration subgroup.
\item For $n\equiv 3\Mod{4}$, the boundary map
$\bd_{n+1} =0$.
\end{enumerate}
\end{lemma}
\begin{proof}
The Mayer{--}Vietoris sequence in $\Linfty$
\begin{multline*}
\dots \to L_{n+1}(\brelcat{M}{U\cup V}) \to
L_n(\brelcat{M}{U\cap V}) \to \\
L_n(\brelcat{M}{U})\oplus L_n(\brelcat{M}{V})\to
L_n(\brelcat{M}{U\cup V}
\end{multline*}
where $U$, $V$ are nice $G${--}invariant subspaces of
the control space $M$,
follows from an excision isomorphism
\[L_{n}(\cc^{>U}_{U\cup V,G}(\bz))
 = L_{n}(\cc^{>U\cap V}_{V,G}(\bz))\]
and standard diagram-chasing.
We apply this to $M = W \times \rminus$, $U = W_1\times \rminus$
and $V = W_2\times\rminus$, where $U\cap V = \rminus$.
The decorations follow from Section \ref{excision}
and Corollary \ref{maxk}.

To see that $\bd_{0}=0$, note that this boundary map
is the composition of
\[\bd\colon L^I_{0}(\brelcat{W\times \rminus}
{W'\times \rminus\cup W''\times \rminus}) \to
L^s_{3}(\brelcat{W'\times \rminus\cup W''\times \rminus}{W_2\times \rminus})\]
and an excision isomorphism
\[L^s_{3}(\brelcat{W'\times \rminus\cup W''\times \rminus}{W_2\times \rminus})
\cong L^s_3(\brelCatrminus{W_1})\ .\]
But $L^s_3(\brelCatrminus{W_1}) =0$ by Lemma \ref{four}.
\end{proof}

\begin{corollary}\label{twocor}
Suppose  $W$ is a  complex $G${--}representation with
no  $\rplus$ summands.
Then the image of
 $\trf{W\times \rminus}(\sigma)$
is zero in
$L^s_{2k+1}(\brelCatrminus{W})$,
where $\dim W =2k$.
\end{corollary}
\begin{proof}
First recall from \pone, Theorem \ref{ktrans} that
$\trf{W\times \rminus}(\sigma) \in L^{c_1}_{2k+1}(\bcatrminus{W})$
where $c_1=\Image(\wh(\ZG) \to \wh(\bcatrminus{W}))$
under the cone point inclusion.
Therefore, the image of $\trf{W\times \rminus}(\sigma)$
``away from $\rminus$" lands in $L^s_{2k+1}(\brelCatrminus{W})$.
We may therefore apply the Mayer{--}Vietoris sequence
from Lemma \ref{two}.

As usual, we  may assume that
$W$ has only $2${--}power isotropy subgroups,
and we will argue by induction on $\dim W$ and on the
orbit type filtration of $W$.
The result is true for $\dim W = 2$  (by Lemma \ref{four})
or for $W$ a free representation (by a direct calculation
following the method of Lemma \ref{fourtors}).
For sets of $2${--}power index
subgroups (ordered by inclusion, with repetitions allowed),
we say that $\{K'_1,\dots,K'_s\} < \{K_1,\dots, K_t\}$
if $s <t$ or $s=t$, $K'_i =K_i$ for $1\leqq i <j\leqq s$, and
$K'_j \subsetneqq K_j$.
Let $\Iso(W) = \{K_1,\dots,K_t\}$ denote the set of
isotropy subgroups of $W$ ordered by inclusion.
Define an ordering by $W' < W$ if (i) $\dim W' <\dim W$
or (ii) $\dim W'=\dim W$ but $\Iso(W') < \Iso(W)$.

Let $W = W_1\oplus W_2$, where $W_1$ has maximal
isotropy subgroup $K$, and suppose the result holds
for all $W' <W$.
If $\dim W\equiv 2\Mod{4}$, we get an injection
\[L^s_3(\brelCatrminus{W}) \to
 L^s_3(\brelcatrminus{W}{W_1})\oplus L^s_3(\brelcatrminus{W}{W_2})\]
from Lemma \ref{two}, where $W_2 <W$ and $W_1 <W$.
If $\dim W \equiv 0\Mod{4}$, we get an injection
\[L^s_3(\brelCatrminus{W\times U}) \to
 L^s_3(\brelcatrminus{W\times U}{W_1})
\oplus L^s_3(\brelcatrminus{W\times U}{W_2\times U})\]
again from Lemma \ref{two}, where $U$ is any $2${--}dimensional free
$G${--}representation.
Notice that $W_2\oplus U < W$ in the ordering above.
Consider the following commutative diagram
(with $\dim W\equiv 2\Mod{4}$):
\[
\xymatrix{
L^h_0(\ZG) \ar[rr]^{\trf{W_2\times \rminus}\ }
\ar[drr]_{\trf{W\times\rminus}}&&
L^{c_1}_1(\bcatrminus{W_2})\ar[r]\ar[d]^{\trf{W_1}}&
L^s_1(\brelCatrminus{W_2})\ar[d]^{\trf{W_1}}\\
&&L^{c_3}_3(\bcatrminus{W})\ar[r]&
L^s_3(\brelcatrminus{W}{W_1})
}
\]
This diagram, and our inductive assumption shows that
the image of  $\trf{W\times \rminus}(\sigma)$  is zero in
$L^s_3(\brelcatrminus{W}{W_1})$.
Similarly, by reversing the roles of $W_1$ and $W_2$ in the diagram,
we see that the image of $\trf{W\times \rminus}(\sigma)$  is zero in
$L^s_3(\brelcatrminus{W}{W_2})$.
Therefore the image of $\trf{W\times \rminus}(\sigma)$  is zero in
$L^s_3(\brelCatrminus{W})$.
If $\dim W \equiv 0\Mod{4}$, we replace $W$ by $W\oplus U$
and make a similar argument, using the observation that
$W_2\oplus U < W$ to justify the inductive step.
\end{proof}

\begin{proof}[The proof of Theorem \ref{one}]
Part (i) follows
from Corollary \ref{twocor}:  we know that the image of our obstruction
 $\trf{W\times\rminus}(\sigma)$ is zero ``away from $\rminus$",
so comes from a torsion element in $\hat\sigma \in L_{2k+1}^h(\bCatrminus)$
(by Lemma \ref{fourtors}, $ L_{2k+2}(\brelCatrminus{W})$ is torsion).

For part (ii) observe that
 $L^h_{2k+1}(\brelcat{\rminus}{0})=L^p_{2k}(\ZH)$ is
torsion{--}free (except for the Arf invariant $\cy 2$ which is
detected by the trivial group, so does not
matter). Hence  the image of $\trf{W\times\rminus}(\sigma)$
vanishes in $L^h_{2k+1}(\brelcat{W\times \rminus}{0})$.
The exact sequence
\[ \dots\to L^{\bu}_{2k+1}(\ZG,w) \to L^h_{2k+1}(\bcatrminus{W}) \to
L^h_{2k+1}(\brelcat{W\times\rminus}{0}),\]
with $\bU=\ker(\widetilde K_0(\ZG) \to \widetilde
K_0(\bcatrminus{W}))$, now shows that there exists a torsion element
\[\hat\sigma \in  \Image(L^{\bu}_{2k+1}(\ZG,w) \to L^p_{2k+1}(\ZG,w))\]
with $c_{\ast}(\hat\sigma) = \trf{W\times \rminus}(\sigma) \in
L^p_{2k+1}(\bcatrminus{W})$.
\end{proof}
\begin{remark}\label{sharper}
In Theorem \ref{sixteenfive} we use the sharper result
$\hat\sigma \in  L^{\bu}_{2k+1}(\ZG,w)$ to
improve our unstable similarity results
by removing extra $\rminus$ factors.
The $L$-group decoration $\bU$ is independent of $W$, since
$K_0(\bCatrminus) \to K_0(\bcatrminus{W}$ is injective
by Corollary \ref{kersimple}, and hence
$\bU=\Image(\widetilde K_0(\ZH)\to \widetilde K_0(\ZG))$.
\end{remark}



\section{ The proof of Theorem C}
 \label{C}
By replacing $V_i$ by $V_i\oplus U$ if necessary,
where $U$ is free and
$2${--}dimensional, we may assume that
$\dim V_i\oplus W\equiv 0\Mod{4}$.
This uses the $s${--}normal cobordism condition.
By the top component argument
 (see \pone, Theorem \ref{twopoweriso})
and Proposition \ref{ninethree},
we may also assume that $W$ has
only $2${--}power isotropy.
We have a commutative diagram analogous to 
\pone\ (\ref{mainda}).

\eqncount
\begin{equation} \label{maindb}
\begin{matrix}
\xymatrix@!C@C-6pc{
H^1(\widetilde K_0(\bcatrminus{W}))    \ar[dr] \ar@/^2pc/[rr]^{\delta^2}      &&
H^1(\wh(\bcatrminus{W})                \ar[dr] \ar@/^2pc/[rr]                 &&
L_0^s(\bcatrminus{W})                                                         \\
&  L_1^h(\bcatrminus{W})               \ar[dr] \ar[ur]                        &&
H^1(\Delta_{W\times\rminus})           \ar[dr] \ar[ur]                        \\
L_1^s(\bcatrminus{W})                  \ar[ur] \ar@/_2pc/[rr]_{}              &&
L_1^p(\bcatrminus{W})                  \ar[ur] \ar@/_2pc/[rr]                 &&
H^0(\widetilde K_0(\bcatrminus{W}))
}\end{matrix}
\end{equation}

\noindent where $H^1(\Delta_{W\times\rminus})$ denotes the relative group
of the double coboundary map.


This diagram for $W$ can
be compared with the one for $W_1=\Wmax$
via the inclusion maps, and we
see that the $K$-theory terms map isomorphically
by Lemma \ref{five}.
Now by Theorem \ref{one} there exists a torsion element
 \[\hat\sigma \in \Image(L^p_1(\bCatrminus)
 \to L^p_1(\bcatrminus{W_1}))\]
 which hits $\trf{W\times\rminus}$ under the cone point inclusion.
In particular, $\hat\sigma$ vanishes ``away from $\rminus$".
The main step in the proof of Theorem C is to show that the torsion
subgroup of $L^p_1(\bcatrminus{W_1})$ essentially injects into the
relative group $H^1(\Delta_{W_1\times\rminus})$ of the double
coboundary.

We have the exact sequence
\[ L^s_2(\brelCatrminus{W_1})
\to L^Y_1(\bCatrminus) \to L^s_1(\bcatrminus{W_1})
\to L^s_1(\brelCatrminus{W_1})\]
and $Y = \ker(\wh(\bCatrminus) \to \wh(\bcatrminus{W_1})$
is zero, by Corollary \ref{kersimple}.
But the first term $ L^s_2(\brelCatrminus{W_1})$ is a torsion group, 
by Lemma \ref{fourtors},
and the next term in the sequence
 $L^{\bY}_1(\bCatrminus)=L^s_1(\bCatrminus)$
 is torsion{--}free by \pone, Lemma \ref{lowA} so the middle map
$L^{\bY}_1(\bCatrminus) \rightarrowtail L^s_1(\bcatrminus{W_1})$
is an injection.

Since $L^s_1(\brelCatrminus{W_1}) =\cy 2$ by Remark \ref{fourex} we conclude
that the previous group $ L^s_1(\bcatrminus{W_1})$
is torsion{--}free modulo this $\cy 2$, which injects into
$\Lminus{1}_1(\brelCatrminus{W_1})
= \cy 2 \oplus H^1(\Kminus{1}(\ZK))$.
Since $\hat\sigma$ vanishes away from $\rminus$, this extra $\cy 2$
may be ignored.

By substituting this computation into the exact sequence
\[
L_1^s(\bcatrminus{W_1})\to L_1^p(\bcatrminus{W_1}) \to
H^1(\Delta_{W_1\times\rminus})\]
of  (\ref{maindb}), we see that the torsion subgroup of
 \[\ker (L_1^p(\bcatrminus{W_1}) \to L_1^p(\brelCatrminus{W_1})) \]
injects into the relative group
$H^1(\Delta_{W_1\times\rminus})$ of the double coboundary.
Therefore $\hat\sigma \in L_1^p(\bcatrminus{W_1})$ vanishes
if and only if
the element
 $\left\{ \Delta(V_1)/\Delta(V_2) \right\}$
is in the image of the double coboundary
\[\delta^2\colon
H^1(\widetilde K_0(\ccat{\Wmax\times \rminus,G}{\bz})) \to
H^1(\wh(\ccat{\Wmax\times \rminus,G}{\bz}))\  \]
This completes the proof of Theorem C.

\medskip
A very similar argument can be used to give an
inductive  criterion
for non{--}linear similarity without the $\rplus$ summand
(generalizing Theorem A).
In the statement we will use the analogue to
$\ck =\ker(\widetilde K_0(\ZH) \to \widetilde K_0(\ZG))$,
 namely
\[\ck_W = \ker(\wh(\brelcat{W\times\rminus}{0}) \to
\widetilde K_0(\ZG)).\]
 By Corollary \ref{kersimple}
 \[\ker(\wh(\ZG) \to 
\wh(\ccat{W\times \rminus,G}{\bz}) \cong  \Image (\wh(\ZH) \to \wh(\ZG)) \cong \wh(\ZH)\]
so we have a short exact sequence
$$0 \to \wh(\ZG)/\wh(\ZH) \to \wh(\ccat{W\times \rminus,G}{\bz})
\to \ck_W \to 0\ .$$
It follows from our $K${--}theory calculations that
$\ck_W = \ck_{\Wmax}$ and that
$\ck_W = \ck$ whenever
$\Kminus{1}(\ZK) =0$ for all $K\in \Iso(W)$.
\begin{theorem}\label{Cprime}
Let $V_1=t^{a_1} + \dots + t^{a_k} $
and $V_2 = t^{b_1} + \dots + t^{b_k}$ be
 free $G${--}representations.
Let $W$ be a  complex $G${--}representation
with no $\rplus$ summands.
Then there exists a topological similarity
 $V_1\oplus W \oplus \rminus \sim_t
V_2\oplus W \oplus \rminus$ if and only if
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $S(V_1)$ is $s${--}normally cobordant to $S(V_2)$,
\item $\Res_H (V_1\oplus W) \oplus \rplus
 \sim_t \Res_H (V_2\oplus W)\oplus \rplus$, and
\item the element
$\left\{ \Delta(V_1)/\Delta(V_2) \right\}$
is in the image of the coboundary
\[\delta\colon
H^0(\ck_{\Wmax}) \to
H^1(\wh(\ZG^{-})/\wh(\ZH)), \]
where $0\subseteq \Wmax \subseteq W$ is a
complex subrepresentation  of real dimension $\leq 2$ with  maximal
isotropy group among the isotropy groups
of $W$ with 2-power index.
\end{enumerate}
\end{theorem}
\begin{proof}
In this case, we have a torsion element $\hat\sigma
\in L_1^h(\bcatrminus{W_1})$ which maps to our surgery
obstruction $\trf{W\times \rminus} $ under the cone point inclusion.
As in the proof of Theorem C, the torsion subgroup of
 $L_1^h(\bcatrminus{W_1})$
injects into 
$H^1(\wh(\bcatrminus{W_1}))\cong H^1(\wh(\bcatrminus{W_1}))$.
Therefore $\hat\sigma \in L_1^h(\bcatrminus{W_1})$ vanishes
if and only if
the element
 $\left\{ \Delta(V_1)/\Delta(V_2) \right\}$
is in the image of the coboundary
$$\delta\colon
H^0(\ck_{\Wmax})  \to
H^1(\wh(\ZG^{-})/\wh(\ZH))$$
as required.
\end{proof}


\section{Iterated $\rminus$ transfers}
\label{rminustwo}

We now apply Theorem \ref{optrick} to show
that iterated $\rminus$ transfers do not lead to any new
similarities.
\begin{theorem}\label{sixteenone}
Suppose that $W$ is a complex $G${--}representation
with no $\rplus$ summands.
Then $V_1\oplus W \oplus \Rminus{l}\oplus \rplus
\sim_t V_2\oplus W \oplus \Rminus{l}\oplus \rplus$
for $l\geqq 1$ implies
$V_1\oplus W \oplus \rminus\oplus \rplus
\sim_t V_2\oplus W \oplus \rminus\oplus \rplus$.
\end{theorem}
The first step in the proof is to show
injectivity of certain transfer maps.
For any homomorphism $w\colon G \to \orient$,
we will use the notation $(\ccat{W,G}{\bz},w)$ to
denote the antistructure where the involution is
$g \mapsto w(g)g^{-1}$ at the cone point.
The standard orientation (\ref{standard}) has $w=\det(\rho_W)$,
but we will need others in this section.
Let $\phi=\det(\rho_\rminus)$ for short, and notice that
$\phi\colon G \to \orient$ is  non{--}trivial.
\begin{lemma}\label{sixteentwo}
The transfer map
$\trfrminus\colon L^p_{2k+1}(\ZG,w) \to
 L^p_{2k+2}(\ccat{\rminus,G}{\bz},w\phi)$
is injective, where $w$ is the non{--}trivial orientation.
\end{lemma}
\begin{proof}
Since
$L^p_{2k+2}(\ccat{\rminus,G}{\bz},w)\cong
 L^p_{2k+2}(\rel{\ZH}{\ZG})$,
we just have to check that  the previous map in the $L^p$ version of (\ref{bdLNseq})
is zero (using the tables in \cite{ht1}).
\end{proof}
Next, a similar result for the $\Rminus{2}$ transfer.
\begin{proposition}\label{sixteenthree}
The bounded transfer
\[\trf{\Rminus{2}}\colon L^p_{2k+1}(\ZG,w) \to
 L^p_{2k+3}(\ccat{\Rminus{2},G}{\bz},w)\]
is injective, where $w$ is the non{--}trivial orientation.
\end{proposition}
\begin{proof}
We can relate the iterated $\rminus$ transfer
$\trf{1}\circ \trf{1}$ to
$\trf{2}=\trf{\Rminus{2}}$ by means of the braid diagram:
\eqncount
\begin{equation}
\begin{matrix}
\xymatrix@!C@C-5.5pc{
L_n(\ZG,w)  \ar[dr]^{\trf{1}} \ar@/^2pc/[rr]^{\trf{2}}     
&& L_{n+2}(\rel{\bz\Gamma_H}{\ZG},w)       \ar[dr]
\ar@/^2pc/[rr]   &&
LNS_{n+3}(\Phi)  \\ 
&L_{n+1}(\rel{\ZH}{\ZG},w\phi) \ar[dr] \ar[ur]^{\trf{1}}  &&
LS_{n+3}(\Phi)  \ar[dr] \ar[ur]  \\
LNS_n(\Phi)   \ar[ur] \ar@/_2pc/[rr]_{}  &&
LN_{n+3}(\rel{\ZH}{\ZG},w\phi) \ar[ur] \ar@/_2pc/[rr]                         
&& L_{n+3}(\ZG,w) }\end{matrix}
\end{equation}

\noindent where the new groups $LNS_{\ast}(\Phi)$ are the
relative groups of the transfer
\[\trf{1}=\trfrminus\colon L^p_k(\bCatrminus,w\phi) \to
 L^p_{k+1}(\bcat{\Rminus{2}},w)\ .\]
The diagram $\Phi$ of groups (as in \cite[Chap.11]{w2})
contains $\Gamma_H \cong \bz\times H$ as the
fundamental group of $S(\rminus\oplus\rminus)/G$
(see \cite[5.9]{hp2}).

The groups $LNS_{\ast}(\Phi)$ have a geometric bordism
description in terms of triples $L^{n-2}\subset N^{n-1} \subset M^n$
where each manifold has fundamental group $G$, $N$ is
characteristic in $M$, and $L$ is characteristic in $N$, with
respect to the index two inclusion $H < G$.
There is also an algebraic description for $LNS_{\ast}(\Phi)$
in terms of ``twisted antistructures" \cite{htw3} as for the other
groups $LN(\rel{\ZH}{\ZG},w\phi)$ \cite[12C]{w2} and
for $LS(\Phi,w)$ \cite[7.8.12]{ra8}.
Substituting these descriptions into the braid gives:

\eqncount
\begin{equation}
\begin{matrix}
\xymatrix@!C@C-6pc{
L_{2k+1}(\ZG,\alpha,u)   \ar[dr]^{\trf{1}} \ar@/^2pc/[rr]^{\trf{2}}      &&
L_{{2k+1}}(\rel{\bz\Gamma_H}{\ZG},\tilde\alpha,\tilde u)      \ar[dr] \ar@/^2pc/[rr]     &&
L_{{2k}}(\rel{\ZH}{\bz\Gamma_H},\tilde \alpha,\tilde u)       \\
&   L_{2k+1}(\rel{\ZH}{\ZG},\alpha,u)                        \ar[dr] \ar[ur]^{\trf{1}}    &&
L_{{2k}}(\bz\Gamma_H,\tilde\alpha,\tilde u)               \ar[dr] \ar[ur]     \\
L_{2k+1}(\rel{\ZH}{\bz\Gamma_H},\tilde \alpha,\tilde u)       \ar[ur] \ar@/_2pc/[rr]_{}    &&
L_{2k}(\ZH,\alpha,u)                                   \ar[ur] \ar@/_2pc/[rr]     &&
L_{2k}(\ZG,\alpha,u)
}\end{matrix}
\end{equation}

\noindent The antistructure $(\ZG,\alpha,u)$ is the twisted antistructure
obtained by scaling with an element  $a\in G - H$.
The antistructure $(\bz\Gamma_H,\tilde\alpha,\tilde u)$ is
the one defined by Ranicki \cite[p.805]{ra8}, then
scaled by $\tilde a \in \Gamma_H$, where $\tilde a$
maps to $a$ under the projection $\Gamma_H \to G$.
Since $\Gamma_H \cong \bz\times H$, we have an exact
sequence (see \cite[Theorem 4.1]{milgran2})
\[\to L^p_n(\ZH,\alpha,u) \to
 L^p_n(\bz\Gamma_H,\tilde\alpha,\tilde u)\to
 \Lminus{1}_{n-1}(\ZH,\alpha,u)\RA{1-w(a)}
L^p_{n-1}(\ZH,\alpha,u)\]
and it follows that
$L^p_n(\rel{\ZH}{\bz\Gamma_H},\tilde \alpha,\tilde u)
\cong \Lminus{1}_{n-1}(\ZH,\alpha,u)$.
It is not difficult to see that
$\Lminus{1}_{2k}(\ZH,\alpha,u)$ is torsion{--}free (except for
the Arf invariant summand)
by a similar argument to \pone, Theorem \ref{Lfacts},
using the $L^p$ to $\Lminus{1}$ Ranicki-Rothenberg sequence.
We  first check that
 $L^p_{2k}(\ZH,\alpha,u)$ is torsion{--}free (again except
for the Arf invariant summand)  from the tables
in \cite[14.21]{ht1}.

Now observe that $L^p_n(\ZG,\alpha,u)\cong L^p_n(\ZG,w)$,
and
\[L^p_{2k+1}(\rel{\ZH}{\ZG},\alpha,u) \cong L^p_{2k+2}(\rel{\ZH}{\ZG})\ .\]
The transfer
 $\trfrminus\colon L^p_{2k+1}(\ZG,w) \to  L^p_{2k+2}(\rel{\ZH}{\ZG})$
is  injective by Lemma \ref{sixteentwo}, and
since $w(a)=-1$, the map
\[ \Lminus{1}_{2k}(\ZH,\alpha,u)\RA{1-w(a)}
L^p_{2k}(\ZH,\alpha,u)\] is also injective (except for the Arf invariant summand).
Therefore $ L^p_{2k+1}(\ZG,w)$ must
inject into the relative group
$L^p_{2k+1}(\rel{\bz\Gamma_H}{\ZG},\tilde\alpha,\tilde u)\cong
L^p_{2k+3}(\bcat{\Rminus{2}},w)$.
\end{proof}
The final step in the proof is to consider the following
 commutative  square of spectra:
\[
\xymatrix{\BL (\ccat{pt}{\ZG},w)
\ar[r]^{c_{*}\ }\ar[d]_{\trfrminus}
&\BL (\bcat{V},w)
\ar[d]^{\trfrminus}\\
\ \BL (\ccat{\rminus,G}{\bz},w\phi)\ar[r]^{c_{*}\ }
&\ \BL (\bcat{V\times \rminus},w\phi)
}
\]
for any $G$-representation $V$ with no $\rplus$ summands.
\begin{lemma}\label{sixteenfour}
This is a pull{--}back square of $\Lminus{\infty}$  spectra.
\end{lemma}
\begin{proof}
The fibres of the vertical $\rminus$ transfers are
\[\mathbb{LN}(\bCatrminus, w\phi) \cong
\BL(\ccat{pt}{\ZH ,\alpha,u})\ ,\] and
$\mathbb{LN}(\bcatrminus{V},w\phi) \cong
\BL(\ccat{V,H}{\bz},\alpha,u)$ respectively by
(\ref{bdLNseq}),
and the fibre of the cone point inclusion
\[\BL(\ccat{pt}{\ZH},\alpha,u) \to \BL(\ccat{V,H}{\bz},\alpha,u)\]
is  $\BL(\Ccat{V,H}{0}{\bz},\alpha,u)$.
But Theorem \ref{optrick} shows that
$\BL(\Ccat{V,H}{0}{\bz},\alpha,u)$ is contractible, and
therefore the cone point inclusion is a homotopy equivalence.
It follows that the fibres of
the horizontal maps are also homotopy equivalent.
\end{proof}
\begin{proof}[The proof of Theorem \ref{sixteenone}]
It is enough to prove that $\trf{W\times\Rminus{l+1}}(\sigma)=0$, for
$l\geqq 1$, implies $\trf{W\times\rminus}(\sigma)=0$ in the top
component of $L^p_{2k+1}(\bcatrminus{W})$. We may assume that
$\Iso(W)$ only contains subgroups of $2${--}power index, and let $\dim
W =2k$. We may also assume that $l$ is even, by crossing
with another $\rminus$ if necessary.
Now by Theorem C, $\trf{W\times\Rminus{2s+1}}(\sigma)=0$
implies $\trf{W\times\Rminus{2s-1}}(\sigma)=0$,
provided that $s\geqq 2$.
It therefore remains to study  $l=2$.

The pullback squares provided by Lemma \ref{sixteenfour} can
be combined as follows. Consider the diagram of 
$\Lminus{\infty}$-spectra

$$\xymatrix{
\BL  (\ccat{pt}{\ZG},w)\ar[r]^{c_{*}\ }\ar[d]_{\trfrminus}&\BL  (\ccat{\rminus,G}{\bz},w\phi)
\ar[r]^{c_{*}\ }\ar[d]_{\trfrminus}&
\BL  (\ccat{W\times\Rminus{2},G}{\bz},w)\ar[d]_{\trfrminus}\cr
\BL  (\ccat{\rminus,G}{\bz},w\phi)\ar[r]^{c_{*}\ }&\BL  (\ccat{\Rminus{2},G}{\bz},w)\ar[r]^{c_{*}\ }&
\BL  (\ccat{W\times\Rminus{3},G}{\bz},w\phi)
}$$
whose outer square ($V=W\times\Rminus{2}$)
 and the left-hand square ($V=\rminus$) are both pullback squares, and hence so is the right-hand
square. Next consider the diagram of $\Lminus{\infty}$-spectra
$$\xymatrix{
\BL  (\ccat{pt}{\ZG},w)\ar[r]^{c_{*}\ }\ar[d]_{\trfrminus}&
\ \BL  (\ccat{W\times\rminus,G}{\bz},w\phi)\ar[d]_{\trfrminus}\cr
\BL  (\ccat{\rminus,G}{\bz},w\phi)
\ar[r]^{c_{*}\ }\ar[d]_{\trfrminus}&
\ \BL  (\ccat{W\times\Rminus{2},G}{\bz},w)\ar[d]_{\trfrminus}\cr
\BL  (\ccat{\Rminus{2},G}{\bz},w)\ar[r]^{c_{*}\ }&
\ \BL  (\ccat{W\times\Rminus{3},G}{\bz},w\phi)
}$$
The lower square was just shown to be a pull-back, and the upper
square is another special case of Lemma \ref{sixteenfour}
(with $V= W\times\rminus$). Therefore the outer square is
a pull-back, and this is the one used for the case $l=2$.


Now we apply homotopy groups to these pull-back squares
(using the fact that $\Lminus{\infty} \equiv \Lminus{1}$  to obtain  the lower
squares in the commutative diagram:
\[
\xymatrix{L^p_{2k+2}(\brelcat{W\times\rminus}{0})\ar[r]\ar[d]^{\cong}
&L^p_{2k+1}(\ZG,w)\ar[r]^{c_{\ast}}\ar[d]
&L^p_{2k+1} (\bcat{W\times\rminus},w)\ar[d]\\
\Lminus{1}_{2k+2}(\brelcat{W\times\rminus}{0})
\ar[r]\ar[d]^\cong
&\Lminus{1}_{2k+1}(\ZG,w)\ar[r]^{c_{\ast}}\ar[d]_{\trf{\Rminus{2}}}
&\Lminus{1}_{2k+1} (\bcat{W\times\rminus},w)\ar[d]^{\trf{\Rminus{2}}} \\
\Lminus{1}_{2k+4}(\brelcat{W\times\Rminus{3}}{0})\ar[r]
&\Lminus{1}_{2k+3}(\ccat{\Rminus{2},G}{\bz},w)\ar[r]^{c_{\ast}}&
\Lminus{1}_{2k+3}(\bcat{W\times \Rminus{3}},w)
}
\]
 The vertical maps in the top row squares are induced by the change of $K$-theory decoration. 
We need some information 
about the maps in this diagram.

\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item The upper left-hand vertical
map is an isomorphism, since 
$\Kminus{1}(\brelcat{W\times\rminus}{0})=0$.

\item The lower left-hand vertical map is an isomorphism, 
by Lemma \ref{sixteenfour}.

\item The transfer map
$$\trf{\Rminus{2}}\colon L^p_{2k+1}(\ZG,w) \to
 L^p_{2k+3}(\bcat{\Rminus{2}},w)$$ is injective, from
Proposition \ref{sixteenthree}.

\item 
The map 
$$L^p_{2k+3}(\ccat{\Rminus{2},G}{\bz},w)
\to \Lminus{1}_{2k+3}(\ccat{\Rminus{2},G}{\bz},w)$$ is
injective, by \pone, Corollary \ref{rstable}.

\item The composite of the middle two vertical maps
in the diagram is injective,  by combining
parts (iii)-(iv).

\end{enumerate}

\noindent
Suppose 
 that  $\trf{W\times\Rminus{3}}(\sigma)=0$.
We have proved that the transfer map
\[\trf{\Rminus{2}}\colon L^p_{2k+1}(\ZG,w) \to
\Lminus{1}_{2k+3}(\ccat{\Rminus{2},G}{\bz},w)\]
is injective in part (v).
Since $\trf{W\times\rminus}(\sigma) = c_{\ast}(\hat\sigma)$
for some $\hat\sigma \in L^p_{2k+1}(\ZG,w)$, 
 it follows  by a diagram chase that 
 $\trf{W\times\rminus}(\sigma)=0$ and we are done.
\end{proof}
We also have a version without $\rplus$ summands.
\begin{theorem}\label{sixteenfive}
Suppose that $W$ is a complex $G${--}representation
with $W^G=0$.
Then $V_1\oplus W \oplus \Rminus{l}
\sim_t V_2\oplus W \oplus \Rminus{l}$
for $l\geqq 3$ implies
$V_1\oplus W \oplus \Rminus{3}
\sim_t V_2\oplus W \oplus \Rminus{3}$.
\end{theorem}
\begin{proof}
This follows from a similar argument, using Theorem \ref{Cprime}
instead of Theorem C.
The injectivity results of Lemma \ref{sixteentwo} and
Proposition \ref{sixteenthree} also hold for $\trf{1}$ and $\trf{2}$
on $L^{\bu}_{2k+1}(\ZG,w)$,
where $\bU=\Image(\widetilde K_0(\ZH) \to \widetilde K_0(\ZG))$
as in Remark \ref{sharper}. This $L$-group decoration  fits
in exact sequences with $L^p_{n}(\ZH)$ so the previous calculations
for injectivity apply again.
The details are left to the reader.
\end{proof}



\section{The proof of Corollary \ref{sixdim}}
\label{sixdiml}
Suppose that $G=\Cy{4q}$, $q$ odd, and that $V_1\oplus \rminus\oplus
\rplus\sim_t V_2\oplus \rminus\oplus
\rplus$ with $\dim V_i =4$.
We will use Theorem B and the assumption about odd class numbers
to prove that $V_1\cong V_2$.
This is the only case we need to discuss to prove Corollary
\ref{sixdim}. If we started instead with
$V_1\oplus \Rminus{2}\sim_t V_2\oplus \Rminus{2}$,
then stabilizing with $\rplus$ and applying Theorem \ref{sixteenone} would reduce
to the case above.


We may assume that $q > 1$ is a prime, and that the free
representations have the form
 $V_1 = t + t^i$ and $V_2=t^{1+2q} + t^{i+2q}$
where $(i,2q)=1$ (not all weights are $\equiv 1 \Mod 4$).
The Reidemeister torsion invariant is
$\Delta(V_1)/\Delta(V_2) = U_{1,i} \in \wh(\ZG)$
in the notation for units of $\ZG$ used in
\cite[p. 732]{cssw1}.
With respect to
 the involution $\Bar{\hphantom{a}}\colon t \mapsto -t^{-1}$, this element
defines a class $u_{1,i} = \{U_{1,i}\} \in H^1(\wh(\ZG^{-}))$.
Since the map $L^p_1(\ZG^{-}) \to L^p_1(\bCatrminus)$ is injective,
it is enough to show that the image of  $u_{1,i}$ is non{--}trivial
in $\Coh{1}{\Delta}$ (see diagram \pone\ (\ref{lpbraid})).

Let $A = \bz[\zeta_{4q}]$ be the ring of cyclotomic integers.,
and $B=\bz[\zeta_{2q}]$.
We will study the top component $\wh(\ZG)(q)$ by
comparing it to $K_1(\cm)$, where $\cm = A \times B\times B$
is the top component of an involution invariant maximal order
in $\QG$ containing $\ZG$.
Notice that the two copies of $B$ are
interchanged under the involution, so
$\Coh{i}{K_1(\cm)} = \Coh{i}{\Units{A}}$.
We have the exact sequence
\[0 \to \wh(\ZG)(q) \to K_1(\cm)
\to K_1( \cmlocal)/\wh(\zlocal G)(q) \to D(\ZG)(q)\to 0\ .\]
where $D(\ZG) =\ker(\widetilde K_0(\ZG) \to \widetilde K_0({\cm}))$.
Note that $H^i(D(\ZG)) =H^i(\widetilde K_0(\ZG))$
since $A$ has odd class number.

Let $i_{\ast}\colon \Coh{1}{\wh(\ZG^{-})(q)} \to \Coh{1}{\Units{A}}$
denote the map on the Tate cohomology
induced by the inclusion $ i\colon\wh(\ZG)(q)\to \Units{A}$.
\begin{lemma}\label{seventeenone}
For  $(j,2q)=1$ and $j\equiv 1\Mod{4}$,
the image $i_{\ast} (u_{1,j}) = \langle -1\rangle \in H^1(\Units{A})$.
\end{lemma}
\begin{proof}
Let $\gamma_j = \frac{t^j-1}{t-1} \in \Units{A}$, and compute
$\bar \gamma_j/\gamma_j$ with respect to the
non{--}oriented involution.
We get $\bar \gamma_j/\gamma_j = u_{1,j}/u_{1,1}$,
so $i_{\ast}(u_{1,j}) = i_{\ast}(u_{1,1})$ for all $i$.
Now let $v = \frac{t+1}{t-1}$.
Since $t=-t^{1+2q}$, this is a cyclotomic
unit in $A$ with $v\bar v = -1$.
But $v/\bar v = -u_{1,1}$, so $i_{\ast} (u_{1,j}) = \langle -1\rangle$
in $H^1(\Units{A})$.
\end{proof}
Next we need a computation:
\begin{lemma}
 $ 0 \neq \langle -1\rangle \in H^1(\Units{A})$.
\end{lemma}
\begin{proof}
Let $E = \bq(\zeta_{4q})$ be the quotient field of $A$.
Since $A$ has odd class number,
$\Coh{i}{\Units{E}/\Units{A}} \cong
\Coh{i}{\localunits{E}{}{}/\localunits{A}{}{}}$.
Next, observe that the extension $E/F$ is evenly ramified at $q$,
where $F$ is the fixed field of the involution
$\Bar{\hphantom{a}}\colon E \to E$.
It follows that the map $\Coh{0}{\localunits{E}{}{q}}
\to \Coh{0}{\localunits{E}{}{q}/\localunits{A}{}{q}}$
is zero.
We finish by considering the commutative diagram
\[
\xymatrix{\Coh{0}{\Units{E}}\ar[r]\ar[d]&
\Coh{0}{\Units{E}/\Units{A}}\ar[r]^{\delta}\ar[d]^{\approx}&
\Coh{1}{\Units{A}}\ar[d]\\
\Coh{0}{\localunits{E}{}{}}\ar[r]&
\Coh{0}{\localunits{E}{}{}/\localunits{A}{}{}}\ \ar@{>->}[r]&
\Coh{1}{\localunits{A}{}{}}
}
\]
and the prime element $w=\zeta_q - \zeta_q^{-1} \in E$
over $q$.
Since $\bar w/w = -1$,  $\omega$  defines an element in
$\Coh{0}{\Units{E}/\Units{A}}$ and
$\delta(w) = \langle -1\rangle \in H^1(\Units{A})$.
On the other hand,  $w$ maps under the middle isomorphism
to a non{--}zero element in
$\Coh{0}{\localunits{E}{}{q}/\localunits{A}{}{q}}$ so
$\delta(w) \neq 0$.
\end{proof}
Now we can complete the proof of Corollary \ref{sixdim} by considering
the commutative braid:

\eqncount
\begin{equation}
\begin{matrix}
\xymatrix@!C@C-4pc{
\Coh{0}{\Units{A}}                                                         
\ar[dr]           \ar@/^2pc/[rr]                &&
H^0\left (\frac{K_1(\cmlocal)}{\wh(\zlocal G)(q)}\right )       \ar[dr]
\ar@/^2pc/[rr]                          &&
\Coh{0}{\widetilde K_0(\ZG^{-})(q)}                                            
                                                \\
&  H^0\left (\frac{K_1(\cm)}{\wh(\ZG)(q)}\right )                  \ar[dr]
\ar[ur]                                 &&
\Coh{1}{\Delta}                                                            
\ar[dr] \ar[ur]                    \\
\Coh{1}{\widetilde K_0(\ZG^{-})(q)}                                             \ar[ur]
\ar@/_2pc/[rr]^{\delta^2}               &&
\Coh{1}{\wh(\ZG^{-})(q)}                                             
\ar[ur] \ar@/_2pc/[rr]^{i_*}                    &&
\Coh{1}{\Units{A}}
}\end{matrix}
\end{equation}

\noindent containing the double coboundary and its relative group
$\Coh{1}{\Delta}$.
By commutativity of the lower right triangle, the image of
our obstruction
element $u_{1,j}$ is non{--}zero in $\Coh{1}{\Units{A}}$
and therefore non{--}zero in $\Coh{1}{\Delta}$.


\section{The normal invariant}
\label{normalinvariant}

In this section we collect some results about
 normal cobordisms of lens spaces with cyclic fundamental group
$G$. Recall that a necessary condition for the existence of
a similarity $V_1\oplus W \sim_t V_2\oplus W$ is that $S(V_1)$ and
$S(V_2)$ must be $s$-normally cobordant.
In \cite[\S 1]{cssww1} it is asserted
that their formula $(A')$ gives necessary and sufficient
conditions for homotopy equivalent lens spaces
to be $s${--}normally cobordant, when $G = \Cy{2^r}$.
However in \cite[1.3]{yo1} it is shown that the given conditions $(A')$
are sufficient but not necessary.
We only use the sufficiency here, and 
 study the subgroups
$\rtildefree{G}$ and $\Rtildefree_h(G)$ of $R(G)$
defined in Section \ref{splitting}.
Recall that 
 $\rtildefree{G} = \ker(\Res\colon\rfree{G} \to \rfree{\Godd})$.
\begin{lemma}\label{twentyone}
Let $G = \Cy{ 2^r q}$ be a finite cyclic group, and
$G_2$ be the 2{--}Sylow subgroup.
If $q >1$ there is an exact sequence
\[0 \to \cy 2 \to  \rtildefree{G}/ \Rtildefree_h(G) \to
\Rtildefree(G_2)/ \Rtildefree_h(G_2) \to 0\]
given by the restriction map $\Res _{G_2}$.
The kernel is generated by any element
$\alpha \in  \rtildefree{G}$ with  $k${--}invariant
$k(\alpha) \in \Units{(\cy{|G|})}/\{\pm 1\}$
in the coset   $ k \equiv 1\Mod{2^r}$ and
$k \equiv -1\Mod{q}$.
\end{lemma}
\begin{proof}
There is a short exact sequence
\[0 \to  \Rtildefree_h(G) \to  \rtildefree{G} \to
 \Units{(\cy{ 2^rq})} /\{ \pm 1\}\]
given by the $k${--}invariant.
Moreover,
the  $k${--}invariants of elements in  $\rtildefree{G} $
lie in
\[\ker \bigl (  \Units{(\cy{2^rq})}/\{ \pm 1\}
 \to  \Units{(\cy{q})} /\{ \pm 1\}\bigr )
\cong \{lq+1\Vertical l \in \bz\}
\subseteq   \Units{(\cy{2^rq})}\]
which injects  under reduction $\Mod{2^r}$ into
$\Units{(\cy{2^r})}$ if $q >1$.
Restriction of the $k${--}invariant to $G_2$ detects only
its value in  $\Units{(\cy{2^r})} /\{ \pm 1\}$, so the
kernel has order 2.
Since the restriction map
$\Rtildefree{G} \to  \Rtildefree(G_2)$
is surjective, we have the required exact sequence.
\end{proof}
The situation for the normal invariant is simpler.
Recall that we write $V \bumpeq V'$  or $(V-V')\bumpeq 0$
if  there exists a homotopy equivalence
$f\colon S(V)/G \to S(V')/G$ of lens
spaces, such that $f$ is $s${--}normally cobordant to the identity.
\begin{lemma}\label{twentytwo}
Let $G = \Cy{2^rq}$ be a finite cyclic group.
Then
\[
\ker \bigl (\Res\colon \Rfree(G) \to
 \Rfree(\Cy{ q})\oplus \Rfree(\Cy{2^r})\bigr )
\subseteq \Rfree_n(G)\ .
\]
\end{lemma}
\begin{proof}
We may assume that $r\geq 2$, since
$\Res\colon  \Rfree(\Cy{2q}) \to  \Rfree(\Cy{q})$
is an isomorphism, and consider an element
\[\sum t^{a_i} - \sum t^{b_i} \in \ker \bigl (\rfree{G} \to
\Rfree(\Cy{q})\oplus\Rfree(\Cy{2^r})\bigr ).\]
With a suitable ordering of the indices, $a_i \equiv b_{i}
\Mod{2^r}$ and
$a_i \equiv b_{\tau(i)}\Mod{q}$ for some
permutation $\tau$.
Since the normal invariants of lens spaces are detected
by a cohomology theory, we can check the condition
at each Sylow subgroup separately.
It follows that $\sum t^{a_i} \bumpeq \sum t^{b_i}$.
\end{proof}
\begin{example}
For $G=\Cy{24}$ we have $\Rtildefree(G) = \{t-t^5, t-t^7, t-t^{11}\}$
and the subgroup 
$\ker\Res_{\Cy{8}}\cap \Rtildefree(G) = \{ t-t^7, t^5 - t^{11}\}$.

\end{example}

\begin{lemma}\label{twentythree}
Let $G = \Cy{2^rq}$ be a finite cyclic group, and
$G_2$ be the 2{--}Sylow subgroup.
If $r \geq 2$ there is an isomorphism
\[ \Rtildefree_h(G)/ \Rtildefree_n(G) \to
\Rtildefree_h(G_2)/ \Rtildefree_n(G_2) \]
given by the restriction map $\Res _{G_2}$.
\end{lemma}
\begin{proof}
We first remark that if
$W = t^{a_1} + \dots + t^{a_n}$ is a free
 $G_2${--}representation (here we use the assumption
that $r\geq 2$), then by choosing
integers $b_i \equiv a_i\Mod{2^r}$ and
$b_i \equiv 1\Mod{q}$ we obtain a free
$G${--}representation
$V = t^{b_1} + \dots + t^{b_n}$  with
$\Res _{G_2}(V) = W$.
It follows that
\[ \Res _{G_2}\colon \Rtildefree_h(G) \to
\Rtildefree_h(G_2) \]
is surjective, and therefore
\[ \Res _{G_2}\colon \Rtildefree_h(G)/ \Rtildefree_n(G)) \to
\Rtildefree_h(G_2)/ \Rtildefree_n(G_2) \]
is also surjective.

Now suppose $\alpha \in \Rtildefree_h(G)$
and $\Res _{G_2}(\alpha) \in \Rtildefree_n(G_2)$.
This means that $\Res _{G_2}(\alpha) = (W -W')$  for some
free ${G_2}${--}representation $W$, $W'$ such
that $W \bumpeq W'$.
By the construction of the last paragraph, we can
find free $G${--}representations $V$, $V'$
such that
(i) $\Res _{G_2}(V) = W$ and $\Res _{G_2}(V') = W'$, and
(ii) $\alpha'=(V - V') \in \Rtildefree_h(G)$.
It follows that
$V\bumpeq V'$ and so $\alpha'  \in
\Rtildefree_n(G))$, and
\[\alpha - \alpha' \in
\ker \bigl (\Rfree_h(G) \to
 \Rfree_h(\Cy{q})\oplus \Rfree_h(\Cy{2^r})\bigr ).\]
By Lemma \ref{twentytwo}, $\alpha - \alpha' \in \Rtildefree_n(G)$
and so $\alpha \in \Rtildefree_n(G)$.
\end{proof}
Our final result is a step towards determining $R_h(G)/R_n(G)$
more explicitly.
\begin{lemma}\label{twentyfour}
Let $ b_{i,s}=(t^i - t^{2^{r-s}q-i}) \in \rtildefree{G}$
for $1\leq i <2^{r-s-1}q$, $1\leq s \leq r-1$, and $ (i,2q) =1$.
Let $l(i,s) = s$ for $1\leq s \leq r-2$, and
$l(i,r-1)$ the order of $i/(2q-i) \in \Units{(\cy{2^r})}$.
Then $b_{i,s}$ is an element of order $2^{l(i,s)}$
in  $R(G)/R_h(G)$, and
 $2^{l(i,s)}\cdot b_{i,s}\bumpeq 0$.
\end{lemma}
\begin{proof}
If $1\leq s\leq r-2$ then $k(b_{i,s}) = i/(2^{r-s}q-i) \in
 \Units{(\cy{2^rq})}/\{ \pm 1\}$ is congruent to $-1\Mod{4q}$.
By Lemma \ref{twentyone}  the linear span of these elements
in $R(G)/R_h(G)$ injects into $R({G_2})/R_h({G_2})$, where
${G_2}$ is the 2{--}Sylow subgroup.
By Lemma \ref{twentythree} the normal invariant  is also detected
by restriction to ${G_2}$, so it is enough to prove the
assertions about these elements ($s\leq r-2$) when $G=\Cy{2^r}$.


For the first part we must show that the $k${--}invariant
of $b_{i,s}$ has order $2^{s}$.
We will use the expression
\[\nu_2(r!) = r-\alpha_2(r),\]
where $\alpha_2(r)$ is the number of non-zero  coefficients
in the 2-adic expansion of $r$, for the 2-adic valuation of $r$.
Then
\[\nu_2(\binom{2^s}{k}) = s-\nu_2(k)\]
for $1\leq k\leq 2^s$.
These formulas and the binomial expansion show that
$k(b_{i,s})^{2^{s}} \equiv 1 \Mod{2^r}$ for $1\leq s\leq r-2$.

Next we consider the normal invariant.
For $1\leq s\leq r-2$ we will take $q=1$ and
 apply the criterion $(A')$ of \cite{cssww1}
to show that $2^{s}\cdot b_{i,s}\approx 0$.
This amounts to a re-labelling of our original
elements $b_{i,s}$ without changing the order
of their $k${--}invariants.

We must now compute the elementary
symmetric functions $\sigma_k(2^s \cdot i^2)$
and $\sigma_k(2^s \cdot (2^{r-s}-i)^2)$,
where the notation $2^s \cdot i^2$ means that the weight
$i^2$ is repeated $2^s$ times in the symmetric function.
The formula $(A')$ is
\[\sigma_k(2^s \cdot (2^{r-s}+i)^2) -
\sigma_k(2^s \cdot i^2) \equiv 2\bigl ( (2^{r-s}-i)^{2^s}
- i^{2^s}\bigr )\binom{2^s-1}{k-1} \Mod{2^{r+3}}\]
so on the right-hand side we have
\[
\begin{cases} +2^{r+1} \Mod{2^{r+3}}& \text{if $k$ odd}\\
 -2^{r+1}\Mod{2^{r+3}} &\text{if $k$ even}
\end{cases}
\]
The formula $(A')$ assumes that the weights are congruent
to $1 \Mod{4}$, or in our case $i \equiv 1\Mod{4}$.
Therefore, if $i \equiv 3\Mod{4}$, we must use the equivalent
weights $-i$ and $-(2^{r-s}-i)$.

To compute the left-hand side we use the Newton polynomials
$s_k$ and their expressions in term of elementary symmetric
functions. We need the property $s_k(2^s\cdot i^2) = 2^s  i^{2k}$
and the coefficient of $\sigma_k$ in $s_k$
which is $(-1)^{k+1}k$.
By induction, we see that the left-hand side is just
\[\frac{(-1)^{k+1}}{k} (s_k(2^s \cdot (2^{r-s}-i)^2) -
s_k(2^s \cdot i^2)\bigr ) \equiv \frac{(-1)^{k+1}2^s}{k}
\bigl ( (2^{r-s}-i)^{2k} - i^{2k}   )\Mod{2^{r+3}}\]
But by writing
\[(2^{r-s}-i)^{2k} - i^{2k}  = (2^{r-s}\theta + i^2)^k - i^{2k}\]
where $\theta = 2^{r-s} - 2i \equiv 2 \Mod{4}$, our expression
becomes
\[\frac{(-1)^{k+1}2^s}{k}(2^{r-s}k\theta)
\equiv (-1)^{k+1}2^{r+1}\Mod{2^{r+3}}\]


If $s = r-1$, then $k(b_{i,s}) = i/(2q-i) \equiv -1 \Mod{q}$,
and $k(b_{i,s}) \equiv 1 \Mod{2^r}$ whenever $i \equiv q
\Mod{2^{r-1}}$. This is for example always the case
for $G = \Cy{4q}$.
Such  elements
$b_{i,r-1}$ lie in the kernel of the restriction map to $R(\Cy{2^r})$.
Moreover, if $b_{i,r-1}
 \in \ker\bigl (R(G) \to R(\Cy{2^r})\bigr )$, then
$2b_{i,r-1} \in R_h(G)$
and $2b_{i,r-1}\bumpeq 0$ by Lemma \ref{twentytwo}.
Otherwise, the order of $b_{i,r-1}\in R(G)/R_h(G)$ is
the same as the order of  its restriction
 $\Res (b_{i,r-1}) $ to the 2{--}Sylow subgroup,
and $\Res (b_{i,r-1}) = \pm \Res (b_{j,s})$ for
$l(i,r-1)=s \leq r-2$ and some $j$.
\end{proof}
\begin{remark} As pointed out by the referee, Lemma 12.1 and
Lemma 12.4 together give a short exact sequence
$$0 \to \cy 2 \to \Rtildefree(G)/ \Rtildefree_n(G) 
\to  \Rtildefree(G_2)/ \Rtildefree_n(G_2) \to 0, $$ assuming
that $G$ is not a cyclic $2$-group. It is not difficult to see from
the proof of Theorem E (given in \pone), that for $G_2=\Cy{2^r}$, 
and $r\geqq 4$, the term
$ \Rtildefree(G_2)/ \Rtildefree_n(G_2)$ is the quotient
of $\rtildetopfree{\Cy{2^r}}$ by the subgroup $\langle \alpha_1 +\beta_1\rangle$.
\end{remark}

\section{The proof of Theorem D}
\label{sD}
We first summarize our information about $\rtop{G}$, obtained
by putting together results from previous sections.
If $\alpha \in \Rtildefree_{h,\Top}(G)$, we define 
the \emph{normal invariant order} of $\alpha$ to be the minimal
$2${--}power such that $2^t\alpha \in \Rtildefree_{n,\Top}(G)$.
\begin{theorem}\label{oneD}
Let $G=\Cy{2^rq}$, with $q$ odd, and $r \geqq 2$.
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item The torsion subgroup of $\rtopfree{G}$ is $\rtildetopfree{G}$.
\item The rank of $\rtopfree{G}$ is $\varphi(q)/2$ for $q >1$
 (resp. rank 1 for $q=1$), and the torsion is at most $2${--}primary.
\item The subgroup $\Rtildefree_{n,\Top}(G)$  has exponent two,
and the Galois action induced by group automorphisms is the identity.
\item For any $\alpha \in \Rtildefree_{h,\Top}(G)$,
if the normal invariant order of $\Res_H(\alpha)$ is $2^t$,
then the normal invariant order of $\alpha$ is $2^{t+1}$.
\end{enumerate}
\end{theorem}
\begin{remark}
In part (ii), $\varphi(q)$ is the Euler function.
The precise number   of $\cy 2$ summands in
$\Rtildefree_{n,\Top}(G)$ is determined by working out
the conditions in Theorem C on the basis elements of
$\Rtildefree_n(G)$.
In cases where the conditions in Theorem C can
actually be evaluated,
the structure of $\rtop{G}$ will thus be determined completely.
\end{remark}
\begin{proof}
Parts (i) and (ii) of Theorem \ref{oneD} have already been proved in
Corollary \ref{cyclicranks}, so it remains to discuss parts (iii) and (iv).
In fact, the assertion that $\Rtildefree_{n,\Top}(G)$
has exponent $2$ is an immediate consequence of
Theorem C.
To see this, suppose that $(V_1 - V_2)$ is any element
 in $\Rtildefree_{n}(G)$.
The obstruction
to the existence of a stable non{--}linear similarity
$V_1\approx_t V_2$ is determined by the class
$\{\Delta(V_1)/\Delta(V_2)\}$ in  the Tate cohomology group
$H^1(\wh(\ZG^{-})/\wh(\ZH))$, which has exponent $2$.
Since the Reidemeister torsion is multiplicative,
$\Delta (V_1 \oplus V_1) = \Delta(V_1)^2$, and
we conclude that $V_1\oplus V_1\approx_t V_2\oplus V_2$
by Theorem C.
Finally, suppose that $\alpha \in \Rtildefree_{n, \Top}(G)$ and
that $\sigma$ is a group automorphism of $G$. By induction, we
can assume that $\Res_H(\alpha - \sigma(\alpha)) = 0$.
We now apply Theorem C to $\beta=\alpha - \sigma(\alpha)$,
with  $W$  a complex
$G${--}representation such that $W^G=0$, containing all the
non-trivial irreducible representations of $G$ with isotropy of 
$2$-power index. Then $\beta$ is
detected by  the image of its Reidemeister torsion invariant
in $H^1(\Delta_{W\times\rminus})$. But by   \pone,  Lemma
\ref{galoisinvariance}, the Galois action on this group is trivial.
Therefore $\beta=\alpha - \sigma(\alpha)=0$.

For part (iv) we recall that the normal invariant for $G$ lies in
a direct sum of groups $H^{4i}(G;\bz_{(2)}) \cong \cy{2^r}$.
Since the map $\Res_H$ induces the natural projection
$\cy{2^r} \to \cy{2^{r-1}}$ on group cohomology, the result follows
from \cite[2.6]{cs1}.
\end{proof}

It follows from our results that
the structure of
$\Rtildefree_{n,\Top}(G)$ is determined
by working out the criteria of Theorem C on a basis of
$\Rtildefree_n(G)$.
Suppose that  $V_1$ is stably topologically equivalent to $V_2$.
Then there exists a similarity
\[V_1\oplus W \oplus \Rminus{l} \oplus \Rplus{s}
\sim_t V_2\oplus W \oplus \Rminus{l} \oplus \Rplus{s}\]
where $W$ has no $\rplus$ or $\rminus$ summands,
and $l,s\geqq1$.
But by \pone, Corollary \ref{rstable} we may assume that $s=1$,
and by Theorem \ref{sixteenone} that $l=1$, so we are reduced
to the situation handled by
Theorem C. The algebraic indeterminacy given there is computable,
but not very easily if the associated cyclotomic fields have
complicated ideal class groups. We carry out the computational details in one
further case of interest (Theorem D).

\begin{proof}[The proof of Theorem D]
We have a basis
\[\mathcal B =\{t^i - t^{i+2q}\mid (i,2q)=1, i\equiv 1\Mod{4}, 1\leqq i < 4q\}\]
for $\rtildefree{G}$, so it remains to work out the relations
given by topological similarity.
Notice that $\Rtildefree_h(G) = \Rtildefree_n(G)$, and that
the sum of any two elements in $\mathcal B$ lies in $\Rtildefree_n(G)$.
Moreover, by Corollary \ref{sixdim} there are no $6${--}dimensional similarities
for $G$.


Now suppose that $V_1\oplus W\oplus \rminus \oplus \rplus
\sim_t V_2\oplus W\oplus \rminus \oplus \rplus$, for some
complex $G${--}representation $W$.
Then $\Res_H V_1 \cong \Res_H V_2$ since $q$ is odd, and
by Theorem C we get a similarity of the form
$V_1\oplus \Rminus{l} \oplus
 \rplus \sim_t V_2\oplus \Rminus{l} \oplus \rplus$.
But by Theorem \ref{sixteenone} this implies that 
$V_1\oplus \rminus\oplus \rplus
\sim_t V_1\oplus \rminus\oplus \rplus$.
Therefore, for any element $a_i=(t^i -t^{i+2q}) \in \mathcal B$
we have $2a_i \notin \Rfree_t(G)$ but $4a_i \in \Rfree_t(G)$.
However, in Section \ref{sixdiml} we determined the bounded
surgery obstructions for all these elements.
Since $i_{\ast}(u_{1,j}) = \langle -1\rangle \in \Coh{1}{\Units{A}}$
for all $j$ with $(j,2q)=1$
by Lemma \ref{seventeenone}, there are further stable relations
$a_1 + a_j \approx a_1 + a_k$, or $a_1 \approx a_j$
for all $j$.
It follows that a basis for $\rtildetopfree{G}$ is
given by $\{ a_1\mid 4a_1 \approx 0\}$.
\end{proof}
\newpage
%\bibliographystyle{amsplain}
%\bibliography{ekpmain}

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
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%\bigskip
%\noindent
%\leftline{\textsc{The first part of this paper is}:}
%\vskip .3cm
%\vbox{\hbox{I. Hambleton and E. K. Pedersen,
%{\it Topological equivalence of linear representations  for}}
%\hbox{{\it cyclic groups}: \partone, Ann. of Math. (2) (to appear).}}
%\bigskip
\end{document}

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