Generalizations of Agol's inequality and nonexistence of tight laminations
Thilo Kuessner
math0506528.tex
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\newtheorem{lem}{Lemma}
\newtheorem{cor} {Corollary}
\newtheorem{exm} {Example}
\newtheorem{pro} {Proposition}
\newtheorem{df} {Definition}
\newtheorem{lam}{Lemma}
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\begin{document}
\title{Generalizations of Agol's inequality and nonexistence of tight
laminations}
\author{Thilo Kuessner}
\date{}
\maketitle
\tableofcontents

%%%% **** The text of the paper starts here **** %%%%

\section{Results}

Agol's inequality (\cite{ag}, Theorem 2.1.) is the following:\\
\\
{\bf Agol's inequality:} {\em If $M$ is a hyperbolic 
3-manifold containing an incompressible, properly embedded surface $F$, 
then $Vol\left(M\right)\ge -2 V_3 \chi\left(Guts\left(\overline{M - F}\right)\right)$.}\\

In \cite{ag2}, this inequality has been improved to \\
$Vol\left(M\right)
\ge Vol\left(Guts\left(\overline{M-F}\right)\right)
\ge -V_{oct}\chi\left(Guts\left(\overline{M-F}\right)\right)$.\\

In this paper we will, building on ideas from \cite{ag}, prove the following general inequality for the (transversal)
Gromov norm $\parallel M\parallel_{\mathcal{F}}$ and the normal Gromov norm
$\parallel M\parallel^{norm}_{\mathcal{F}}$ of laminations.\\

We recall that for a manifold $Q$ with a
given decomposition $\partial Q=\partial_0Q\cup\partial_1Q$
of its boundary, $\left(Q,\partial_1Q\right)$ is pared acylindrical if
every $\pi_1$-injective map 
$\left({\Bbb S}^1\times\left[0,1\right],{\Bbb S}^1\times\left\{0,1\right\}\right)\rightarrow\left(Q,
\partial_1Q\right)$ can be homotoped (as a map of pairs) into $\partial Q$. 

Moreover, for a pair $\left(N,N^\prime\right)$ 
we say that a decomposition $N=R\cup Q$ is an essential decomposition of $\left(N,N^\prime\right)$
if, with $Q^\prime:=N^\prime\cap Q,R^\prime:=N^\prime\cap R, Q_0^\prime:=Q\cap R$, we have
that $Q\cap R=Q^\prime\cap R^\prime$ and the inclusions $Q_0^\prime\rightarrow Q\rightarrow N, Q_0^\prime
\rightarrow R\rightarrow N$
and
$Q_0^\prime\rightarrow Q^\prime\rightarrow N^\prime,
Q_0^\prime\rightarrow R^\prime\rightarrow N^\prime$
are each $\pi_1$-injective (for each connected component).\\
\\
{\bf Theorem 1}:  {\em 
Let $M$ be a compact, orientable, connected n-manifold and $\mathcal{F}$ a lamination (of codimension one)
of
$M$. Assume that $N:=\overline{M - {\mathcal{F}}}$
has a decomposition $N=Q\cup R$ into orientable n-manifolds with boundary $Q,R$ such that, if we
let $
\partial_0Q=Q\cap R,
\partial_1Q=\partial
Q\cap\partial N, N^\prime=\partial N\cup\partial_0Q
$, then\\
i) each connected component of
$\partial_0Q$ has
amenable fundamental group, \\
ii) $\left(Q,\partial_1Q\right)$ is pared acylindrical, $\partial_1Q\rightarrow Q$ is $\pi_1$-injective, $\partial_0Q\cap\partial_1Q\rightarrow\partial_1Q$ is not $\pi_1$-surjective\\
iii) $Q,\partial Q,\partial_1Q$ are aspherical,\\
iv) the decomposition $N=Q\cup R$
is an essential decomposition of $\left(N,N^\prime\right)$.

Then
$$\parallel M,\partial M\parallel^{norm}_{\mathcal{F}}
\ge\frac{1}{n+1}\parallel \partial Q\parallel.$$}
%$$\parallel M
%\parallel_{\mathcal{F}}\ge
%\frac{1}{\left[\frac{n}{2}\right]+1}\parallel \partial Q\parallel.$$}

In the case of 3-manifolds $M$ carrying an essential lamination $\mathcal{F}$, considering $Q=Guts\left(\overline{M - {\mathcal
{F}}}\right)$ yields then as a special case:\\

{\bf Theorem 2}: {\em Let $M$ be a compact 3-manifold and $\mathcal{F}$ an essential lamination of 
$M$. Then $$\parallel M,\partial M\parallel_{\mathcal{F}}^{norm}\ge
-\chi\left(Guts\left(\overline{M - {\mathcal{F}}}\right)\right).$$
%, \parallel M\parallel_{\mathcal{F}} \ge -2\chi\left(Guts\left(\overline{M - {\mathcal{F}}}\right)\right).$$
More generally, if $P$ is a polyhedron with $f$ faces, then
 $$ \parallel M,\partial M\parallel_{{\mathcal{F}},P}^{norm}\ge
-\frac{2}{f-2}\chi\left(Guts\left(\overline{M - {\mathcal
{F}}}\right)\right).$$}

\noindent The following corollary applies, for example, to all hyperbolic manifolds $M$
obtained by Dehn-filling the complement of the figure-eight knot in ${\Bbb S}^3$. (It is known that each of these $M$ contains tight laminations. By the following corollary, all these tight laminations have empty guts.)\\

{\bf Corollary 4}: {\em If $M$ is a closed hyperbolic 3-manifold with 
$Vol\left(M\right)< 2V_3 =2.02...$, then $M$ carries no essential lamination $\mathcal{F}$
with $\parallel M\parallel_{{\mathcal{F}},P}^{norm}=\parallel M\parallel_P$ 
for all polyhedra P, and nonempty guts. In particular, there is no tight essential
lamination with nonempty guts.}\\

It was observed by Calegari-Dunfield in \cite{cd} that a 
generalization of Agol's inequality to the case of tight laminations, 
together with the results in \cite{cd} about tight laminations with empty guts, would imply the following corollary.\\

{\bf Corollary 5} (\cite{cd}, Conjecture 9.7.): {\em The Weeks manifold admits no tight 
lamination $\mathcal{F}$.}\\

Putting this together with the main result of a recent paper by Tao Li (\cite{li}), one can even improve this result as follows.\\

{\bf Corollary 6}: {\em The Weeks manifold admits no transversely orientable essential lamination.}\\

Finally, we also have an application of Theorem 1 to higher-dimensional manifolds.\\

{\bf Corollary 7}:
{\em  Let $M$ be a compact Riemannian $n$-manifold of negative sectional curvature and finite volume. Let $F\subset M$ be a geodesic $n-1$-dimensional
hypersurface of finite volume. Then $\parallel
F\parallel\le \frac{n+1}{2}\parallel M\parallel$.}\\

\noindent
The basic of idea of Theorem 1, say for simplicity in the special situation of Corollary 7, is the following: a simplex which contributes to a normalized fundamental
cycle of $M$ should intersect $\partial Q=2F$ at most $n+1$ times. This is of course not true in general: simplices can wrap around $M$ many times and
intersect $F$ arbitrarily often, and even a homotopy rel.\ vertices will not change this. As an obvious examle, look at the following situation: let $\gamma$ be a closed geodesic transverse to $F$,
and for some large $N$ let $\sigma$ be a straight simplex contained in a small neighborhood of $\gamma^N$. Then $\sigma$ intersects $F$ $N$ times and, since $\sigma$ is already straight, this number of intersections can of 
course not be reduced by straightening. This shows that some more involved straightening must take place, and that
the acylindricity of $F$ is an essential condition. The way to use acylindricity will be 
to find a normalization such that many subsets of simplices are mapped to cylinders, which degenerate and thus can be removed without changing the homology class.

We remark that many technical points, in particular the use of multicomplexes, can be omitted if (in the setting of Theorem 2) one does not consider incompressible surfaces or essential laminations,
but just geodesic surfaces in hyperbolic manifolds. In this case, all essential 
parts of the proof of Theorem 1 enter without the notational complications caused by the use of multicomplexes. Therefore we have given a fairly detailed outline of the proof for this special case in section 6.1. This should help to motivate the general proof in section 6.2.

Acknowledgements: It is probably obvious that this paper is strongly influenced by Agol's preprint
\cite{ag}. We will give in section 2.3 a somewhat detailed description of the relation between the proof of Theorem 1
in this paper and the proof of Lemma 6.1 in \cite{ag}. We mention that Theorem 1 is not true without assuming amenability of $\pi_1\partial_0Q$. This indicates that the use of multicomplexes in the proof of Theorem 1 seems unavoidable.

Moreover, 
the argument that a generalization of Agol's inequality would imply 
Corollary 2 is due to \cite{cd}.
\section{Preliminaries}
\subsection{Laminations}
Let $M$ be an n-manifold, possibly with boundary. In this paper 
all manifolds will be smooth and orientable. (Hence they are triangulable by Whitehead's theorem and
possess a locally finite fundamental class.)
A (codimension 1)
lamination ${\mathcal{F}}$ of $M$
is a foliation of a closed subset $\F$ of $M$, i.e., a decomposition
of a closed subset $\F\subset M$ into immersed codimension 1 submanifolds (leaves) 
so that $M$ is covered by charts $\phi_j:{\Bbb R}^{n-1}\times{\Bbb R}\rightarrow 
M$, the intersection of any leaf with the image of any chart $\phi_j$ 
being a union of plaques of the form $\phi_j\left({\Bbb R}^{n-1}\times\left\{*\right\}\right)$. (We
will denote by $\F$ both the lamination and the laminated subset of $M$, i.e.\ the union of leaves.)
If $M$ has boundary, we will always assume without further mentioning that $\mathcal{F}$ is transverse or tangential to $\partial M$. (If this were not the case, then the transverse and normal Gromov norm would be infinite, and all lower bounds will be trivially true.)

To construct the leaf space $T$ of $\F$,
one considers the pull-back lamination $\widetilde{\F}$
on the universal covering $\widetilde{M}$.
The space of leaves $T$ is defined as
the quotient of $\widetilde{M}$ under the following equivalence
relation $\sim$. Two points $x,y\in \widetilde{M}$ are equivalent if either they belong to 
the same leaf of $\widetilde{\F}$, or they belong to the same
connected component of the metric completion $\overline{\widetilde{M} - \widetilde{\F}}$ (for the path metric inherited by $\widetilde{M} - \widetilde{\F}$ from an arbitrary Riemannian metric on $\widetilde{M}$).\\
\\
{\em Laminations of 3-manifolds.}
A {\bf lamination} $\F$
of a {\bf 3-manifold} $M$
is called {\bf essential}
if no leaf is a sphere or a torus bounding a solid torus, $\overline{M - \F}$
is irreducible, and $\partial\left(\overline{M - \F}\right)$ is incompressible
and end-incompressible in $\overline{M - \F}$, where again the metric completion
$\overline{M- \F}$ of $M-\F$ is taken w.r.t.\ the path metric inherited from any Riemannian metric on $M$,
see \cite{goe}, ch.1. (Note that $\overline{M- \F}$ is immersed in $M$, the leaves of
$\mathcal{F}$ in the image of the immersion are called boundary leaves.)

Examples
of essential laminations are taut foliations or compact, incompressible, boundary-incompressible surfaces
in compact 3-manifolds.
(We always consider laminations without isolated leaves. If a lamination has isolated 
leaves, then it can be converted into a lamination without isolated leaves by replacing each two-sided isolated leaf $S_i$ with the
trivially foliated product $S_i\times\left[0,1\right]$, resp.\ each one-sided 
isolated leaf with the canonically foliated normal $I$-bundle, without changing the topological type of $M$.)

If $\mathcal{\F}$ is an essential lamination, then the
leaf space $T$ is an order tree, with segments 
corresponding to directed, transverse, efficient arcs. (An order tree $T$ is a set $T$ with a collection of linearly ordered subsets, called segments, such that the axioms of \cite{goe}, Def.\ 6.9., are satisfied.)
Moreover, $T$ is an ${\Bbb R}$-order tree, that is, it is a countable union of segments
and each segment is order isomorphic to a closed interval in ${\Bbb R}$. $T$ can be topologized by the order topology on segments (and declaring that a set is closed if the intersection with each segment is closed). 
For this topology, $\pi_0T$ and $\pi_1T$ are trivial (see, for example, \cite{rss}, chapter 5, and its references).

The order tree $T$ comes 
with a fixed-point 
free action of $\pi_1M$. Fenley (\cite{fe}) 
has exhibited hyperbolic 3-manifolds whose fundamental groups do not admit any fixed-point free
action on ${\Bbb R}$-order
trees. Thus there are hyperbolic 3-manifolds not carrying any essential lamination.  

If $M$ is hyperbolic and $\mathcal{F}$ an essential lamination, then $\overline{M - {\mathcal{F}}}$
has a characteristic submanifold which is the maximal submanifold 
that can be decomposed into $I$-bundles and solid tori, respecting boundary patterns (see \cite{js}, \cite{joh} for precise definitions). The complement of this characteristic submanifold
is denoted by $Guts\left({\mathcal{F}}\right)$. It admits a hyperbolic 
metric with geodesic boundary 
and cusps. (Be aware that some authors, like \cite{cd}, include the solid tori into the guts.)
If ${\mathcal{F}}=F$ is a properly embedded, incompressible, boundary-incompressible surface, then
{\bf Agol's inequality} states that $Vol\left(M\right)\ge -2 V_3\chi\left(Guts\left(F\right)\right)$. This implies, for example, that a hyperbolic manifold of volume $< 2V_3$ can
not contain any geodesic surface of finite area. Recently, this inequality has been improved to $Vol\left(M\right)
\ge Vol\left(Guts\left(F\right)\right)
\ge -V_{oct}\chi\left(Guts\left(F\right)\right)$ in \cite{ag2}, using estimates coming from
Perelman's work on the Ricci flow.\\

Assume that $\mathcal{F}$ is a codimension one lamination
of an n-manifold $M$ such that its leaf space $T$ is an ${\Bbb R}$-order tree. (For example this is the case if $n=3$ and $\mathcal{F}$ is essential.)
An essential lamination is called {\bf tight} if $T$
is Hausdorff. It is called unbranched if  
$T$
is homeomorphic to $
{\Bbb R}$. It is said to have two-sided branching (\cite{cal}, Def.\ 2.6.1) 
if there are leaves $\lambda,\lambda_1,\lambda_2,\mu,\mu_1,\mu_2$ such that the corresponding points in the
$T$ satisfy $\lambda<\lambda_1,\lambda<\lambda_2,\mu>\mu_1,\mu>\mu_2$ but $\lambda_1,\lambda_2$ are incomparable and $\mu_1,\mu_2$ are incomparable.
%if it is not unbranched but there is some point $x\in T$ such that one connected component 
%of $T-\left\{x\right\}$ is order-isomorphic to a connected subset of ${\Bbb R}$. 
It is said to have one-sided branching if it is neither unbranched nor has two-sided branching.\\

If $M$
is a hyperbolic 3-manifold and carries a tight lamination with empty guts, then
Calegari and Dunfield have shown (\cite{cd}, Theorem 3.2.) that $\pi_1M$ acts effectively on the circle, i.e., there is an injective homomorphism
$\pi_1M\rightarrow Homeo\left({\Bbb S}^1\right)$. This
implies that the Weeks manifold (the closed hyperbolic manifold of smallest
volume) can not carry a tight lamination with empty guts (\cite{cd}, Corollary 9.4.). The aim of this paper
is to find obstructions to existence of laminations with nonempty guts.

\subsection{Simplicial volume and refinements}

Let $M$ be a compact, orientable, connected n-manifold, possibly with boundary. Its top integer (singular) homology
group $H_n\left(M,\partial M;{\Bbb Z}\right)$ is cyclic. The image of a generator under the change-of-coefficients homomorphism $H_n\left(M,\partial M;{\Bbb Z}\right)\rightarrow
H_n\left(M,\partial M;{\Bbb R}\right)$ is called a fundamental class and is 
denoted $\left[M,\partial M\right]$. If $M$ is not connected, we define $\left[M,\partial M\right]$ to be the formal sum of the fundamental classes of its connected components.

The simplicial volume $\parallel M,\partial M\parallel$ is defined as $\parallel M,
\partial M\parallel=inf\left\{\sum_{i=1}^r\mid a_i\mid\right\}$ 
where the infimum is taken over all singular chains $\sum_{i=1}^r a_i\sigma_i$
(with real coefficients) representing the fundamental 
class in $H_n\left(M,\partial M;{\Bbb R}\right)$. 

If $M - \partial M$ carries a complete hyperbolic metric of finite volume $Vol\left(M\right)$,
then $\parallel M\parallel=\frac{1}{V_n}Vol\left(M\right)$ with $V_n=
sup\left\{Vol\left(\Delta\right):\Delta\subset{\Bbb H}^n\ \mbox{geodesic simplex}\right\}$ (see \cite{gro},\cite{thu},\cite{bp}, \cite{fr}).

More generally, let $P$ be any polyhedron. Then the invariant $\parallel M,\partial M\parallel_P$
is defined in \cite{ag} as follows: denoting by $C_*\left(M,\partial M;P;{\Bbb R}\right)$ the complex of $P$-chains with real coefficients, and by $H_*\left(
M,\partial M;P;{\Bbb R}\right)$ its homology, there is a canonical chain homomorphism $\psi:C_*\left(
M,\partial M;P;{\Bbb R}\right)\rightarrow C_*\left(M,\partial M;{\Bbb R}\right)$, given by some 
triangulations of $P$ which is to be chosen such that all possible cancellations of boundary faces are preserved. 
$\parallel M,\partial M\parallel_P$ is defined as the infimum of $\sum_{i=1}^r\mid a_i\mid$
over all $P$-chains $\sum_{i=1}^r a_iP_i$ such that $\psi\left(\sum_{i=1}^r a_iP_i\right)$ represents
the fundamental class $\left[M,\partial M\right]$. Let $V_P:=sup\left\{Vol\left(\Delta\right)\right\}$,
where the supremum is taken over all straight $P$-polyhedra $\Delta\subset{\Bbb H}^3$. The following 
proposition is Lemma 4.1. in \cite{ag}. (The proof in \cite{ag} is quite short, and it does not give details for the cusped case. However, the
proof in the cusped case can be completed using the arguments in sections 5 and 6
of Francaviglia's paper \cite{fr}.)

\begin{pro}: If $M - \partial M$ admits a hyperbolic metric of finite volume $Vol\left(M\right)$, then
$$\parallel M,\partial M\parallel_P=\frac{1}{V_P}Vol\left(M\right).$$\end{pro}

Let $M$ be a manifold and $\mathcal{F}$ a {\bf codimension one lamination} of $M$.
Let $\Delta^n$ be the standard simplex in ${\Bbb R}^{n+1}$, and $\sigma:\Delta^n\rightarrow M$ some (continuous, not
necessarily differentiable) singular simplex.
The lamination $\mathcal{F}$ induces an equivalence relation on $\Delta^n$ by:
$x\sim y\Longleftrightarrow \sigma\left(x\right)$ and $\sigma\left(y\right)$ belong to the same connected component
of $L\cap\sigma\left(\Delta^n\right)$ for some leaf $L$ of $\mathcal{F}$.
We say that a singular simplex $\sigma:\Delta^n\rightarrow M$ is laminated if
the equivalence relation $\sim$
is induced by a lamination ${\mathcal{F}}\mid_{\sigma}$
of $\Delta^n$.
We call a lamination $\mathcal{F}$ of $\Delta^n$ affine if there is an affine mapping $f:\Delta^n\rightarrow{\Bbb R}                                                                                                                  $ such that $x,y\in\Delta^n$ belong to the same leaf if and only if $f\left(x\right)=f\left(y\right)$. We 
say that a lamination $\mathcal{G}$ of $\Delta^n$ is conjugate to an affine foliation if there is a 
simplicial homeomorphism $H:\Delta^n                   
\rightarrow\Delta^n$ such that $H^*\mathcal{G}$ is an affine foliation.\\   
We say that a singular $n$-simplex $\sigma:\Delta^n\rightarrow M$, $n\ge 2$, is {\bf transverse} to 
$\mathcal{F}$ if it is foliated and  it is 
either contained in a leaf,
or ${\mathcal{F}}\mid_\sigma$ is conjugate to 
an affine foliation $\mathcal{G}$ of $\Delta^n$.\\
For $n=1$, we say that a singular 1-simplex $\sigma:\Delta^1\rightarrow M$ is transverse to $\mathcal{F}$ if it is either contained in a leaf,                                                                                        or for each foliation chart $\phi:U\rightarrow {\Bbb R}^{m-1}\times{\Bbb R}^1$ (with m-th coordinate map           $\phi_m:U\rightarrow{\Bbb R}^1$) one has that $\phi_m\circ\sigma\mid_{\sigma^{-1}\left(U\right)}:                  \sigma^{-1}\left(U\right)\rightarrow{\Bbb R}^1$ is locally surjective at                                           all points of $int\left(\Delta^1\right)$, i.e.\ for all                                                            $p\in int\left(\Delta^1\right)\cap \sigma^{-1}\left(U\right)$, the                                                 image of $\phi_m\circ\sigma\mid_{\sigma^{-1}\left(U\right)}$ contains a neighborhood of $\phi_m\circ\sigma\left(p\right)$.\\ 
We say that
the simplex $\sigma:\Delta^n\rightarrow M$ is {\bf normal} to ${\mathcal{F} }$ if, for each leaf $F$,
$\sigma^{-1}\left(F\right)$ consists of normal disks, i.e.\ disks meeting each edge of $\Delta^n$ at most once.
(If $F=\partial M$ is a leaf of ${\mathcal{F}}$ we also admit that $\sigma^{-1}\left(F\right)$
can be a face of $\Delta^n$). In particular, any transverse simplex is normal.\\
In the special case of 
{\bf foliations} $\mathcal{F}$
one has that
the transversality of a singular simplex $\sigma$ is implied by (hence equivalent to) the normality of $\sigma$,
as can be shown along the lines of \cite{k3}, section 1.3.\\
More generally, let $P$ be any polyhedron. Then we say that a singular polyhedron $\sigma:P\rightarrow M$ is normal to $\mathcal{F}$ if, 
for each leaf $F$,
$\sigma^{-1}\left(F\right)$ consists of normal disks, i.e.\ disks meeting each edge of $P$ at most once
(or being equal to a face of $P$, if $F$ is a boundary leaf).
%each edge is either contained in a leaf or
%intersects each leaf of $\mathcal{F}$ at most once.

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\begin{df}: Let $M$ be a compact, oriented, connected n-manifold, possibly with boundary, and ${\mathcal{F}}$ a foliation or lamination on $M$. Let $\Delta^n$ be the standard simplex and $P$ any polyhedron. Then $$\parallel M,\partial M\parallel_{{\mathcal{F}}}:=inf\left\{\sum_{i=1}^r\mid a_i\mid: \psi\left(\sum_{i=1}^r a_i\sigma_i\right) \mbox{ represents }\left[M,\partial M\right], \sigma_i:\Delta^n\rightarrow M \mbox{ transverse to }{\mathcal{F}}\right\}$$
and 
$$\parallel M,\partial M\parallel^{norm}_{{\mathcal{F}},P}:=inf\left\{
\sum_{i=1}^r\mid a_i\mid: \psi\left(\sum_{i=1}^r a_i\sigma_i\right) \mbox{ represents
 }\left[M,\partial M\right], \sigma_i:P\rightarrow M \mbox{ normal to }{
\mathcal{F}}\right\}.$$
In particular, we define $\parallel M,\partial M\parallel_{\mathcal{F}}^{norm}
=\parallel M,\partial M
\parallel_{{\mathcal{F}},\Delta^n}^{norm}$.\end{df}
All norms are finite, under the assumption that $\mathcal{F}$ is transverse
or tangential to $\partial M$.\\
There is an obvious inequality $$\parallel M,\partial M\parallel\le
\parallel M,\partial M\parallel^{norm}_{{\mathcal{F}}}\le
\parallel M,\partial M\parallel_{{\mathcal{F}}}.$$
In the case of foliations, equality $\parallel M,\partial M\parallel^{norm}_{{\mathcal{F}}}=\parallel M,\partial M\parallel_{{\mathcal{F}}}$ holds.\\
(We remark that all definitions extend in an obvious way to disconnected manifolds by summing over the connected components.)\\


The following proposition and lemma are
a straightforward generalisation of \cite{cal}, Theorem 2.5.9, and of arguments in \cite{ag}.
\begin{pro}: Let $M$ be a compact, oriented 3-manifold.\\
a) If ${\mathcal{F}}$ an
essential lamination which is either unbranched or has 
one-sided branching such that the induced lamination of $\partial M$ is unbranched, then
$$\parallel M,\partial M\parallel^{norm}_{
{\mathcal{F}},P}=\parallel M,\partial M\parallel_P$$ for each polyhedron $P$.\\
b) If ${\mathcal{F}}$ is a tight essential lamination, then
$$\parallel M,\partial M\parallel^{norm}_{
{\mathcal{F}},P}=\parallel M,\partial M\parallel_P$$ for each polyhedron $P$.\end{pro}
\begin{pf} Since ${\mathcal{F}}$ is an essential lamination, we know from \cite{goe}, Theorem 6.1., that the leaves are $\pi_1$-injective, 
the universal covering $\widetilde{M}$ is homeomorphic
to ${\Bbb R}^3$ and that the leaves of the pull-back lamination are planes, in particular aspherical. 
%Moreover, it follows with $\widetilde{M}={\Bbb R}^3$ and $\widetilde{F}\simeq{\Bbb R}^2$ from
%Alexander duality that
%$\widetilde{M}-\widetilde{F}$ has
%two connected components for each leaf $F$. 
Therefore the proposition is a special case of the following lemma.\end{pf}

\begin{lem}: Let $M$ be a compact, oriented, 
aspherical manifold, and $\F$ a lamination of codimension 
one. \\
Assume that the leaves are $\pi_1$-injective and aspherical, and that the leaf space $T$ is an ${\Bbb R}$-order tree.\\
a) If the leaf space $T$ is either ${\Bbb R}$ or branches 
in only one direction, such that the induced lamination of $\partial M$ has leaf space ${\Bbb R}$, then $$\parallel M,\partial M
\parallel^{norm}_{{\mathcal{F}},P}=\parallel M,\partial M\parallel_P$$ for each polyhedron $P$.\\
b) If the leaf space is a Hausdorff tree, 
%and $\widetilde{M}-\widetilde{F}$ has
%two connected components for each leaf $F$, 
then $$\parallel M,\partial M
\parallel^{norm}_{{\mathcal{F}},P}=\parallel M,\partial M\parallel_P$$ for each polyhedron $P$.\end{lem}

\begin{pf}
To prove the wanted equalities, it suffices in each case to show that any (relative)
cycle can be homotoped to a cycle consisting of normal polyhedra.\\
We denote $\widetilde{\mathcal{F}}$ the pull-back lamination of $\widetilde{M}$ and $p:\widetilde{M}\rightarrow T=\widetilde{M}/\widetilde{\mathcal{F}}$ the projection to the leaf space.\\
a)
First we consider the case that $P$=simplex (\cite{cal}, section 4.1) and $\mathcal{F}$ unbranched. 
For this case, we can repeat the argument in \cite{cal}, Lemma 2.2.8. Namely, let us be given a 
(relative) cycle $\sum_{i=1}^r a_i\sigma_i$, lift it to a $\pi_1M$-equivariant (relative) cycle on $\widetilde{M}$ and then perform an (equivariant) straightening, by induction on the dimension of subsimplices of the lifts $\widetilde{\sigma_i}$ as follows: for each edge $\tilde{e}$
of any 
lift $\widetilde{\sigma_i}$, its projection $p\left(\tilde{e}\right)$
to the leaf space $T$ is homotopic to a unique straight
arc $str\left(p\left(\tilde{e}\right)\right)$
in $T\simeq{\Bbb R}$. It is easy to see (covering the arc by foliation charts and then extending the
lifted arc stepwise) that $str\left(p\left(\tilde{e}\right)\right)$ 
can be lifted to an arc $str\left(\tilde{e}\right)$ with 
the same endpoints as $\tilde{e}$, and that 
the homotopy between $str\left(p\left(\tilde{e}\right)\right)$
and $p\left(\tilde{e}\right)$ can be lifted to a homotopy between $str\left(\tilde{e}\right)$ and $\tilde{e}$.
$str\left(\tilde{e}\right)$ is transverse to $\mathcal{F}$, because its projection is a straight arc in $T$. 
These homotopies of edges can be extended to a homotopy of the whole (relative) cycle. Thus we have straightened the 1-skeleton of the given (relative) cycle.

Now let us be given a 2-simplex $\tilde{f}:
\Delta^2\rightarrow\widetilde{M}$ with transverse edges. There is 
an obvious straightening $str\left(p\left(\tilde{f}\right)\right)$
of $p\left(\tilde{f}\right):\Delta^2\rightarrow T$ as 
follows: if, for $t\in T$, $\left(p \tilde{f}\right)^{-1}\left(t\right)$ 
has two preimages $x_1,x_2$ on edges of $\Delta^2$ (which are necessarily unique), then 
$str\left(p\left(\tilde{f}\right)\right)$ maps the line which connects $x_1$ and $x_2$ in
$\Delta^2$ constantly to $t$. It is clear that this defines a continuous map $str\left(p\left(\tilde{f}\right)\right):\Delta^2\rightarrow T$.

Since leaves $\widetilde{F}$
of $\widetilde{\mathcal{F}}$ are connected ($\pi_0\widetilde{F}=0$), $str\left(p\left(\tilde{f}\right)\right)$ can be lifted to a map $str\left(\tilde{f}\right):\Delta^2\rightarrow\widetilde{M}$ with $p\left(str\left(\tilde{f}\right)\right)=
str\left(p\left(\tilde{f}\right)\right)$. $str\left(\tilde{f}\right)$ is transverse to $\mathcal{F}$, because its projection is a straight simplex in $T$.

There is an obvious homotopy between $p\left(\tilde{f}\right)$ and
$str\left(p\left(\tilde{f}\right)\right)$. For each $t\in T$, the restriction of the homotopy to $\left(p \tilde{f}\right)^{-1}\left(t\right)$
can be lifted to a homotopy in $\widetilde{M}$, because $\pi_1\widetilde{M}=0$.
Since $\pi_2\widetilde{M}=0$, these homotopies for various $t\in T$ fit together continuously to give a homotopy between $\tilde{f}$ and $str\left(\tilde{f}\right)$.

These homotopies of 2-simplices leave the (already transverse) boundaries pointwise fixed, thus they
can be extended to a homotopy of the whole (relative) cycle. 
Hence we have straightened the 2-skeleton of the given (relative) cycle.

Assume that we have already straightened the $k$-skeleton, for some $k\in{\Bbb N}$.
The analogous procedure, using $\pi_{k-1}\widetilde{F}=0$ for all leaves, and $\pi_k\widetilde{M}=0, \pi_{k+1}\widetilde{M}=0$, allows to straighten the $\left(k+1\right)$-skeleton of the (relative) cycle. This finishes the proof in
the case that ${\mathcal{F}}$ is unbranched.

The generalization to the case that $\mathcal{F}$ has one-sided branching such that the induced lamination of $\partial M$ is unbranched works as in \cite{cal}, Theorem 2.6.6.

We remark that in the case $P=simplex$ we get not only a normal cycle, but even a transverse cycle.

Now we consider the case of arbitrary polyhedra $P$. Let $\sum_{i=1}^r a_i\sigma_i$ be a P-cycle. It can be 
subtriangulated to a simplicial cycle $\sum_{i=1}^r a_i\sum_{j=1}^s\tau_{i,j}$. Again the argument in \cite
{cal}, Lemma 2.2.8 (resp.\ its version for manifolds with boundary), shows that this simplicial cycle can be homotoped such that each $\tau_{i,j}$ is transverse (and such that boundary cancellations are preserved).
But transversality of each $\tau_{i,j}$ implies by definition that $\sigma_i=\sum_{j=1}^s\tau_{i,j}$ is normal (though in general not transverse) to $\mathcal{F}$.\\
b) By assumption $\widetilde{M}/\widetilde{\mathcal{F}}$ is a Hausdorff tree. We observe that its branching points are 
the projections of complementary regions. Indeed, let $F$ 
be a leaf of $\F$, then $\widetilde{F}$ is a submanifold of 
the contractible manifold $\widetilde{M}$.
By asphericity and $\pi_1$-injectivity of $F$, $\widetilde{F}$ must be contractible.
By Alexander duality follows that
$\widetilde{M}-\widetilde{F}$ has 
two connected components. Therefore the complement of the point $p\left({\widetilde{F}}\right)$ in the leaf space
has (at most) two connected components, thus $p\left(\widetilde{F}\right)$
can not be a branch point.\\
Again, to define a straightening of $P$-chains it suffices to
define a canonical straightening of singular polyhedra $P$ such that straightenings of common boundary faces will agree. Let $\tilde{v}_0,\ldots,\tilde{v}_n$ be the vertices of the image of $P$.
For each pair $\left\{\tilde{v}_i,\tilde{v}_j\right\}$ 
there exists at most one edge $\tilde{e}_{ij}$ with
vertices $\tilde{v}_i,\tilde{v}_j$
in the image of $P$. Since the leaf space is a tree, we have a unique 
straight arc $str\left(p\left(\tilde{e}_{ij}\right)\right)$ 
connecting the points $p\left(\tilde{v}_i\right)$ and $p\left(\tilde{v}_j\right)$ 
in the leaf space. As in a), one can lift this straight arc
$str\left(p\left(\tilde{e}_{ij}\right)\right)$ to an arc $str\left(\tilde{e}_{ij}\right)$ 
in $\widetilde{M}$, connecting $\tilde{v}_i$ and $\tilde{v}_j$, which
is transverse to $\mathcal{F}$. 
We define this arc $str\left(\tilde{e}_{ij}\right)$ to be
the straightening of $\tilde{e}_{ij}$.
As in a), we have homotopies of 1-simplices, which extend to a homotopy of the whole (relative) cycle.
Thus we have straightened the 1-skeleton. 

Now let us be given the 3 vertices $\tilde{v}_0,\tilde{v}_1,\tilde{v}_2$ of a 
2-simplex $\tilde{f}$
with straight edges. If the projections $p\left(\tilde{v}_0\right),p\left(\tilde{v}_1\right),p\left(\tilde{v}_2\right)$
belong to a subtree isomorphic to a connected subset of
${\Bbb R}$, then we can straighten $\tilde{f}$
as in a). If not, we have that the projection of the 1-skeleton of this simplex has exactly one branch point, which corresponds to a complementary region. (The projection may of course meet many branch points of the tree, but the image of the projection, considered as a subtree, can have at most one branch point. In general, a subtree with $n$ vertices can have at most $n-2$ branch points.)
The preimage of the complement of this complementary region
consists of three connected subsets of the 2-simplex ("corners around the vertices").
We can straighten each of these subsets and do not need to care about the complementary region corresponding to the branch point. Thus we have straightened the 2-skeleton.\\
Assume that we have already straightened the $k$-skeleton, for
some $k\in{\Bbb N}$.
Let us be given the $k+2$ vertices $\tilde{v}_0,\tilde{v}_1,\ldots,\tilde{v}_{k+1}$ of a $\left(k+1\right)$-simplex 
with straight faces. Then we have (at most $k$) branch points in the projection of
the simplex, which correspond to complementary regions. Again we can straighten 
the parts of the simplex which do not belong to these 
complementary regions as in a), since they are projected to linearly ordered subsets of the tree. Thus we have straightened the $\left(k+1\right)$-skeleton.

Since, by the recursive construction,
we have defined straightenings of simplices with common faces by first 
defining (the same) straightenings of their common faces,
the straightening of a 
(relative)
cycle will be again a (relative) cycle, in the same (relative) homology class.
\end{pf}\\
\\
Remark: For $\parallel M\parallel_{\mathcal{F}}$ instead of
$\parallel M\parallel^{norm}_{\mathcal{F}}$, equality b) is in general wrong, and equality a) is unknown (but presumably wrong). \\
If $\mathcal{F}$ is essential but not tight, one may still try to homotope cycles to be transverse, 
by possibly changing the lamination. In the special case 
that the cycle is coming from a triangulation, this has been done in \cite{bri}
and \cite{gab} by Brittenham resp.\ Gabai. It is not obvious how to generalize their arguments to cycles with overlapping simplices.






\subsection{Plan of proof and relation with \cite{ag}}


We give a very rough explanation of the proof. A much more detailed and readable
exposition of the proof in a special case, which already contains all essential ideas, is given in section 6.1.

The most optimistic goal would be to show that each fundamental cycle $\sum_{i=1}^r a_i\sigma_i$ of $M$ can be brought into
some form such that each $\sigma_i$ intersects $\partial Q$ in at most $n+1$ subsimplices. 
However, in general, even after straightening and crushing cusps, the number 
of intersections will be arbitrarily large. 

Let us first indicate the proof in the case $Q=N$. We are given a fundamental cycle for $M$, normal to the lamination, thus
after further triangulation giving a new fundamental cycle $z_N$ for $N$.
In section 5 we define a straightening on $Q$.
The nonobvious point of this straightening is that,
for each ordered pair of connected components of $\partial_1Q$,
we fix one 'distinguished' 1-simplex connecting them, and
that those edges of the new fundamental cycle $z_N$, which
are subarcs of edges of some $\sigma_i$, are
only allowed to be straightened into
'distinguished' 1-simplices.
(In particular, this straightening changes the homotopy classes relative boundary.
However, the straightening homotopy extends to a homotopy of the whole
$z_N$ because no closed cycle of subarcs of $\sigma_i$
exists.)
After this straightening one removes all degenerate simplices without changing the homology class.
In particular, the boundary
of the (straightened and nondegenerate) fundamental cycle represents the
fundamental class of $\partial Q$.
On the other hand, using acylindricity,
one can show that each simplex, of the fundamental cycle we started with,
can contribute at most $n+1$ simplices to the (straightened
and nondegenerate) fundamental cycle of $\partial Q$. (In fact, acylindricity implies that there are after straightening no nondegenerate simplices with 'white-parallel' arcs. The content of the combinatorial arguments in section 4 is then to show that there are per
simplex at most n+1 affine subsimplices without 'white-parallel' arcs.)

In the case $Q\not=N$, we need as another ingredient
a retraction from $N$ to $Q$. Such a retraction unfortunately does not exist,
but it is known 
by work of Gromov (which we recall in section 3)
that
such a retraction can be defined in the following weak sense: there are multicomplexes $K\left(N\right)$ and $K\left(Q\right)$, with isomorphic bounded cohomology to $N$ resp.\ $Q$, and an action of a 
group $G$ on them, such that there is defined a retraction $r:C_*\left(K\left(N\right)
\right)\otimes_{{\Bbb Z}G}{\Bbb Z}
\rightarrow C_*\left(K\left(Q\right)
\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$. In other words, there is an ambiguity up to some group action. 
If $\partial_{0}Q$ has amenable fundamental group, then $G$ is amenable and this allows in some sense to 
remove the ambiguity (using bounded cohomology). Unfortunately, the necessary use of multicomplexes with a group action also complicates some of the later arguments, concerning the straightening.\\
\\
\\
In \cite{ag}, Theorem 1 has been proven for incompressible surfaces in hyperbolic 3-manifolds. We compare the steps of the proof in \cite{ag} with the arguments in our paper.\\
\\
Step 1 in \cite{ag} is the normalization procedure, which we have restated in Lemma 1. \\
Step 2 in \cite{ag} consists in choosing compatible triangulations of the
polytopes $\sigma_i^{-1}\left(N\right)$.\\
Step 3 in \cite{ag} boils down to the statement that, for each component $Q_i$ of $Q$,
there exists a retraction $r:\hat{N}\rightarrow p^{-1}\left(Q_i\right)$, for the covering $p:\hat{N}\rightarrow N$ corresponding to $\pi_1Q_i$. 
Such a statement can not be correct because it would (together with step 7 from \cite{ag}) imply $\parallel N\parallel
\ge \parallel Q\parallel$ whenever $Q$ is a $\pi_1$-injective submanifold 
of $N$. This inequality is true for
submanifolds with amenable boundary, but not in general. In fact, one only has the more complicated retraction
$r:C_*\left(K\left(N\right),
K\left(N^\prime\right)\right)\otimes_{{\Bbb Z}G}{\Bbb Z}
\rightarrow C_*\left(K\left(Q\right),K\left(\partial Q\right)
\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$, with $G=\Pi\left(K\left(\partial_{0}Q\right)\right)$. 
This more complicated retraction is the reason that much of the latter arguments
become notationally awkward, although conceptually not much is changing. Moreover, the action of 
the group $G$ is basically the reason that Theorem 1 is true only for amenable $G$.\\
Basically, the reason why the retraction $r:\hat{N}\rightarrow Q$ does not exist, is as follows. Let $R_j$ be the connected components of $\hat{N}-p^{-1}\left(Q_i\right)$.
Then $R_j$ is homotopy equivalent to each connected component of $\partial R_j$. If $\partial R_j$ were
connected for each $j$, this homotopy equivalence could be extended to a homotopy equivalence $r:\hat{N}\rightarrow p^{-1}\left(Q_i\right)$. However, in most cases $\partial R_j$ 
will be disconnected, and then such an $r$ can not exist.\\
We note that also the weaker construction of cutting off simplices does not work. A simplex may intersect $Q_i$ in many components and it is not clear which component to choose.\\
Step 5 in \cite{ag} is the straightening procedure, it corresponds to sections 5.1-5.4 in this paper.
We remark that the straightening procedure must be slightly more complicated than in \cite{ag} because it 
is not possible, as suggested in \cite{ag}, to homotope 
all edges between boundary components of $\partial Q$ into shortest geodesics.
This is the reason why we can only straighten chains with an admissible 0-1-labeling of their edges
(and why 
our straightening homomorphism in section 5.3.\ is only defined on $C_*^{simp}\left(\mid\Upsilon\mid\right)$ and not on all of
$C_*^{sing}\left(Q\right)$).\\
Step 6 in \cite{ag} consists in removing degenerate simplices. This corresponds to section 5.5 in this paper.\\
Step 7 in \cite{ag} proves that each triangle in $\sigma_i^{-1}\left(\partial N\right)$ contributes only once 
to the constructed fundamental cycle of $\partial Q$. Since, in our argument, we do not work with the covering $p:\hat{N}\rightarrow N$, we have no need 
for this justification.\\
Step 8 in \cite{ag} counts the remaining triangles per simplex (after removing degenerate simplices).
It seems to have used the combinatorial arguments which we work out for arbitrary dimensions in section 4.\\
We mention that the arguments of section 4 are the only part of the proof which gets easier 
if one restricts to 
3-manifolds rather than arbitrary dimensions. Moreover, the proof for laminations 
is the same as for hypersurfaces except for Lemma 1. Thus, upon these two points it seems that even in the
case of incompressible surfaces in 3-manifolds the proof of Theorem 1 can not be further simplified.



 
\section{Retracting chains to codimension zero submanifolds}




\subsection{Definitions}
 
The results of this section are essentially all due to Gromov, but we follow mainly our exposition in \cite{k2}.
We start with some recollections about multicomplexes (cf.\ \cite{gro}, section 3, or
\cite{k2}, section 1).\\
\\
A multicomplex $K$ is a topological space $\mid K\mid$ with a decomposition into simplices,
where each $n$-simplex is attached to the $n-1$-skeleton $K_{n-1}$
by a simplicial homeomorphism $f:\partial \Delta^n\rightarrow K_{n-1}$.
(In particular, each $n$-simplex has $n+1$ distinct vertices.) 

As opposed to simplicial complexes, in a multicomplex
there may be $n$-simplices with
the same $n-1$-skeleton.

We call a multicomplex minimally complete if the following holds:
whenever $\sigma:\Delta^n\rightarrow \mid K\mid$ is a singular n-simplex, such that
$\partial_0\sigma,\ldots,\partial_n\sigma$ are distinct simplices of $K$, then $\sigma$ is homotopic relative $\partial\Delta^n$ to
a {\em unique} simplex in $K$.

We call a minimally complete
multicomplex $K$ aspherical if all simplices $\sigma\not =\tau$ in $K$
satisfy $\sigma_1\not=\tau_1$.        \\
Orientations of multicomplexes are defined as
usual in the simplicial theory. For a simplex $\sigma$, $\overline{\sigma}$ will denote the simplex with the
opposite orientation.\\
A submulticomplex $L$ of a multicomplex $K$ consists of a subset of the set of simplices closed under face maps.
$\left(K,L\right)$ is a pair of multicomplexes if $K$ is a multicomplex and $L$ is a submulticomplex of $K$.\\
A group $G$ acts simplicially on a pair of multicomplexes $\left(K,L\right)$ if it acts on the set of simplices of $K$, mapping simplices in $L$ to simplices in $L$, such that the action commutes with all face maps. For $g\in G$ and $\sigma$ a simplex in $K$, we denote by
$g\sigma$ the simplex obtained by this action.\\
%Notational remark: for a connected multicomplex $K$ we write
%$\pi_1K$ for $\pi_1\left(\mid K\mid,p\right)$ with some $p\in\mid K\mid$. The choice of $p$ will not be of importance for our arguments.






\subsection{Construction of $K\left(X\right)$}

For a topological space $X$, we denote by $S_*\left(X\right)$ the simplicial set of all singular simplices in $X$ and $\mid S_*\left(X\right)\mid$ its geometric realization.

For a topological space $X$, a 
multicomplex $K\left(X\right)\subset \mid S_*\left(X\right)\mid$ is constructed as follows. The 0-skeleton $K_0\left(X\right)$ equals $S_0\left(X\right)$. The 1-skeleton $K_1\left(X\right)$
contains one element in each 
homotopy class (rel.\ $\left\{0,1\right\}$) of singular 1-simplices $f:\left[0,1\right]\rightarrow X$ 
with $f\left(0\right)\not=f\left(1\right)$.
For $n\ge 2$,
assuming by recursion
that the n-1-skeleton is defined, the n-skeleton $K_n\left(X\right)$ contains {\em one} 
singular n-simplex with any given boundary
if singular n-simplices with this given boundary exist at all.

$K\left(X\right)$ is closed under face maps, minimally complete, and aspherical.

Another multicomplex $\widetilde{K}\left(X\right)\subset \mid S_*\left(X\right)\mid$ is constructed as follows. The 
1-skeleton of $\widetilde{K}\left(X\right)$ agrees with the 1-skeleton of
$K\left(X\right)$. For $n\ge 2$,
assuming by recursion
that the n-1-skeleton is defined, the n-skeleton $\widetilde{K}_n\left(X\right)$ contains one element in each homotopy class 
(rel.\ boundary) of singular n-simplices $f:\Delta^n\rightarrow X$ with $\partial f\in 
\widetilde{K}_{n-1}\left(X\right)$.

According to \cite{gro}, $\mid \widetilde{K}\left(X\right)\mid $ is weakly homotopy equivalent to $X$.

If $X^\prime\subset X$ is a subspace, then 
%we will simplices in $K\left(X\right)$ to have image in $X^\prime$ whenever this is possible.
% is $\pi_1$-injective (for each connected component), then
%$K\left(X\right)$ (resp.\ $\widehat{K}\left(X\right)$) can be constructed
%such that $K\left(X^\prime\right)$ naturally embeds into $K\left(X\right)$. If $X^\prime\subset X$ is 
%not $\pi_1$-injective, 
we have (not necessarily injective) simplicial mappings $\tilde{j}:\widetilde{K}\left(X^\prime\right)
\rightarrow\widetilde{K}\left(X\right)$ and  
$j:K\left(X^\prime\right)\rightarrow K\left(X\right)$. 
We will, by abuse of notation, denote the images of $\tilde{j}$ resp.\ $j$ by $\widetilde{K}\left(X^\prime\right)$ resp.\ $K
\left(X^\prime\right)$. We will assume that we have constructed $\widetilde{K}\left(X\right)$ such that
simplices in $\widetilde{K}\left(X\right)$ have image in $X^\prime$ whenever this is possible. This means in particular that
simplices in $\widetilde{K}\left(X^\prime\right)$ indeed have image in $X^\prime$.
We will denote $C_*^{simp}\left(K\left(X\right), K\left(X^\prime\right)\right)$ (resp.\ $C_*^{simp}\left(\widetilde{K}\left(X\right), \widetilde{K}
\left(X^\prime\right)\right)$ the
relative chain complex of simplices in $K\left(X\right)$ (resp.\ $\widetilde{K}\left(X\right)$), 
modulo simplices in the image of $K\left(X^\prime\right)$ (resp.\ $\widetilde{K}\left(X^\prime\right)$).

There are obvious inclusions
$$i_*:C_*^{simp}\left(K\left(X\right),K\left(X^\prime\right)\right)\rightarrow 
C_*^{simp}\left(\widetilde{K}\left(X\right),\widetilde{K}\left(X^\prime\right)\right)
\rightarrow
C_*^{sing}\left(X,X^\prime\right)$$
into the relative singular chain complex of $\left(X,X^\prime\right)$.

In section 5 we will consider manifolds $Q$ with a decomposition $\partial
Q=\partial_0Q\cup \partial_1Q$. In this case we assume
to have chosen the simplices in $K\left(Q\right)$ such that, whenever a simplex $\sigma\in K\left(Q\right)$ is homotopic rel.\ boundary 
to a singular simplex with image in $\partial Q$ resp.\ $\partial_0Q$ resp.\ $\partial_1Q$, then $\sigma$ has image
in $\partial Q$ resp.\ $\partial_0Q$ resp.\ $\partial_1Q$. 
Such a choice is indeed possible. Namely, 
if a simplex $\sigma\in S_n\left(Q\right)$ is homotopic rel.\ boundary to simplices $\sigma_0\in S_n\left(\partial_0Q\right)$
and $\sigma_1\in S_n\left(\partial_1Q\right)$,
then we get a homotopy rel.\ boundary between $\sigma_0$ and $\sigma_1$, giving a map
${\Bbb D}^{n+1}\rightarrow Q$ such that
$\partial {\Bbb D}^{n+1}$
is divided into two halves, one mapped by $\sigma_0$ to $\partial_0Q$, the other mapped by $\sigma_1$
to $\partial_1Q$. Hence there must be an n-simplex in the given homotopy class
rel.\ boundary mapped to $\partial_0Q\cap\partial_1Q$. This shows that we can
choose $\sigma_0=\sigma_1\in S_n\left(\partial_0Q\right)\cap S_n\left(
\partial_1Q\right)$ in the homotopy class rel.\ boundary of $\sigma$.\\


{\bf Infinite and locally finite chains.} In this paper we will also work with infinite chains, and in particular
with locally finite chains on noncompact manifolds, as introduced in \cite{gro}, section 0.2. \\
For a topological space $X$, a formal sum $\sum_{i\in I}a_i\sigma_i$ of singular k-simplices with real coefficients
(with a possibly infinite index set $I$, and the convention $a_i\not=0$ for $i\in I$)
is an infinite singular k-chain. It is
said to be a locally finite chain if each point of $X$ is contained in 
the image of finitely many $\sigma_i$. Infinite resp.\ locally finite k-chains form 
real vector spaces $C_k^{inf}\left(X\right)$ resp.\ $C_k^{lf}\left(X\right)$.
The boundary operator maps locally finite k-chains
to locally finite k-1-chains, hence, for a pair of spaces $\left(X,X^\prime
\right)$ the homology $H_*^{lf}\left(X,X^\prime\right)$ of the complex of locally finite chains can be defined.\\
For a noncompact, orientable $n$-manifold $X$ with (possibly noncompact)
boundary $\partial X$, one has a fundamental class $\left[X,\partial X\right]
\in H_n^{lf}\left(X,\partial X\right)$. We will say that an infinite 
chain $\sum_{i\in I}a_i\sigma_i$ represents $\left[X,\partial X\right]$ 
if it is homologous to a locally finite chain representing $\left[X,\partial 
X\right]\in H_n^{lf}\left(X,\partial X\right)$.\\
For a simplicial complex $K$, we denote by $C_k^{simp,inf}\left(K\right)$ the ${\Bbb R}$-vector space
of (possibly infinite) formal sums $\sum_{i\in I}a_i\sigma_i$ with 
$a_i\in{\Bbb R}$ and $\sigma_i$ k-simplices in $K$.
%is a locally finite simplicial chain if each $\sigma_i$ is a simplex of $K$ and 
%each vertex of $K$ is adjacent to only finitely many $\sigma_i$. 
%We can define the homology of the complex of locally finite simplicial chains. \\
We have again the obvious inclusion $i_*:
C_*^{simp,inf}\left(K\left(X\right),K\left(X^\prime\right)\right)
\rightarrow
C_*^{inf}\left(X,X^\prime\right)$.\\
The following observation is of course well-known, but we will use it so often that we state it here for reference.
\begin{obs}: Let $X$ be a topological space and
$\sigma_0:\Delta^n\rightarrow  X$ a singular simplex. Let $H:\partial\Delta^n\times I\rightarrow X$ be a homotopy
with $H\left(x,0\right)=\sigma_0\left(x\right)$ for all $x\in\partial \Delta^n$. Then there exists a homotopy $\overline {H}:\Delta^n\times I\rightarrow X$ with $\overline{H}\mid_{\partial\Delta^n\times I}=H$ and
$\overline{H}\mid_{\Delta^n\times\left\{0\right\}}=\sigma_0$.

If $X^\prime\subset X$ is a subspace and the images of $\sigma_0$ and $H$ belong to $X^\prime$, then we can choose $\overline{H}$ such that its image belongs to $X^\prime$.
\end{obs}




\begin{lem}: Let $\left(X,X^\prime\right)$ be a pair of topological spaces. \\
%Assume that each path-connec
%ted component of $X$ and $X^\prime$ contains infinitely many points or is empty. 
a) Let $\sum_{i\in I}a_i\tau_i\in C_n^{inf}\left(X,X^\prime\right)$ be a (possibly infinite) singular n-chain.
Assume that $I$ is countable, and that each path-component of $X$ and each non-empty path-component
of $X^\prime$ contain uncountably many points.
Then $\sum_{i\in I}a_i\tau_i\in C_n^{inf}\left(X,X^\prime\right)$ 
is homotopic to a (possibly infinite) simplicial chain $\sum_ia_i\tau_i^\prime\in
C_n^{simp,inf}\left(\widetilde{K}\left(X\right),\widetilde{K}\left(X^\prime\right)\right)$. 
In particular, $\sum_ia_i\tau_i^\prime\in
C_n^{simp,inf}\left(\widetilde{K}\left(X\right),\widetilde{K}\left(X^\prime\right)\right)\subset C_*^{inf}\left(X,X^\prime\right)$ is homologous to $\sum_{i\in I}a_i\tau_i$. \\
b) Let $\sigma_0\in \widetilde{K}\left(X\right)$ and $H:\Delta^n\times I\rightarrow X$ a homotopy with $H\left(.,0\right)=\sigma_0$. Consider a minimal triangulation $\Delta^n\times I= \Delta_0\cup\ldots\Delta_n$ of $\Delta^n\times I$ into n+1 n+1-simplices. Assume that $H\left(\partial\Delta^n\times I\right)$ consists of simplices in $\widetilde{K}\left(X\right)$. Then $H$ is homotopic (rel.\ $\Delta^n\times\left\{0\right\}\cup\partial\Delta^n\times I$) to a map $\overline{H}:\Delta^n\times I\rightarrow X$ such that $\overline{H}\mid_{\Delta_i}\in\widetilde{K}\left(X\right)$, in particular $\sigma_1:=\overline{H}\left(.,1\right)\in\widetilde{K}\left(X\right)$.\\
c) There exists a simplicial map \\
$p: C_*^{simp,inf}\left(\widetilde{K}\left(X\right),\widetilde{K}\left(X^\prime\right)\right)\rightarrow C_*^{simp}\left(K\left(X\right),K\left(X^\prime\right)\right)$ such that: \\
if $\left(X,\partial X
\right)$ is an n-manifold with boundary, satisfying $\parallel X,\partial X\parallel\not=0$, 
and if $\sum_ia_i\tau_i^\prime\in
C_n^{simp,inf}\left(\widetilde{K}\left(X\right),\widetilde{K}\left(\partial X
\right)\right)$ represents $\left[X,\partial X\right]$, then $p\left(\sum_ia_i\tau_i^\prime\right)$ 
represents $\left[X,\partial X\right]$. 
\end{lem}
\begin{pf} 
%Fix a covering of $X$
%by compact subsets $K_j$ such that each point $x\in X$ has a neigborhood $U_x$ intersecting only finitely many $K_j$. (This is possible beacuse $X$ is a finite-dimensional manifold.) 
%First homotope
%$\sum_ia_i\tau_i$ such that after the homotopy each n-simplex $\tau_i$ has n+1 distinct vertices. This homotopy can be performed such that each vertex $v$ remains in its fixed neighborhood $U_v$. In particular, local finiteness is preserved.

a) From the assumptions it follows that there exists a homotopy of the 0-skeleton such that each vertex is moved
into a distinct point of $X$, and such that vertices in $X^\prime$ remain in $X^\prime$ during the homotopy.
By Observation 1, this homotopy can by induction be extended to a homotopy of the whole chain. 

Now we prove the claim by induction on $k$ ($0\le k <n$). We assume that the $k$-skeleton of $\sum_{i\in I}a_i\tau_i$
consists of simplices in $\widetilde{K}\left(X\right)$ and we want to homotope 
$\sum_{i\in I}a_i\tau_i$
such that the homotoped k+1-skeleton consists of simplices in $\widetilde{K}\left(X\right)$. 

By construction, each singular k+1-simplex $\sigma$ in $X$ with boundary a simplex in 
$\widetilde{K}\left(X\right)$ is homotopic (rel.\ boundary)
to a unique k+1-simplex in $\widetilde{K}\left(X\right)$. Since the homotopy keeps the boundary fixed, the homotopies of different k+1-simplices are compatible.
By Observation 1, the homotopy of the k+1-skeleton can by induction be extended to a homotopy of the whole chain.

If the image of the k+1-simplex $\sigma$ is contained in $X^\prime$, then it is homotopic 
rel.\ boundary to a simplex in $\widetilde{K}\left(X^\prime\right)$, 
for a homotopy with image in $X^\prime$. Thus we can realise 
the homotopy such that all simplices with image in $X^\prime$ are homotoped inside $X^\prime$.\\
b) follows by the same argument as a), succesively applied to $\Delta_0,\ldots,\Delta_n$.\\
c) $p$ is defined by induction. It is defined to be the identity on the 1-skeleton. If it is defined on the n-1-skeleton, for $n\ge 2$, then, for an n-simplex $\tau$, $p\left(\tau\right)\in K\left(X\right)$ is the unique simplex with
$\partial_i p\left(\tau\right)=p\left(\partial_i\tau\right)$ for $i=0,\ldots,n$. It is proved in \cite{gro} that $p^*$ is an isometric isomorphism of bounded cohomology. By duality this implies $p_*\left[X,\partial X\right]=\left[X,\partial X\right]$.\end{pf}




\subsection{Action of $G=\Pi\left(A\right)$}
We first repeat the definitions from \cite{k2}, section 1.5. Thereafter we will describe a straightforward extension, needed for our definition of straightening in section 5.

Let $\left(P,A\right)$ be a pair
of aspherical, minimally complete multicomplexes. \\
We define its nontrivial-loops space $\Omega^* A$ as the
set of homotopy classes (rel.\ $\left\{0,1\right\}$)
of continuous maps $\gamma:\left[0,1\right]\rightarrow \mid A\mid$ with $\gamma\left(0\right)=\gamma\left(1\right)$ and not homotopic (rel.\ $\left\{0,1\right\}$) to a
constant map.

We define
$$
\Pi\left(A\right):=\left\{\begin{array}{c}\left\{\gamma_1,\ldots,\gamma_n
\right\}
:n\in N,
\gamma_1,\ldots,\gamma_n\in A_1\cup \Omega^* A\\
\gamma_i\left(0\right)\not=\gamma_j\left(0\right),\gamma_i\left(1\right)\not=\gamma_j\left(1\right)\mbox{\ for\ }i\not= j,\\
\left\{\gamma_1\left(
0\right),\ldots,\gamma_n\left(0\right)\right\}=\left\{
\gamma_1\left(1\right),\ldots,\gamma_n\left(1\right)\right\}.
\end{array}\right\}.$$
If $\gamma,\gamma^\prime$ are elements of $A_1$ with $\gamma^\prime\not=\overline{\gamma}$ and $\gamma\left(0\right)=\gamma^\prime\left(1\right)$,
we denote $\gamma*\gamma^\prime\in A_1$ to be the unique edge of $A$ in 
the homotopy class of the concatenation. If $\gamma\in A_1$ and 
$\gamma^\prime\in\Omega^*A$ (or vice versa), 
with $\gamma\left(1\right)\not=\gamma\left(0\right)=\gamma^\prime\left(1\right)=\gamma^\prime\left(0\right)$, we also
denote $\gamma*\gamma^\prime\in A_1$ the unique edge in the homotopy class of the concatenation. 
If $\gamma,\gamma^\prime\in\Omega^*A$ with $\gamma\left(1\right)
=\gamma\left(0\right)=\gamma^\prime\left(1\right)=\gamma^\prime\left(0\right)$, we denote $\gamma*\gamma^\prime\in\Omega^*A$ the concatenation of loops.\\
\\
This can be used to define a multiplication on
$\Pi\left(A\right)$ as follows:\\
\\
given $\left\{\gamma_1,\ldots,\gamma_m\right\}$
and $\left\{\gamma_1^\prime,\ldots,\gamma_n^\prime\right\}$,
we chose a reindexing of the unordered sets $\left\{\gamma_1,\ldots,\gamma_m\right\}$
and $\left\{\gamma_1^\prime,\ldots,\gamma_n^\prime\right\}$
such that we have: $\gamma_j\left(1\right)
=\gamma_j^\prime\left(0\right)$ for $1\le j\le i$ and $\gamma_j\left(1\right)
\not=\gamma_k^\prime\left(0\right)$ for $j\ge i+1, k\ge i+1$.
(Since we are assuming that all $\gamma_j\left(1\right)$ are pairwise distinct, and also all
$\gamma_j^\prime\left(0\right)$ are pairwise distinct, such a reindexing exists for some $i\ge0$, and
it is unique up to permuting the indices $\le i$ and permuting separately the
indices of $\gamma_j$'s and $\gamma_k^\prime$'s with $j\ge i, k\ge i$.) \\
Moreover we permute the indices $\left\{1,\ldots,i\right\}$ such that there exists some $h$ with $0\le h\le i$ satisfying the following conditions: \\
- for $1\le
j\le h$ we have either $\gamma_j^\prime\not=\overline{\gamma_j}\in A_1$ or $\gamma_j^\prime
\not=\gamma_j^{-1}\in\Omega^*A$,\\
- for $h<
j\le i$ we have either $\gamma_j^\prime=\overline{\gamma_j}\in A_1$ 
or $\gamma_j^\prime=\gamma_j^{-1}\in\Omega^*A$.\\

With this fixed reindexing we define
$$\left\{\gamma_1,\ldots,\gamma_m\right\}
\left\{\gamma_1^\prime,\ldots,\gamma_n^\prime\right\}:=
\left\{\gamma_1^\prime*\gamma_1,\ldots,\gamma_h^\prime*\gamma_h,\gamma_{i+1},\ldots,
\gamma_m,\gamma_{i+1}^\prime,
\ldots,\gamma_n^\prime\right\}.$$
(Note that we have omitted all $\gamma_j^\prime *\gamma_j$ with $j>h$. The choice of $\gamma_j^\prime *\gamma_j$ rather than $\gamma_j*\gamma_j^\prime$ is just because we want to define a left action on $\left(P,A\right)$.)\\
This multiplication is independent of the chosen reindexing. It is clearly associative.
A neutral element is given by the empty set. The inverse to $\left\{\gamma_1,\ldots,\gamma_n\right\}$ is given by $\left\{\gamma_1^\prime,\ldots,\gamma_n^\prime\right\}$ with $\gamma_i^\prime=\overline{\gamma_i}$ if 
$\gamma_i\in A_1$ resp.\ $\gamma_i^\prime=\gamma_i^{-1}$ if $\gamma_i\in\Omega^*A$. 
(Indeed, in this case $h=0$, thus $\left\{\gamma_1,\ldots,\gamma_n\right\}
\left\{\gamma_1^\prime,\ldots,\gamma_n^\prime\right\}$ is the empty set.)
Thus we have defined a group law on $\Pi\left(A\right)$.\\
\\
We remark that there is an
inclusion $\Pi\left(A\right)\subset map_0\left(A_0,\left[\left[0,1\right],\mid A\mid\right]_{\mid P\mid}\right)$,
where $\left[\left[0,1\right],\mid A\mid\right]_{\mid P\mid}$ is the set of homotopy
classes (in $\mid P\mid$) rel.\ $\left\{0,1\right\}$ of maps from $\left[0,1\right]$ to $\mid A\mid$,
and $map_0\left(A_0,\left[\left[0,1\right],\mid A\mid\right]_{\mid P\mid}\right)$
is the set of maps $f:A_0\rightarrow \left[\left[0,1\right],\mid A\mid\right]_{\mid P\mid}$ with \\
- $f\left(y\right)\left(0\right)=y$ for all $y\in A_0$ and \\
- $f\left(.\right)\left(1\right):A_0\rightarrow A_0$ is a bijection.\\
This inclusion is given by sending $\left\{\left[\gamma_1\right],\ldots,\left[\gamma_n\right]\right\}$
to the map $f$ defined by $f\left(\gamma_i\left(0\right)\right)=\left[\gamma_i\right]$ for $i=1,\ldots,n$, and $f\left(y\right)=\left[c_y\right]$
(the constant path) for $y\not\in \left\{\gamma_1\left(0\right),\ldots,\gamma_n\left(0\right)\right\}$.\\
The inclusion is a homomorphism with respect to the group law defined on
$map_0\left(A_0,\left[\left[0,1\right],\mid A\mid\right]_{\mid P\mid}\right)$
by $\left[gf\left(y\right)\right]:=\left[f\left(y\right)\right]*\left[g\left(f\left(
y\right)\left(1\right)\right)\right]$.\\


{\bf Action of $\Pi\left(A\right)$ on $P$:}\\
We define an action of $map_0\left(A_0,\left[\left[0,1\right],\mid A\mid\right]_P\right)$ on $P$.
This gives, in particular, an action of $\Pi\left(A\right)$ on $P$.\\
Let $g\in map_0\left(A_0,\left[\left(0,1\right),\mid A\mid\right]_P\right)$.
Define $gy=g\left(y\right)\left(1\right)$ for $y\in A_0$ and $gx=x$ for $x\in P_0-A_0$. This defines the action on the 0-skeleton of $P$.\\
As a next step, we extend this to an action on the 1-skeleton of $P$.
Recall that, by minimal completeness of $P$, $1$-simplices $\sigma$ are in 1-1-correspondence with homotopy classes (rel.\ $\left\{0,1\right\}$)
of (nonclosed) singular 1-simplices in $\mid P\mid$ with vertices in $P_0$.
Using this correspondence,
define $g\sigma:=\left[\overline{g\left(\sigma\left(0\right)\right)}\right]*\left[\sigma\right]*\left[g\left(\sigma\left(1\right)\right)\right]$, where $*$ denotes concatenation of (homotopy classes of) paths.

We remark that we get elements in $P_1$
because $g\left(\sigma\left(0\right)\right)^{-1}*\sigma*g\left(\sigma\left(
1\right)\right)$ is a singular 1-simplex with distinct endpoints.
Indeed, if the vertices of $g\sigma$ agreed, then 
we would have
$g\left(\sigma\left(0\right)\right)\left(1\right)=g\left(\sigma\left(1\right)\right)\left(1\right)$.
But, since $g\left(.\right)\left(1\right)$ is a bijection, this would contradict $\sigma\left(0\right)\not=\sigma\left(1\right)$.\\
One checks easily that $\left(gf\right)\left(\sigma\right)=g\left(f\left(\sigma\right)\right)$ for all $g,f\in
map_0\left(A_0,\left[\left[0,1\right],\mid A\mid\right]_{\mid P\mid}\right), \\
\sigma\in P_1$.
Thus we defined an action of $map_0\left(A_0,\left[\left[0,1\right],\mid A\mid\right]_{\mid P\mid}\right)$ on $P_1$. We claim that this action
extends (uniquely) to a simplicial action on $P$.
To define the action of $map_0\left(A_0,\left[\left[0,1\right],\mid A\mid\right]_{\mid P\mid}\right)$ on all of $P$,
we claim that for a simplex
$\sigma\in P$ with 1-skeleton $\sigma_1$, and $g\in
map_0\left(A_0,\left[\left[0,1\right],\mid A\mid\right]_{\mid P\mid}\right)$,
there exists {\em some} simplex in $P$ with 1-skeleton $g\sigma_1$. Since $P$ 
is aspherical, this will allow a unique extension of the group action from $P_1$ to $P$.

The claim is proved by induction on the dimension of $\sigma$. First, let $\sigma$ be a 2-simplex and denote $g\partial_0\sigma=e_0,g\partial_1\sigma=e_1,g\partial_2\sigma=e_2$. Then the concatenation of the oriented 1-simplices $e_2,\overline{e_1}$
and $e_0$ is homotopic to the concatenation of $\partial_2\sigma,\overline{\partial_1\sigma}$ and $\partial_0\sigma$, thus null-homotopic, therefore bounds a singular 2-simplex. By (minimal) completeness it bounds then a (unique) 2-simplex in $P$.

Similarly, if we already have extended the action to the n-1-skeleton and let $\sigma$ be an n-simplex, we denote $g\partial_i\sigma=\tau_i$ for $i=0,\ldots,n$ and we observe that the composition of the $\tau_i$ is 0-homotopic, hence bounds a singular simplex and, by completeness, even a simplex in 
$P$ which we define to be $g\sigma$.
(Note also that all vertices of $g\sigma$ are distinct, since $g\left(.\right)\left(1\right)$ permutes vertices.)
The equality $\left(gf\right)\left(\sigma\right)=g\left(f\left(\sigma\right)\right)$ extends from $P_1$ to $P$ because $\left(gf\right)\left(\sigma\right)$ is uniquely determined by $\left(gf\right)\left(\sigma_1\right)$.

This finishes the definition of the action.
We remark, because this will be one of the assumptions to apply Lemma 7, that the action of any 
element $g\in\Pi\left(A\right)$ is homotopic to the identity. The homotopy between 
the action of $\left\{\gamma_1,\ldots,\gamma_r\right\}$ and 
the identity is given by the action of
$\left\{\gamma_1^t,\ldots,\gamma_r^t\right\}, 0\le t\le 1$, 
with $\gamma_i^t\left(s\right)=\gamma_i\left(\frac{s}{t}\right)$.

The next Lemma follows directly from the construction, but we will use it so often that we want to
explicitly state it.

\begin{lem}: Let $\left(P,A\right)$ be a pair of aspherical, minimally complete multicomplexes, with the action of $G=\Pi\left(A\right)$. If $\sigma\in P$ is a simplex, all of whose vertices are not in $A$, then 
$g\sigma=\sigma$ for all $g\in G$.\end{lem}

%\begin{obs}: Assume that a simplex $\sigma$ has an edge $e$ in $A$. Then $\sigma$ is in the $\Pi\left(A\right)$-orbit of some degenerate simplex.\end{obs}
%\begin{pf} Let $g=\left\{e\right\}$. Then $g^{-1}\sigma$ has a constant edge.\end{pf}

For a topological space and a subset $P\subset S_*\left(X\right)$,
closed under face maps,
the (antisymmetric)
bounded cohomology $H_b^*\left(P\right)$
and its pseudonorm are defined literally like for multicomplexes 
in \cite{gro}, section 3.2. The following well-known fact will
be needed for applications of Lemma 7 (to the setting of Theorem 1)
with $P=K^{str}\left(\partial Q\right),G=\Pi\left(K\left(\partial_0Q\right)\right)$.

\begin{lem}: a) Let $\left(P,A\right)$ be a pair of minimally complete multicomplexes.
If each connected component of $\mid A\mid$ has amenable fundamental group, 
then $\Pi\left(A\right)$ is amenable.\\
b) Let $X$ be a topological space, $P\subset S_*\left(X\right)$ a subset closed under face maps, 
and $G$ an amenable group acting on $P$.
Then the canonical homomorphism $$id\otimes 1:
C_*^{simp}\left(P\right)\rightarrow C_*^{simp}\left(P\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$$
induces an isometric monomorphism in bounded cohomology.\end{lem}
\begin{pf} a) The proof is an obvious adaptation of the proof of \cite{k2}, Lemma 4.\\ 
b) This is proved by averaging bounded cochains, see \cite{gro}.\end{pf}







\subsection{Retraction to central simplices}


\begin{lem}:\\
a) Let $\left(N,\partial N\right)$ be a pair of
topological spaces with $N=Q\cup R$ for two subspaces
$Q,R$. Let $\partial_0Q=Q\cap R, \partial Q=\partial_0Q\cup\left(Q\cap\partial N\right),\partial R=
\partial_0Q\cup\left(R\cap\partial N\right), N^\prime=\partial Q\cup\partial R$.
Assume 
that the inclusions      
$\partial_0Q\rightarrow Q\rightarrow N, \partial_0Q\rightarrow R\rightarrow N$ as well   
as the inclusions                              
$\partial_0Q\rightarrow \partial Q\rightarrow N^\prime, \partial_0Q\rightarrow \partial 
R\rightarrow N^\prime$           
induce injections of the fundamental groups (for                              
each connected component).            
Consider the simplicial action of $G=\Pi\left(K\left(\partial_0Q\right)\right)$ on 
$\left(K\left(N\right),K\left(N^\prime\right)\right)$.\\         
Then there is a chain homomorphism $$r:C_*^{simp,inf}\left(K\left(N\right)
\right)\otimes_{{\Bbb Z}G}{\Bbb Z}
\rightarrow C_*^{simp,inf}\left(K\left(Q\right)                             
\right)\otimes_{{\Bbb Z}G}{\Bbb Z}                      
$$
in degrees $*\ge2$, mapping 
$C_*^{simp,inf}\left(K\left(N^\prime\right)
\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$ to $C_*^{simp,inf}\left(K\left(\partial Q\right)
\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$, such that\\                
- if $\sigma$ is a simplex in $K\left(N\right)$, then $r\left(\sigma\otimes 1 \right)=\kappa\otimes 1$, where either                    
$\kappa$ is a simplex in $K\left(Q\right)$        
or $\kappa=0$,\\              
- if $\sigma$ is a simplex in $K\left(Q\right)$, then $r\left(\sigma\otimes 1\right)=\sigma\otimes 1$; if 
$\sigma$ is a simplex in $K\left(R\right)$, not contained in $K\left(Q\cap R\right)$, then $r\left(\sigma\otimes 1\right)=0$.\\
%r\left(i_Q\otimes id\right)=id_{C_*\left(K\left(Q\right),K\left(\partial Q\right)\right) \otimes_{{\Bbb Z}G}{\Bbb Z}}, r\left(i_R\otimes id\right)=0$, where $i_Q:K\left(Q\right)\rightarrow K\left(N\right),i_R:K\left(R\right)\rightarrow
%K\left(N\right)$ denote the inclusions. \\
b) Let $C$ be a path-component of $\partial Q$ with $C\subset \partial_0Q$.
If $\sigma\in K\left(\partial N\right)$
%(the image under the canonical mapping $K\left(N\right)\rightarrow N$ of) a simplex
%$\sigma\in K\left(N\right)$ does not intersect $C$,
and if
$r\left(\sigma\otimes 1\right)=\kappa
\otimes 1$ with $\kappa\in K\left(Q\right)$, then no vertex of $\kappa$ is contained in $C$. \\
%In particular, if the image of $\sigma$ is contained in $\partial N$, then no vertex of $\kappa$ is contained in $C$.
c) If $\sigma\in K\left( N\right)$ and $r\left(\sigma\otimes 1\right)=\kappa
\otimes 1$,
then each edge of $\kappa$ is \\
- either in the G-orbit of some edge of $\sigma$,\\
- or connects a vertex of $\sigma$ to a vertex in $\partial_0Q$\\
- or connects vertices in two distinct path-components of $\partial_0Q$.
\end{lem}

\begin{pf}
a) is \cite{k2}, Proposition 6. The Lemma is there stated for locally finite chains, but of course $r$ extends linearly to infinite chains. b) and c) are \cite{k2}, Observation 3.
\end{pf}\\
\\
{\bf Fundamental cycles in $K\left(N\right)$ and $K\left(Q\right)$.} 
Let $N$ be a (possibly noncompact) connected,
orientable n-manifold with (possibly noncompact) boundary $\partial N$. Then $H_n^{lf}\left(N,\partial N\right)\simeq{\Bbb Z}$ by Whitehead's theorem and a generator is called $\left[N,\partial N\right]$. (It is only defined up to sign, but this will not concern our arguments.)
Recall that an infinite chain is said to represent
$\left[N,\partial N\right]$ if it is homologous to a locally finite chain representing $\left[N,\partial N\right]$.

For the proof of Theorem 1 we will need the following fact. (Recall that $C_*^{simp,inf}\left(K\left(N\right),K\left(\partial N\right)\right)
\subset
C_*^{sing,inf}\left(N,\partial N\right)$, thus it makes sense to say that some chain $z\in 
C_*^{simp,inf}\left(K\left(N\right),K\left(\partial N\right)\right)$ represents the 
fundamental class $\left[N,\partial N\right]$. Moreover remark that, 
with $N^\prime:=\partial N\cup\partial Q$, $\left[N,\partial N\right]$ 
has an image $\left[N,N^\prime\right]\in
C_*^{sing}\left(N,N^\prime\right)$.)


\begin{lem}: Let $N$ be an orientable n-manifold with boundary, $n\ge 2$, such that $N=Q\cup R$ with
orientable n-manifolds
with 
boundary $Q,R$, such that the assumptions of Lemma 5 are satisfied with $N^\prime=\partial N\cup\partial Q, R^\prime=\partial R,
Q^\prime=\partial Q$. \\
If $\sum_i a_i\sigma_i\in C_n^{simp,inf}\left(K\left(N\right),K\left(N^\prime\right)\right)$ represents 
$\left[N,N^\prime\right]$, then \\
a) $$\sum_i a_ir\left(\sigma_i\otimes 1\right)\in
C_n^{simp,inf}\left(K\left(Q\right),K\left(\partial Q\right)\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$$ represents
the image of $\left[Q,\partial Q\right]\otimes 1$ under the canonical homomorphism\\
$H_n^{sing,inf}\left(Q,\partial Q\right)\otimes_{{\Bbb Z}G}{\Bbb Z}\rightarrow H_n\left(
C_*^{sing,inf}\left(Q,\partial Q\right)\otimes_{{\Bbb Z}G}{\Bbb Z}\right)$, \\
b) $$\partial \sum_i a_ir\left(\sigma_i\otimes 1\right)\in
C_n^{simp,inf}\left(K\left(\partial Q\right)\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$$
represents
the image of $\left[\partial Q\right]\otimes 1$ under the canonical homomorphism\\
$H_n^{sing,inf}\left(\partial Q\right)\otimes_{{\Bbb Z}G}{\Bbb Z}\rightarrow H_n\left(
C_*^{sing,inf}\left(\partial Q\right)\otimes_{{\Bbb Z}G}{\Bbb Z}\right)$.\end{lem}

\begin{pf}
By Lemma 5, we have $r\left(i_Q\otimes id\right)=id, r\left(i_R\otimes id\right)=0\ mod\ C_n^{simp,inf}\left(K\left(Q\cap R\right)\right)$. 
Together with $\left[N,N^\prime\right]=\left(i_Q\right)_*\left[Q,Q^\prime\right]
+\left(i_R\right)_*\left[R,R^\prime\right]$ this implies a). 

It is well-known that the boundary operator $\partial:Z_n\left(Q,\partial Q\right)\rightarrow
Z_{n-1}\left(\partial Q\right)$ (mapping relative cycles to genuine cycles in $\partial Q$)
induces the connecting morphism (of the long exact homology sequence) in
homology and maps the relative fundamental class
$\left[Q,\partial Q\right]\in H_n\left(Q,\partial Q\right)$ to
the fundamental class $\left[\partial Q\right]\in H_{n-1}\left(\partial Q\right)$.
%Hence the boundary
%operator on $C_*^{simp}\left(K\left(Q\right)\right)$ maps (any) relative cycle representing
%$\left[Q,\partial Q\right]$ to
%a cycle representing $\left[\partial Q\right]$.
%Since $\partial$ commutes with $i_Q$,
After tensoring with ${\Bbb Z}$, we obtain claim b). 
\end{pf}



\subsection{Using amenability}


The following Lemma 7 is well-known in slightly different formulations and we reprove it here
only for completeness.
We will apply Lemma 7 in the proof of Theorem 1
with $K=\widehat{K}\left(\partial Q\right)$ and $G=\Pi\left(K\left(\partial_0Q\right)\right)$.
%We recall that, for a closed, orientable manifold $X$, we 
%say that \\
%$z\in C_*^{simp}\left(K\left(X\right)\right)$ is called a fundamental cycle if $i_*\left(z\right)$ represents the fundamental class $\left[X\right]$ (see section 3.2).
%Moreover, for a group $G$ acting on
%$C_*^{simp}\left(K\left(X\right)
%\right)$,
%we denote $$p:C_*^{simp}\left(K\left(X\right)
%\right)\rightarrow C_*^{simp}\left(K\left(X\right)
%\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$$ the homomorphism defined by 
%$$p\left(\sigma\right)=\sigma\otimes 1.$$
(The following lemma has of course also a relative version, but we will not need that for our argument.)

\begin{lem}: Let $X$ be a closed, orientable manifold and $K\subset S_*\left(X\right)$ closed under face maps.
Assume that \\
- there is an amenable group $G$ acting on $K$, such that the action of each $g\in G$ on $\mid K\mid$
is homotopic to the identity\\
- there is a fundamental cycle
$z\in C_*^{simp}\left(K\right)$
such that $z\otimes 1$ is homologous to a cycle
$h=\sum_{j=1}^s b_j \tau_j\otimes 1\in C_*^{simp}
\left(K\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$. \\
Then 
$$\parallel X\parallel\le \sum_{j=1}^s \mid b_j\mid.$$\end{lem}


\begin{pf}
If $\parallel X\parallel=0$, there is nothing to prove. \\
Thus we 
may assume $\parallel X\parallel\not=0$, which implies (\cite{gro}, p.17) that there is
$\beta\in H_b^n\left(X\right)$, a bounded cohomology 
class dual to 
$\left[X\right]\in H_n\left(X\right)$,
with $\parallel\beta\parallel=
\frac{1}{\parallel X\parallel}$. 

Let $p:C_*^{simp}\left(K
\right)\rightarrow C_*^{simp}\left(K
\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$ be the homomorphism defined by
$p\left(\sigma\right)=\sigma\otimes 1$.
Since $G$ is amenable we have, by the proof of Lemma 4b) in \cite{gro},
an 'averaging homomorphism' $Av:H_b^*
\left(K\right)\rightarrow 
H_b^*\left(C_*\left(K\right)
\otimes_{{\Bbb Z}G}{\Bbb Z}\right)$ such that $Av$ is left-inverse to $p^*$
and $Av$ is an isometry.
Hence we have $$\parallel Av\left(\beta\right)\parallel=\parallel\beta\parallel=\frac{1}{\parallel X\parallel}.$$
Moreover, denoting by $\left[\sum_{j=1}^sb_j\tau_j\otimes 1\right]$ the homology class of 
$\sum_{j=1}^sb_j\tau_j\otimes 1$, we have obviously
$$\mid Av\left(\beta\right)\left[\sum_{j=1}^sb_j\tau_j\otimes 1\right] \mid \le
\parallel Av\left(\beta\right)\parallel \sum_{j=1}^s\mid b_j\mid$$
and therefore
$$\parallel X\parallel=\frac{1}{\parallel Av\left(\beta\right)\parallel}\le\frac{
\sum_{j=1}^s\mid b_j\mid}{\mid Av\left(\beta\right)\left[\sum_{j=1}^sb_j\tau_j\otimes 1\right]\mid}.$$
It remains to prove $Av\left(\beta\right)\left[\sum_{j=1}^sb_j\tau_j\otimes 1\right]= 1$.\\
For this we have to look at the definition of $Av$, which is as follows:
by amenability there exists a bi-invariant mean $av: B\left(G\right)\rightarrow{\Bbb R}$ on the bounded functions on $G$ with $inf_{g\in G} \delta
\left(g\right)\le av\left(\delta\right)\le sup_{g\in G} 
\delta\left(g\right)$ for all $\delta\in B\left(G\right)$.
Then, given any $p\left(\sigma\right)\in C_*\left(K\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$ one can fix an identification between $G$ and
the set of all $\sigma^\prime$ with $p\left(\sigma^\prime\right)=p\left(\sigma\right)$ and thus consider the restriction of $\gamma$ to this set as a
bounded cochain on $G$. The average $av$ of this bounded cochain is defined to be $Av\left(\gamma\right)\left(p\left(\sigma\right)\right)$. (This definition is independent of all choices, see \cite{iva}.)\\
Now, if $z=\sum_{j=1}^sb_j\tau_j$ is a fundamental cycle, 
then we have $\beta\left(z\right)=1$. \\
If $g\in G$ is arbitrary, then left multiplication with $g$ is a chain map
on $C_*^{simp}\left(K\right)$, as well as on $C_*^{sing}\left(X\right)$. Since the
action of $g$ on $\mid K\mid$ is
homotopic to the identity, it induces the identity on the image of $C_*^{simp}\left(K\right)\rightarrow C_*^{sing}\left(X\right)$.
%This chain map has an inverse, given by left multiplication with $g^{-1}$. Both chain maps are preserving the
%integer lattice in real homology. Since $H_n\left(X\right)$ is one-dimensional for $n=dim\left(X\right)$, 
Thus, for each cycle $z\in
C_*^{simp}\left(K\right)$ representing $\left[X\right]\in H_*^{sing}\left(X\right)$, the cycle $gz\in
C_*^{simp}\left(K\right)$ must also 
represent $\left[X\right]$. \\
If $gz$ represents $\left[X\right]$, 
then $\beta\left(gz\right)=\beta\left(\left[X\right]\right)=1$. 
%If $gz$ represents $-\left[X\right]$, then,
%since the bounded cohomology of the multicomplex $K\left(X\right)$ is defined using {\em antisymmetric}
%bounded cochains, we have $\beta\left(gz\right)=-1$. Moreover, since the action of every element $g$ is 
%homotopic to the identity, $\beta\left(gz\right)$ must not depend on $g$, for each cycle $z$.
In conclusion, we have $\beta\left(p\left(z^\prime\right)\right)=1$ for each $z^\prime$
with $p\left(z^\prime\right)=p\left(z\right)$.
%or $\beta\left(p\left(z^\prime\right)\right)=-1$ for each $z^\prime$
%with $p\left(z^\prime\right)=p\left(z\right)$. 
By definition of 
$Av$, this implies $Av\left(\beta\right)\left(p\left(z\right)\right)= 1$ for 
each fundamental cycle $z$.\\
In particular, $Av\left(\beta\right)\left[\sum_{j=1}^sb_j\tau_j\otimes 1\right]=1$, what finishes the proof of the lemma.
\end{pf}\\
\\
Remark: In the proof of Theorem 1, we will work with $C_*^{simp}\left(K\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$
rather than $C_*^{simp}\left(K\right)$. This is analogous to
Agol's construction of "crushing the cusps to points"
in \cite{ag}. However 
$C_*^{simp}\left(K\left(Q\right)\right)\otimes_{{\Bbb Z}\Pi\left(\partial_0Q
\right)}{\Bbb Z}\not= C_*^{simp}\left(K\left(Q/\partial_0Q\right)\right)$,
thus one can not simplify our arguments by working directly with $Q/\partial_0Q$.





\section{Disjoint planes in a simplex}

In this section, we will discuss the possibilities how a simplex can be cut by planes 
without producing parallel arcs in the boundary. (More precisely, 
we pose the additional condition that the components of the complement can be coloured by black and white such that all vertices belong to black components, and we actually 
want to avoid only parallel arcs in the boundary of white components.)
For example, for the 3-simplex, it
will follow that there is essentially only the possibility in Case 1, pictured below, meanwhile in Case 2
each triangle has a parallel arc with another triangle, regardless how the quadrangle is triangulated. 
%(Of course, in case 1 one may have less than 4 triangles.)

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Let $\Delta^n\subset{\Bbb R}^{n+1}$ be the standard simplex with vertices $v_0,\ldots,v_n$. It is contained in the plane $E=\left\{\left(x_1,\ldots,x_{n+1}\right)\in{\Bbb R}^{n+1}:x_1+\ldots+x_{n+1}=1\right\}$.

In this section we will be interested in n-1-dimensional affine planes $P\subset
E$ whose intersection with $\Delta^n$ either contains no vertex, consists of exactly one vertex, or consists of a face of $\Delta^n$.
For such planes we define their type as follows.


\begin{df}: 
Let $P\subset E$ be an n-1-dimensional affine plane such that $P\cap\Delta^n$ either contains no vertex,
consists of exactly one vertex, or consists of a face of $\Delta^n$.\\
If $P\cap\Delta^n=\partial_0\Delta^n$, then we say that $P$ is of type $\left\{0\right\}$.\\
If $P\cap\Delta^n=\partial_j\Delta^n$ with $j\ge 1$, then we say that $P$ is of type $\left\{01\ldots \hat{j}\ldots n\right\}$.\\
If $P\cap\left\{v_0,\ldots,v_n\right\}=\left\{v_0\right\}$, then we say that $P$ is of type $\left\{0\right\}$.\\
If $P\cap\left\{v_0,\ldots,v_n\right\}=\emptyset$ or $P\cap\left\{v_0,\ldots,v_n\right\}=\left\{v_j\right\}$ with $j\ge 1$, then we say that $P$ is of type $\left\{0 a_1\ldots a_k\right\}$ with $a_1,\ldots,a_k\in\left\{1,\ldots,n\right\}$ if: 

$v_i$ belongs to the same connected component of $\Delta^n - \left(P\cap\Delta^n\right)$ as 
$v_0$ 

if and only if $i\in\left\{a_1,\ldots,a_k\right\}$.
\end{df}

%\begin{obs}: If $P$ is of type $\left\{0a_1\ldots a_k\right\}$,
%then the polytope $Q$ can be triangulated into $C_k:=min\left\{k+1,n-k\right\}$ simplices.\end{obs}

\begin{obs}: Let $P_1,P_2$ be two planes of type $\left\{0a_1\ldots a_k\right\}$ resp.\ $\left\{0b_1\ldots b_l\right\}$ and let $Q_1=P_1\cap\Delta^n\not=\emptyset, Q_2=P_2\cap\Delta^n\not=\emptyset$. 
Then $Q_1\cap Q_2=\emptyset$ implies that 
either $\left\{a_1,\ldots,a_k\right\}= \left\{b_1,\ldots,b_l\right\}$ or exactly one of the following conditions holds:\\
- $\left\{a_1,\ldots,a_k\right\}\subset \left\{b_1,\ldots,b_l\right\}$,\\ 
- $\left\{b_1,\ldots,b_l\right\}\subset
\left\{a_1,\ldots,a_k\right\}$,\\
- $\left\{a_1,\ldots,a_k\right\}\cup \left\{b_1,\ldots,b_l\right\} =\left\{1,\ldots,n\right\}$.\end{obs}
%(equivalently
%$\left\{1,\ldots,n\right\}-\left\{b_1,\ldots,b_l\right\}\subset
%\left\{a_1,\ldots,a_k\right\}$).\end{obs}
\begin{pf} $\Delta^n-Q_1$ consists of two connected components, $C_1$ 
and $C_2$. W.l.o.g.\ assume that $v_0\in C_1$. $\Delta^n-Q_2$ 
consists of two connected components, $D_1$ and $D_2$. W.l.o.g.\ assume that $v_0\in D_1$. In particular, $C_1\cap D_1\not=\emptyset$.

If $Q_1\cap Q_2=\emptyset$, then $Q_2$ is contained in one of $C_1$ or
$C_2$, and
$Q_1$ is contained in one of $D_1$ or $D_2$.\\
Case 1: $Q_1\subset D_1$. Then either we have $C_1\subset D_1$, which implies $\left\{a_1,\ldots,a_k\right\}\subset \left\{b_1,\ldots,b_l\right\}$, or we have $C_2\subset D_1$, which implies $\left\{1,\ldots,n\right\}-
\left\{a_1,\ldots,a_k\right\}\subset \left\{b_1,\ldots,b_l\right\}$, hence 
$\left\{a_1,\ldots,a_k\right\}\cup \left\{b_1,\ldots,b_l\right\} =\left\{1,\ldots,n\right\}$.\\
Case 2: $Q_1\subset D_2$. This implies $Q_2\subset C_1$ and after interchanging $Q_1$ and $Q_2$ we are in Case 1.\end{pf}\\
\\
Notational remark: 'arc' will mean the intersection of 
an n-1-dimensional affine plane $P\subset E$ (such that $P\cap\Delta^n\not=\emptyset$ 
either contains no vertex, consists
of exactly one vertex or consists of a face) with a 2-dimensional subsimplex $\tau^2\subset\Delta^n$. If an arc consists of only one vertex, we call it a degenerate arc.


\begin{df}{\bf (Parallel arcs)}: Let $P_1,P_2\subset E$ be n-1-dimensional affine planes. Let $\tau^2$
be a 2-dimensional subsimplex of $\Delta^n$ with vertices $v_r,v_s,v_t$.
We say that disjoint arcs $e_1,e_2$ obtained 
as intersections of $P_1$ resp.\ $P_2$ with (the same) $\tau^2$ are parallel arcs if \\
- both are nondegenerate and any two of $\left\{v_r,v_s,v_t\right\}$ belong to the same connected component of $\tau^2-e_1$ if and only if they belong to the same
connected component of $\tau^2 - e_2$,\\
- one, say $e_1$ is nondegenerate, the other, say with vertices $v_s,v_t$ is contained in a face, and $v_r$ belongs to another connected component of $\tau^2-e_1$ as both $v_s$ and $v_t$,\\ 
- one, say $e_1$, is nondegenerate, the other is degenerate, say equal to $v_r$, and both $v_s,v_t$ belong to another
connected component of $\tau^2-e_1$ as $v_r$,\\
- both are degenerate and equal,\\
- both are contained in a face and equal,\\
- one is degenerate, the other is contained in a face.\end{df}


\begin{lem}: Let $\Delta^n\subset{\Bbb R}^{n+1}$
be the standard simplex. Let $P_1,P_2\subset E$ be n-1-dimensional affine planes with $Q_i=P_i\cap\Delta^n\not=\emptyset$ for
$i=1,2$.\\
Let $P_1$ be of type $
\left\{0a_1\ldots a_k\right\}$ with $1\le k\le n-2$ and $P_2$ of type $\left\{0b_1\ldots b_l\right\}$ with $l$ arbitrary.\\
Then either $Q_1\cap Q_2\not=\emptyset$, or $Q_1$ and $Q_2$ have a parallel arc.\end{lem}

\begin{pf}
Assume that $Q_1\cap Q_2=\emptyset$.\\
By Observation 2, there are 4 possible cases if $Q_1\cap Q_2=\emptyset$. \\
Case 1: $\left\{0a_1\ldots a_k\right\}=\left\{0b_1\ldots b_l\right\}$. Then we clearly have parallel arcs.\\
Case 2: $\left\{0a_1\ldots a_k\right\}$ is a proper subset of $\left\{0b_1\ldots b_l\right\}$, i.e.\ 
$1\le k<l\le n-1$ and $a_1=b_1,\ldots,a_k=b_k$. There is at least one index, say $i$, not contained in 
$\left\{0b_1\ldots b_l\right\}$. Consider the 2-dimensional subsimplex $\tau^2\subset
\Delta^n$ with vertices $v_0,v_{a_1},v_i$. It intersects $P_1$ and $P_2$ in parallel arcs.\\
Case 3: $\left\{0b_1\ldots b_l\right\}$ is a proper subset of $\left\{0a_1\ldots a_k\right\}$, i.e.\ 
$ 0\le l <k\le n-2$ and $a_1=b_1,\ldots,a_l=b_l$. There are two indices $i,j$ not contained in 
$\left\{0a_1\ldots a_k\right\}$. Consider the 2-dimensional subsimplex $\tau^2\subset
\Delta^n$ with vertices $v_0,v_i,v_j$. It intersects $P_1$ and $P_2$ in parallel arcs.\\
Case 4: 
$\left\{a_1,\ldots,a_k\right\}\cup\left\{b_1,\ldots,b_l\right\}=\left\{1,\ldots,n\right\}$.
By $k\le n-2$, there are two indices $i,j$ with
$i,j\not\in\left\{0a_1\ldots a_k\right\}$. Hence $i,j\in \left\{b_1,\ldots,b_l\right\}$. Moreover, there exists
an index $h$ such that $h\in\left\{a_1,\ldots,a_k\right\}$ but
$h\not\in\left\{b_1,\ldots,b_l\right\}$. (If not, we would have $\left\{a_1,\ldots,a_k\right\}\subset 
\left\{b_1,\ldots,b_l\right\}$, hence $\left\{1,\ldots,n\right\}=
\left\{a_1,\ldots,a_k\right\}\cup 
\left\{b_1,\ldots,b_l\right\}\subset\left\{b_1,\ldots,b_l\right\}$,
contradicting $Q_2\not=\emptyset$.) Consider the 2-dimensional subsimplex $\tau^2\subset
\Delta^n$ with vertices $v_i,v_j,v_h$. It intersects $P_1$ and $P_2$ in parallel arcs.

\end{pf} 

\begin{df} {\bf (Canonical colouring of complementary regions)}:\\
Let $P_1,P_2,\ldots\subset E$ be a (possibly infinite)
set of n-1-dimensional affine planes with
$Q_i:=P_i\cap \Delta^n\not=\emptyset$ and $Q_i\cap Q_j=\emptyset$ for all $i\not=j$. Assume that each $Q_i$ either
contains no vertices or consists of exactly one vertex.

A colouring of\\
- the connected components
of $\Delta^n-\cup_i Q_i$ by colours black and white, and \\
- of all $Q_i$ by black,\\
is called a canonical colouring (associated to $P_1,P_2,\ldots$) if:\\
- all vertices of $\Delta^n$ are coloured black,\\
- each $Q_i$ is incident to at least one white component. \end{df}
%It is clear that a canonical colouring is unique when it exists.

\begin{df} {\bf (White-parallel arcs)}: Let $\left\{P_i: i\in I\right\}$ be a set of
of $n-1$-dimensional affine planes $P_i\subset E$, with $Q_i:=P_i\cap\Delta^n\not=\emptyset$ for $i\in I$.
Assume that $Q_i\cap Q_j=\emptyset$ for all $i\not=j\in I$, and that we
have a canonical colouring associated to $\left\{P_i: i\in I\right\}$.
We say that arcs $e_i,e_j$ obtained
as intersections of $P_i,P_j$ ($i,j\in I$) with some 2-dimensional subsimplex
$\tau^2$ of $\Delta^n$ are white-parallel arcs if they are parallel arcs
and, moreover, belong to the boundary of the closure of the same white component.\end{df}

The following two corollaries will not be needed for the proof of Lemma 10, but they will be necessary for the proof of Theorem 1.

\begin{cor}: Let $\Delta^n\subset{\Bbb R}^{n+1}$ 
be the standard simplex. Let $P_1,\ldots,P_m\subset E$ be a finite set of
n-1-dimensional
affine planes and let $Q_i=P_i\cap\Delta^n$ for 
$i=1,\ldots,m$. 

Assume that $Q_i\cap Q_j=\emptyset$ for all $i\not=j$, and that
we have an associated 
canonical colouring, such that
$Q_i$ and $Q_j$ 
do not have a white-parallel arc for $i\not=j$.

Then either $m=0$, or\\
$m=n+1$ and $P_1$ is of type $\left\{0\right\}$, $P_{n+1}$ is of type $\left\{0,1\ldots n-1\right\}$,
and $P_i$ is of type
$\left\{0,1\ldots i-2,i,\ldots n\right\}$ for $i=2,\ldots,n$.\end{cor}



\begin{pf}
If the conclusion were not true, there would exist a plane $P_1$ of type $\left\{0a_1\ldots a_k\right\}$ with $1\le k\le n-2$. Let $W$ be the white component of the canonical colouring, which is incident to $P_1$. Because, for a canonical colouring, no vertex belongs to a white component, there must be at least one more plane $P_2$ incident to $W$. Since $Q_1\cap Q_2=\emptyset$, from Lemma 8
we get that $Q_1$ and $Q_2$ have a parallel arc. Because $Q_1$ and $Q_2$ are incident to $W$, the arc is white-parallel.\end{pf}

\begin{cor}: Let $\Delta^n\subset{\Bbb R}^{n+1}$                                                                   be 
the standard simplex. Let $P_1,P_2,\ldots\subset E$ be a (possibly infinite) set of
n-1-dimensional                                                                                                    affine 
planes and let $Q_i=P_i\cap\Delta^n$ for             
$i=1,2,\ldots$.  
Assume that we have an associated canonical colouring.
\\
Let $P_i$ be of type 
$                        \left\{0a_1^i\ldots a^i_{c\left(i\right)}\right\}$, for $i=1,2,\ldots$.
Then \\
- either $c\left(1\right)\in\left\{0,n-1\right\}$,\\
- or whenever, for some $i\in\left\{2,3,\ldots\right\}$,
$P_1$ and $ P_i$ bound a white component of $\Delta^n-\cup_jQ_j$, then they must have a white-parallel arc.\end{cor}

\begin{pf} Assume that $c\left(1\right)\not \in\left\{0,n-1\right\}$. 
Let $W$ be the white component bounded by $P_1$.
%In particular, 
%there is no connected component of $\Delta^n-\cup_i Q_i$ bounded by n+1 planes of type $\left\{0\right\},
%\left\{023\ldots n\right\},\left\{013\ldots n\right\},\ldots,\left\{012\ldots n-2,n\right\},\left\{012\ldots n-2,n-1\right\}$. 
%Thus each white component $W$
%is bounded by at least one plane $P_1$ of type $\left\{0a_1^1\ldots a^1_{c\left(1\right)}\right\}$ with $1\le c\left(1\right)\le n-2$. 
$W$ is bounded by a finite number of planes, thus we can
apply Corollary 1, and conclude that $P_1$ has a 
white-parallel arc with each other plane adjacent to $W$.  \end{pf}

\begin{df}: Let $P\subset E$ be an n-1-dimensional affine plane, and $T$ a triangulation of the 
polytope $Q:=P\cap\Delta^n$. We say that $T$ is minimal, if all vertices of $T$ are vertices of $Q$. 
We say that an edge of some simplex in $T$ is an exterior edge if it is an edge of $Q$.\end{df}

\begin{obs}: Let $P\subset E$ be an n-1-dimensional affine plane, and $T$ a triangulation of the
polytope $Q:=P\cap\Delta^n$. If $T$ is minimal, then each edge of $Q$ is an (exterior)
edge of (exactly one) simplex in $T$.\end{obs}
\begin{pf} By minimality, the triangulation does not introduce new vertices. Thus every edge of $Q$ is an edge of some simplex.\end{pf}

\begin{obs}: Let $P\subset E$ be an n-1-dimensional affine plane with $Q:=P\cap \Delta^n\not=\emptyset$.
Assume that $P$ is of type
$\left\{0a_1\ldots a_k\right\}$.\\
a) Each vertex of $Q$ arises as the intersection of $P$ with an edge $e$ of $\Delta^n$. The vertices of $e$ are $v_i$ and $v_j$ with $
i\in\left\{0,a_1,\ldots,a_k\right\}$ and
$j\not\in \left\{0,a_1,\ldots,a_k\right\}$. (We will denote such a vertex by $\left(v_iv_j\right)$.)\\
b) Two vertices
$\left(v_{i_1}v_{j_1}\right)$ and $\left(v_{i_2}v_{j_2}\right)$ of $Q$ are connected by an edge of $Q$ (i.e.\ an exterior edge of any triangulation)
if either $i_1=i_2$ or $j_1=j_2$. \end{obs}
\begin{pf} a) holds because $e$ has to connect vertices in distinct components of $\Delta^n-Q$. b) holds because the edge of $Q$ has to belong to some 2-dimensional subsimplex of $\Delta^n$, with vertices either $v_{i_1},v_{j_1},v_{j_2}$ or $v_{i_1},v_{i_2},v_{j_1}$.\end{pf}\\
\\
Remark: if, for an affine hyperplane $P\subset E$, $Q=P\cap \Delta^n$ consists of exactly one vertex, then we will consider the minimal triangulation
of $Q$ to consist of one (degenerate) n-1-simplex. This convention helps to avoid needless case distinctions.

%\begin{obs}: Let $P\subset E$ be an n-1-dimensional affine plane, and $T$ a minimal triangulation of the
%polytope $Q:=P\cap\Delta^n$. Then each n-1-simplex of $T$ contains an
%exterior edge.\end{obs}
%\begin{pf}
%Let $P$ be of type $\left\{0a_1\ldots a_k\right\}$. 
%If $k=0$ or $k=n-1$, then $Q$ is an n-1-simplex and there is nothing to prove.
%
%So assume $1\le k\le n-2$, and assume that some n-1-simplex $\tau$ in a minimal triangulation of $Q$ has no exterior edge. 
%Let $\left(v_{i_1}v_{j_1}\right),\ldots\left(v_{i_n}v_{j_n}\right)$ be the vertices of $\tau$. If $\tau$ has no ecterior edge, then $i_1,\ldots,i_n$ are pairwise distinct.
%However, $i_1,\ldots,i_n\in\left\{0,a_1,\ldots,a_k\right\}$, and this contradicts $k\le n-2$.\end{pf}




\begin{lem}: 
Let $\left\{P_i\subset E:i\in I\right\}$ be a set of n-1-dimensional affine
planes and let $Q_i:=P_i\cap\Delta^n$ for $i\in I$.
Assume that $Q_i\cap Q_j=\emptyset$ for all $i\not=j$ and that
we have an associated
canonical colouring.
Assume that we have fixed, for each $i\in I$, a minimal triangulation $Q_i=\cup_a\tau_{ia}$ of $Q_i$. 

If $P_1$ is of type $\left\{0a_1^1\ldots a_{c\left(1\right)}^1\right\}$ with $1\le c\left(1\right)\le n-2$, then for each simplex $\tau_{1a}\subset Q_1$
there exists some $j\in I$ and some simplex $\tau_{jb}\subset Q_j$ (of the fixed triangulation of $Q_j$)
such that $\tau_{ia}$ and $\tau_{jb}$
have a white-parallel arc.\end{lem}

\begin{pf} 
%Let $P_i$ be of type
%$\left\{0a_1^i\ldots a_{c\left(i\right)}^i\right\}$. 
Let $w_1,\ldots,w_n$ be the $n$ vertices of the n-1-simplex $\tau_{1k}$.  
By Observation 4a), each 
$w_l$ arises as intersection of $Q_1$ with some edge $\left(v_{r_l}v_{s_l}\right)$ of $\Delta^n$, and the 
vertices $v_{r_l},v_{s_l}$ 
satisfy $r_l\in \left\{0,a_1^1,\ldots, a_{c\left(1\right)}^1\right\}$ and
$s_l\not\in \left\{0,a_1^1,\ldots, a_{c\left(1\right)}^1\right\}$. 

For the canonical colouring, there must be a white component 
$W$ bounded by $P_1$. We distinguish the cases whether $W$ and $v_0$ belong to the same connected component
of $\Delta^n-Q_1$ or not.\\
Case 1: $W$ and $v_0$ belong to the same connected component
of $\Delta^n-Q_1$. 

Since $c\left(1\right)\le n-2$, there exist at most $n-1$ possible values for $r_l$.
Hence there exists $l\not=m\in\left\{1,\ldots,n\right\}$
such that $v_{r_l}=v_{r_m}$. 

Let $e$ be the edge of $\tau_{1k}\subset Q_1$ connecting $w_{l}$ and $w_{m}$. By Observation 4b), $e$ is an exterior edge.
Consider the 2-dimensional subsimplex $\tau^2\subset\Delta^n$ with vertices $v_{r_l},v_{s_l},v_{s_m}$. We have that $P_1$ intersects $\tau^2$ in $e$, i.e.\ in an
arc separating $v_{r_l}$ from the other two vertices of $\tau^2$. 

Note that $r_l\in \left\{0,a_1^1,\ldots, a_{c\left(1\right)}^1\right\}$, hence $v_{r_l}$
belongs to the same component of $\Delta^n-Q_1$ as $v_0$. In particular,
$v_{r_l}$
belongs to the same component of $\Delta^n-Q_1$ as $W$. On the other hand, since the colouring is canonical, all vertices are coloured black and
$v_{r_l}$ can not belong to the white component $W$. Thus 
there must be some plane $P_j$ such that $Q_j$ bounds $W$ and separates $v_{r_l}$ from $Q_1$. (The possiblity $P_j\cap\Delta^n=\left\{v_{r_l}\right\}$ is allowed.)
In particular, some (possibly degenerate)
exterior edge $f$ of $Q_j$ separates $v_{r_l}$ from $v_{s_l},v_{s_m}$. Thus $e$ and $f$ are white-parallel arcs.
By Observation 3, $f$ is an edge of some $\tau_{jl}$.\\
\\
Case 2:
$W$ and $v_0$ don't belong to the same connected component
of $\Delta^n-Q_1$.

Since $n-c\left(1\right)\le n-1$, there exist some $l\not=m\in\left\{1,\ldots,n\right\}$
such that $v_{s_l}=v_{s_m}$.

Let $e$ be the edge of $\tau_{1k}\subset Q_1$ connecting $w_{l}$ and $w_{m}$. $e$ is an 
exterior edge by Observation 4b).
Consider the 2-dimensional subsimplex $\tau^2\subset\Delta^n$ with vertices $v_{r_l},v_{r_m},v_{s_l}$.
$P_1$ intersects $\tau^2$ in $e$, i.e.\ in an
arc separating $v_{s_l}$ from the other two vertices of $\tau^2$. 

We have that $s_l\not\in \left\{0,a_1^1,\ldots, a_{c\left(1\right)}^1\right\}$, hence $v_{s_l}$
does not belong to the same component of $\Delta^n-Q_1$ as $v_0$. This implies that
$v_{s_l}$
belongs to the same component of $\Delta^n-Q_1$ as $W$. On the other hand, since the colouring is canonical, $v_{s_l}$ 
can not belong to the white component $W$ and there must be 
some plane $P_j$ such that $Q_j$ bounds $W$ and separates $v_{s_l}$ from $Q_1$. 
In particular, some exterior edge $f$ of $Q_j$ separates $v_{s_l}$ from $v_{r_l},v_{r_m}$. Thus $e$ and $f$ are white-parallel arcs.
By Observation 3, $f$ is an edge of some $\tau_{jl}$.\end{pf}









\begin{lem}: 
Let $\left\{P_i:i\in I\right\}$ be a set of n-1-dimensional affine
planes with $Q_i:=P_i\cap\Delta^n\not=\emptyset$ for $i\in I$. Let $P_i$ be of type $\left\{
0a_1^{\left(i\right)}\ldots a^{\left(i\right)}_{k_i}\right\}$ for $i\in I$.
Assume that
$Q_i\cap Q_j=\emptyset$ for $i\not=j\in I$,
and that we have an associated canonical colouring.
Assume that for each $Q_i$ one has fixed a minimal triangulation $Q_i=\cup_{k=1}^{t\left(i\right)}\tau_{ik}$.

For each $i\in I$, let $$D_i=\sharp\left\{\tau_{ik}\subset Q_i:\mbox{\ there\ is\ no\ }\tau_{jl}\subset Q_j\mbox{\ such\ that\ }\tau_{ik},\tau_{jl}\mbox{\ have\ a\ white-parallel\ arc}\right\}.$$
Then $$\sum_{i\in I} D_i=0\ \ or\ \ \sum_{i\in I} D_i = n+1.$$
%In particular, the minimal triangulations of $Q_1,Q_2,\ldots$ contain altogether at most $n+1$ simplices
%without white-parallel arcs.
\end{lem}

 \begin{pf}
First we remark that the number of planes may be infinite, but we may of course remove pairs
of planes $P_i,P_j$ whenever they are of the same type and bound the same white component. This removal (of $P_i,P_j$ and the common white component) 
does not affect $\sum_{i\in I} D_i$. Since there are only finitely many different types of planes,
we may w.l.o.g.\ assume that we start with a finite number $P_1,\ldots,P_m$ of planes. (It may happen 
that after this removal no planes and no white components remain. In this case 
$\sum_{i\in I} D_{i\in I}=0$.) So we assume now that we have a finite number 
of planes $P_1,\ldots,P_m$, and no two planes of the same type bound a white region.

The first case to consider is that all planes are of type $\left\{0a_1\ldots a_k\right\}$ with 
$k=0$ or $k=n-1$. Since all vertices are coloured black, this means that $m=n+1$ and (upon renumbering) 
$P_1$ is of type $\left\{0\right\}$, $P_{n+1}$ is of type $\left\{0,1\ldots n-1\right\}$,
and $P_i$ is of type
$\left\{0,1\ldots i-2,i,\ldots n\right\}$ for $i=2,\ldots,n$. 
Hence $D_1=\ldots=D_{n+1}=1$ and $\sum_{i=1}^{n+1} D_i=n+1$.

Now we assume that there exists $P_i$, w.l.o.g.\ $P_1$,
of type 
$\left\{0a_1^{\left(1\right)}\ldots a_{k_1}^{\left(1\right)}\right\}$ with $1\le c\left(1\right)\le n-2$. Let $W$ be the 
white component bounded by $P_1$ and let w.l.o.g.\ $P_2,\ldots,P_l$ be the other planes bounding $W$. 
Then Lemma 9 says that each simplex in the chosen triangulation of $Q_1$ has a parallel arc with some simplex in the chosen triangulation of each of $Q_2,\ldots,Q_l$. 
In particular, $D_1=0$. For $j\in\left\{2,\ldots,l\right\}$,
if $1\le c\left(j\right)\le n-2$, the same argument shows that $D_j=0$. 
If $j\in\left\{2,\ldots,l\right\}$ and
$c\left(j\right)=0$ or $c\left(j\right)=n-1$, 
then $Q_j$ consists of only one simplex. By Corollary 2, this simplex has a parallel arc with 
(some exterior edge of) $Q_1$ and thus
(by Observation 3) with (some) simplex of the chosen triangulation of $Q_1$. This shows $D_j=0$ 
also in this case. Altogether we conclude $\sum_{j=1}^l D_j=0$ and 
thus $\sum_{i=1}^m D_i=\sum_{i=l+1}^m D_i$. Hence we can remove the white component $W$ and its
bounding planes $P_1,\ldots,P_l$ to obtain a smaller number of planes and
a new canonical colouring without changing $\sum_{i=1}^m D_i$.
Since we start with finitely many planes, we can repeat this reduction finitely many times and will end up either with an empty set of planes or with a set of planes of type $\left\{0a_1\ldots a_k\right\}$ with $k=0$ or $k=n-1$. Thus either $\sum_{i=1}^m D_i=0$ or $\sum_{i=1}^mD_i=n+1$.
\end{pf}\\
\\
We have thus proved that, in presence of a canonical colouring, the number of n-1-simplices without white-parallel arcs in a minimal triangulation of the $Q_i$'s is $0$ or $n+1$.
We remark that in the proof of Theorem 1 we will actually count only those triangles which neither have a white-parallel arc nor a degenerate arc. Thus, in general, we may remain with 
even less than $n+1$ n-1-simplices. 










\section{A straightening procedure}

In this section we will always work with the following set of assumptions.\\
\\
{\bf Assumption I}: {\em $Q$ is an aspherical manifold with aspherical boundary $\partial Q$.
We have subsets $\partial_0Q,\partial_1Q\subset\partial Q$ such that $\partial Q=\partial_0Q\cup\partial_1Q$ and
$\partial_1Q\not=\emptyset$. If $\partial_0Q\cap\partial_1Q\not=\emptyset$, we assume $\partial_0Q\cap\partial_1Q\rightarrow\partial_1Q$ is not $\pi_1$-surjective.}\\
\\
(The example we have in mind is a nonpositively curved manifold $Q$ with totally geodesic boundary $\partial_1Q$ and cusps corresponding to $\partial_0Q$.) 

\subsection{A motivating example}

Let $M$ be a connected hyperbolic manifold, $F$ a geodesic hypersurface, $Q=\overline{M-F}$. For simplicity we assume
that $M$ and $F$ are closed, thus $Q$ is a hyperbolic manifold with geodesic boundary 
$\partial_1Q\not=\emptyset$ consisting of two copies of $F$, and $\partial_0Q=\emptyset$.

Of course, there is a well-known 
straightening procedure (see \cite{bp}, Lemma C.4.3.), which homotopes each relative cycle into a straight 
relative cycle. However, we will need a more subtle construction, which considers 
relative cycles with a certain 0-1-labeling of their edges and straightens the 1-labeled edges into certain distinguished 1-simplices.\\
\\
{\bf Definition of $S_*^{str}\left(Q\right)\subset S_*\left(Q\right)$.}\\
\\
$\st$: Choose one vertex in each connected component of $\partial_1Q=\partial Q$. \\
\\$S_n^{str}\left(Q\right), n\ge1,$ consists of all straight (i.e.\ geodesic, parametrized according to the order of vertices) simplices with vertices in $S_0^{str}\left(Q\right)$.\\
(Remark: the definition differs from usual definitions in that we make restrictions on 
the allowed vertices.) \\
\\
{\bf Definition of distinguished 1-simplices.} 

Among the simplices of $\su$ choose, for each ordered
pair $\left(x,y\right)\in \st
\times \st$, one distinguished 1-simplex $e_{xy}$, e.g.\ the shortest geodesic connecting $x$ and $y$. (If there
are several shortest geodesics, choose only one of them.)\\
\\
Now we describe a straightening procedure for relative cycles $\sum_{i=1}^n a_i\tau_i$, under the 
assumption that there exists an admissible 0-1-labeling of $\sum_{i=1}a_i\tau_i$. (We say, 
see Definition 10 below, 
that a labeling of all edges of all $\tau_i$, by 0 and 1, is admissible if no cycle of edges is labeled with 1.) The straightening will be such that 1-labeled edges are straightened to distinguished 1-simplices.

First homotope $\sum_{i=1}^n a_i\tau_i$ such that each vertex is moved into $\st$. Vertices in $\partial Q$ 
must remain in $\partial
Q$ during the homotopy. After 
that, perform classical straightening (\cite{bp}, Lemma C.4.3) s.t. one gets a cycle of geodesic simplices 
with vertices in $\st$. (Note that the so constructed cycle consists of simplices 
contained in $S_n^{str}\left(Q\right)$, but it need not be the case
that 1-labelled edges are homotoped into distinguished 1-simplices.)\\
Let $e$ be an edge labelled with 1. Let $x=e\left(0\right),y=e\left(1\right)$ be the vertices of $e$.
Homotope $e$ into the unique distinguished edge $e_{xy}$ with $e_{xy}\left(0\right)=x,
e_{xy}\left(1\right)=y$. (Of course, 
we change the homotopy class relative boundary, so we can not keep the endpoints fixed during the homotopy.) For example this homotopy can be realized by first 
homotoping $e_{xy}$ into $e_{xy}*e^{-1}*e$ (using the canonical homotopy of $e^{-1}*e$ into its start point $y$)
and then homotoping $e_{xy}*e^{-1}*e$ into $e$ (using the canonical homotopy of $e_{xy}*e^{-1}$ into its endpoint $x$). \\
We succesively perform this homotopy for all 1-labeled edges (This is possible because no cycle of 
1-labeled edges exists.)
Then we 
extend this homotopy to a homotopy of the whole 1-skeleton. (If a 0-labeled edge has one or two vertices adjacent to a 1-labeled edge, then these vertices are just moved around during this homotopy.) Thus we have defined a new
straightening of the 1-skeleton. Now we can again
apply the usual straightening procedure (\cite{bp}, Lemma C.4.3) with fixed 1-skeleton to get a new straightening of 
$\sum_{i=1}^n a_i\tau_i$. By construction, 1-labeled edges are straightened into distinguished edges.\\

We remark that an analogous definition of $S_*^{str}\left(Q\right)$ and of distinguished 1-simplices,
and an analogous straightening procedure,
works whenever $M$ is of (not necessarily constant) nonpositive sectional curvature and $F$ is a geodesic hypersurface. 


\subsection{Definition of $K^{str}\left(Q\right)$}

Let $Q,\partial Q,\partial_0Q,\partial_1Q$ satisfy Assumption I.

Recall that we have defined
in section 3.2 a multicomplex $K\left(Q\right)\subset
 S_*\left(Q\right)$ with the property that (for aspherical $Q$) each singular simplex in $Q$, with boundary in $K\left(Q\right)$ and pairwise distinct vertices, is homotopic rel.\ boundary to a unique simplex 
in $K\left(Q\right)$. 
%We remark that, if $Q$ is aspherical, then $K\left(Q\right)$ can be chosen such that the following Assumption II is satisfied.\\
%\\
%{\bf Assumption II:} {\em
%If $\sigma\in K_n\left(Q\right)$ is homotopic rel.\ boundary to a simplex $\tau\in S_n\left(\partial Q\right)$, then $\sigma\in K_n\left(\partial
%Q\right)$. If
%$\sigma\in K_n\left(Q\right)$ is homotopic rel.\ boundary to a simplex $\tau\in S_n\left(\partial_1 Q\right)$, then
%$\sigma\in K_n\left(\partial_1 Q\right)$. If
%$\sigma\in K_n\left(Q\right)$ is homotopic rel.\ boundary to a simplex
%$\tau\in S_n\left(\partial_0 Q\right)$, then $\sigma\in K_n\left(\partial_0 Q\right)$}\\
%\\
%Indeed, we start with choosing simplices in $K\left(\partial_0Q\cap \partial_1Q\right)$ whenever possible.
%Then, if a simplex $\sigma$ with $\partial\sigma\in K_{n-1}\left(\partial_0Q\cap\partial_1 Q\right)$ is homotopic rel.\ boundary to different simplices $\sigma_0\in K\left(\partial_0Q\right)$ and $\sigma_1\in K\left(
%\partial_1Q\right)$, the homotopy between $\sigma_0$ and $\sigma_1$ gives a map ${\Bbb D}^{n+1}\rightarrow Q$
%such that 
%$\partial {\Bbb D}^{n+1}$ 
%is divided into two halves, one mapped by $\sigma_0$ to $\partial_0Q$, the other mapped by $\sigma_1$
%to $\partial_1Q$. Hence there must be an n-simplex in the given homotopy class 
%rel.\ boundary mapped to $\partial_0Q\cap\partial_1Q$. This shows that we can 
%choose simplices in $K\left(\partial_0Q\right)$ and $K\left(\partial_1Q\right)$ 
%without contradiction. Afterwards we choose simplices in $K\left(\partial
%Q\right)$ whenever possible, and continue the construction of
%jjjjjjjj$K\left(Q\right)$ arbitrarily.\\

The aim of this subsection is to describe a selection
procedure yielding
a subset $K_*^{str}\left(Q\right)\subset S_*\left(Q\right)$.
The final purpose of the straightening procedure will be to produce a large number of 
(weakly) degenerate simplices, in the sense of the following definition.
\begin{df}: Let $Q$ be an compact manifold with boundary $\partial Q$.
We say that a simplex in $S_n\left(Q\right)$ is degenerate
if
one of its edges is a constant loop. We say that it is weakly degenerate if it is degenerate 
or its image is contained in $\partial Q$.
\end{df}

Notational remark: for subsets $K_*^{str}\left(Q\right)\subset
S_*\left(Q\right)$ we will denote $K_*^{str}\left(\partial
Q\right):=K_*^{str}\left(Q\right)\cap S_*\left(\partial Q\right), K_*^{str}\left(\partial_1
Q\right):=K_*^{str}\left(Q\right)\cap S_*\left(\partial_1 Q\right),
K_*^{str}\left(\partial_0
Q\right):=K_*^{str}\left(Q\right)\cap S_*\left(\partial_0 Q\right)$.

\begin{lem}: Let $Q,\partial Q,\partial_0Q,\partial_1Q$ satisfy Assumption I.
Let $K\left(Q\right)\subset  S_*\left(Q\right)$ be as defined in section 3.2.

Then there exists a subset $K_*^{str}\left(Q\right)\subset
S_*\left(Q\right)$,
closed under face maps, 
such that:\\
i) $K_0^{str}\left(Q\right)$ contains each point in $\partial_0Q$,\\
ii) $K_0^{str}\left(Q\right)$ contains exactly one 0-simplex in each path-component $C$ of $\partial Q-\partial_0Q$,\\
iii) $K_0^{str}\left(Q\right)=K_0^{str}\left(\partial Q\right)$,\\
iv) $K_1^{str}\left(Q\right)$ consists of \\
- all 1-simplices $e\in K\left(Q\right)$ with $\partial e\in K_0^{str}\left(Q\right)$, and\\
- exactly one 1-simplex for each homotopy class (rel.\ boundary) of loops $e$ with $\partial_0e=\partial_1e\in K_0^{str}\left(Q\right)$,\\
v) for $n\ge 2$, if $\sigma\in S_n\left(Q\right)$ is an n-simplex with $\partial\sigma\in K^{str}_{n-1}
\left(Q\right)$,
then $\sigma$ is homotopic rel.\ boundary to a unique $\tau\in K_n^{str}\left(Q\right)$,\\
vi) if $\sigma\in K_n^{str}\left(Q\right)$ is homotopic rel.\ boundary to some $\tau\in K_n\left(Q\right)$, then $\sigma=\tau$,\\
vii) for $n\ge 2$, if $\sigma\in K_n^{str}\left(Q\right)$ is homotopic rel.\ boundary to a simplex $\tau\in S_n\left(\partial Q\right)$, then 
$\sigma\in K_n^{str}\left(\partial Q\right)$,\\
%viii) for $n\ge 2$, if 
%$\sigma\in K_n^{str}\left(Q\right)$ is homotopic rel.\ boundary to a simplex 
%$\tau\in S_n\left(\partial_0 Q\right)$, then $\sigma\in K_n^{str}\left(\partial_0 Q\right)$,\\
viii) $K^{str}_*\left(Q\right)$
is aspherical, i.e.\ if $\sigma,\tau\in K^{str}_*\left(Q\right)$ have the same 1-skeleton, then $\sigma=\tau$.

\end{lem}

\begin{pf}\\
$K_*^{str}\left(Q\right)$ is
defined by induction on the dimension of simplices as follows.\\
\\
Definition of $\stk$:\\
If $C$ is a path-component of $\partial Q-\partial_0Q$, then
choose exactly one point in $C$.\\
Moreover choose each point in $\partial_0Q$.\\
Note that we have chosen a nonempty set of 0-simplices since we are 
assuming $\partial_1Q\not=\emptyset$. Moreover, by construction, 
$K_0^{str}\left(Q\right)=K_0^{str}\left(\partial Q\right)$. \\
\\
Definition of $\suk$:\\
For an ordered pair $$\left(x,y\right)\in \stk
\times \stk$$ with $x\not=y$, there exists 
in each homotopy class (rel.\ boundary) of arcs $e$ with $$e\left(0\right)=x, e\left(1\right)=y$$ a unique simplex in $K_1\left(Q\right)$. Choose this simplex. (By the construction in section 3.2. these simplices belong to $\partial_0Q$ and/or $\partial_1Q$ if they are homotopic into one of them.)\\
Moreover, for pairs $$\left(x,x\right)\in \stk
\times \stk$$ choose one simplex in each homotopy class (rel.\ boundary) of loops $e$ with $$e\left(0\right)=e\left(1\right)=x.$$
For the homotopy class of the constant loop choose the constant loop. Choose the simplices in $\partial Q$ 
%and/or $\partial_1Q$ 
whenever this is possible. Hence vii) is satisfied for $n=1$.\\
%(Note that these choices do not contradict each other. Indeed, since 
%$S_0^{str}\left(\partial_0Q\right)\cap S_0^{str}\left(\partial_1Q\right)=\emptyset$, a 
%straight 1-simplex can not be homotopic rel.\ boundary
%into both $\partial_0Q$ and $\partial_1Q$. Moreover, if a 1-simplex,
%with vertices in $S_0^{str}\left(Q\right)\subset S_0\left(\partial Q\right)$, is homotopic to a degenerate 1-simplex, then this degenerate 1-simplex necessarily 
%has image in $\partial Q$. \\
\\
Definition of $K_n^{str}\left(Q\right)$ for $n\ge 2$, assuming that $K_{n-1}^{str}\left(Q\right)$ is defined:\\
For an $n+1$-tuple $\kappa_0,\ldots,\kappa_n$
of $n-1$-simplices in $K_{n-1}^{str}\left(Q\right)$, satisfying $$\partial_i\kappa_j=\partial_{j-1}\kappa_i$$ 
for all $i,j$, there are two possibilities:\\
- if no edge of any $\kappa_i$ is a loop, then there is 
%by asphericity of $Q$ 
a unique n-simplex
$$\sigma\in K_n\left(Q\right)$$
with $$\partial_i\sigma=\kappa_i$$ 
for $i=0,\ldots,n$. In this case set $\kappa:=\sigma$.\\
- otherwise, choose one arbitrary
$n$-simplex $$\kappa\in S_n\left(Q\right)$$  with $$\partial_i\kappa=\kappa_i$$ 
for $i=0,\ldots,n$. (By asphericity of $Q$, $\kappa$ exists
and is unique up to homotopy
rel.\ boundary.) Choose the simplices in $\partial Q$ 
%and/or $\partial_1Q$ 
whenever this is
possible. Hence vii) is satisfied for $n$.\\
\\
By construction, $K_*^{str}\left(Q\right)$ 
is closed under face maps and satisfies the conditions i)-vii). 
Since $Q$ is aspherical,
condition viii) follows by induction on the dimension of subsimplices of $\sigma$ and $\tau$ from condition v). 
\end{pf}\\

The simplices in $K_*^{str}\left(Q\right)$
will be called the {\bf straight simplices}. \\
We remark that $K_*^{str}\left(Q\right)$ is 
not a multicomplex because simplices in $K_*^{str}\left(Q\right)$ need not have pairwise distinct vertices.
%We emphasize that for $e=e_1*e_2*e_3\in
%\widehat{K_1^{str}}\left(Q\right)-K_1^{str}\left(Q\right)$ the decomposition $e_1*e_2*e_3$ can be recovered from $e$.\\





\begin{obs}: Let $Q,\partial Q,\partial_0Q,\partial_1Q$ satisfy Assumption I. 
Let $K_*^{str}\left(Q\right)
\subset  S_*\left(Q\right)$ 
satisfy the conditions i)-viii) from Lemma 11.

Then 
%$\left(K_*^{str}\left(Q\right),K_*^{str}\left(\partial
% Q\right)\right)$ is invariant under 
the action of
$$G:=\Pi\left(K\left(\partial_0Q\right)\right)$$ 
on $K\left(Q\right)$ as defined in section 3.3.\ can be extended to 
an action of $G$ on 
$$\left(K_*^{str}\left(Q\right),K_*^{str}\left(\partial Q\right)\right).$$

\end{obs}
\begin{pf} It is straightforward to check that $K_0^{str}\left(Q\right)$ is invariant under the $G$-action. Since $\sigma\in K\left(Q\right)$ belongs to $K^{str}\left(Q\right)$ if and only if all vertices of $\sigma$ belong to $K_0^{str}\left(Q\right)$, this implies that $K_*^{str}\left(Q\right)\cap K\left(Q\right)$ is invariant under the action of $G$.

It remains to extend the action to all of $K_*^{str}\left(Q\right)$. (Note that a simplex in $K_*^{str}\left(Q\right)$ does not belong to $K\left(Q\right)$ if and only if one of its edges is a loop.) By asphericity it suffices to 
extend the action on the 1-skeleton, i.e.\ on loops. This is done in the obvious way.
Let $e$ be a loop with $e\left(0\right)=e\left(1\right)=x$ 
and $g=\left\{\left[\gamma_1\right],\left[\gamma_2\right],\ldots\right\}$. Assume 
that after renumbering we either have $\gamma_1\left(0\right)=\gamma_1\left(1\right)=x$, or $\gamma_1\left(0\right)=\gamma_2\left(1\right)=x$.
(If no such renumbering exists, set $ge=e$.) In the first 
case set $ge:=\left[\gamma_1*e*\gamma_1\right]$, in the second case set 
$ge:=\left[\gamma_2*e*\gamma_1\right]$, where as usual $\left[.\right]$ means the unique element of $K_1^{str}\left(Q\right)$ in the homotopy class rel.\ boundary. It is straightforward to check that we have defined an action on $\left(K_*^{str}\left(Q\right),K_*^{str}\left(\partial Q\right)\right)$.\end{pf}


\subsection{Definition of the straightening}

\begin{df}: Let $\left(Q,\partial Q\right)$ be a pair
of topological spaces and 
let $z=\sum_{i\in I} a_i\tau_i\in C_n^{inf}\left(Q\right)$ a (possibly infinite)
singular chain.\\ 
a) A set of cancellations of $z$ is a symmetric set ${\mathcal{C}}
\subset S_{n-1}\left(Q\right)\times S_{n-1}\left(Q\right)$
with $\left(\eta_1,\eta_2\right)\in{\mathcal{C}}\Rightarrow \eta_1=\eta_2$ and $\eta_1=\partial_{a_1}\tau_{i_1},
\eta_2=\partial_{a_2}\tau_{i_2}$ for some $i_1,i_2\in I, a_1,a_2\in \left\{0,\ldots,n\right\}$.\\
b) If $z =\sum_{i\in I} a_i\tau_i\in C_n^{inf}\left(Q\right)$ and $\mathcal{C}$ is a set of cancellations for $z$, then the associated 
simplicial set $\Upsilon_{z,{\mathcal{C}}}$ is the set consisting of $\mid I\mid$ n-simplices $\Delta_i,
i\in I$, together with all their iterated faces and degenerations, subject to the identifications 
$\partial_{a_1}\Delta_{i_1}=\partial_{a_2}\Delta_{i_2}$ if and only if $\left(\partial_{a_1}\tau_{i_1},\partial_{a_2}\tau_{i_2}\right)\in{\mathcal{C}}$. \\
c) Let $z =\sum_{i\in I} a_i\tau_i\in C_n^{inf}\left(Q\right)$.  
Choose a minimal presentation for $\partial z$ (i.e.\ no further cancellation is possible). Let 
$$J=J_{\partial z}:=\left\{\begin{array}{c}\left(i,a\right)\in I\times\left\{0,\ldots,n\right\}:\\
\partial_a\tau_i\mbox{\ occurs\ with\ non-zero\ coefficient\ in\ the\ chosen\ presentation\ of\ }\partial z\end{array}\right\}.$$ 
Let $\mathcal{C}$ be a set of cancellations for $z$. 
Then the simplicial set $\partial\Upsilon_{z,{\mathcal{C}}}$ is defined as the set consisting of $\mid J\mid$ 
n-1-simplices $\Delta_{i,a},
\left(i,a\right)\in J$, together with all their iterated faces and degenerations, subject to the identifications
$\partial_a\partial_{a_1}\tau_{i_1}=\partial_a\partial_{a_2}\tau_{i_2}$ for all $a=0,\ldots,n-1$, 
whenever $\left(\partial_{a_1}\tau_{i_1},\partial_{a_2}\tau_{i_2}\right)\in{\mathcal{C}}$
and $ \left(i_1,a_1\right)\in J$ .\\
d) If $z =\sum_{i\in I} a_i\tau_i\in C_n^{inf}\left(Q\right)$ is a relative cycle, then a set of 
cancellations $\mathcal{C}$ is called sufficient if the formal sum
$\sum_{i\in I}\sum_{k=0}^n \left(-1\right)^k a_i\partial_k\tau_i$ can be reduced to a chain in 
$C_{n-1}^{inf}\left(\partial Q\right)$ by substracting (possibly infinitely many) multiples of $\left(\partial_{a_1}\tau_{i_1}-\partial_{a_2}\tau_{i_2}\right)$ with $\left(\partial_{a_1}\tau_{i_1},\partial_{a_2}\tau_{i_2}\right)\in{\mathcal{C}}$.
\end{df}
\begin{obs}: Let $\left(Q,\partial Q\right)$ be a pair of topological spaces.\\
a) If $z=\sum_{i\in I} a_i\tau_i\in C_n^{inf}\left(Q\right)$ is a singular chain, $\mathcal{C}$ a set of cancellations, and $\Upsilon:=\Upsilon_{z,{\mathcal{C}}}$ the associated simplicial set, 
then the geometric realisation $\mid\Upsilon\mid$ is obtained from $\mid I\mid $ copies of the standard n-simplex $\Delta_i,i\in I$, with identifications $\partial_{a_1}\Delta_{i_1}=\partial_{a_2}\Delta_{i_2}$ if and only if 
$\left(\partial_{a_1}\tau_{i_1},\partial_{a_2}\tau_{i_2}\right)\in{\mathcal{C}}$. 
Moreover, for a minimal presentation of $\partial z$ and $\partial\Upsilon:=\partial\Upsilon_{z,{\mathcal{C}}}$, $\mid\partial
\Upsilon\mid$ is the subspace of $\mid\Upsilon\mid$ containing all simplices $\partial_{a_1}\Delta_{i_1}$ with
$\left(i_1,a_1\right)\in J$. \\
b) There exists an associated continuous map $\tau:\mid\Upsilon\mid\rightarrow Q$ with $\tau\mid\Delta_i=\tau_i$ (upon the identification $\Delta_i=\Delta^n$). If $z$ is a relative cycle, i.e.\ 
$ \partial z\in C_{n-1}^{inf}\left(\partial Q\right)$,
then $\tau$ maps $\mid\partial\Upsilon\mid$ to $\partial Q$.\\
c) Let $z_1=\sum_{i\in I}a_i\tau_i,
z_2=\sum_{i\in I}a_i\sigma_i\in C_n^{inf}\left(Q,\partial Q\right)$ be relative cycles and 
${\mathcal{C}}_1,{\mathcal{C}}_2$ sufficient sets of cancellations of $z_1$ resp.\ $z_2$.
Assume that $\left(\partial_{a_1}\tau_{i_1},\partial_{a_2}\tau_{i_2}\right)\in{\mathcal{C}}_1$ if and only if
$\left(\partial_{a_1}\sigma_{i_1},\partial_{a_2}\sigma_{i_2}\right)\in{\mathcal{C}}_2$, and that there exist minimal presentations of $\partial z_1,\partial z_2$ such that $J_{z_1}=J_{z_2}$. \\
If the associated continuous maps \\
$\tau,\sigma:\mid\Upsilon\mid\rightarrow Q$ are homotopic, for a homotopy mapping $\mid\partial \Upsilon\mid$ to $\partial Q$, then $\sum_{i\in I}a_i\tau_i$
and $
\sum_{i\in I}a_i\sigma_i\in C_*^{inf}\left(Q,\partial Q\right)$ are relatively homologous.
\end{obs}
We emphasize that we do not assume that $\mathcal{C}$ is a complete list of cancellations, 
the simplicial map $\tau_*:C_*^{simp}\left(\Upsilon\right)\rightarrow C_*^{sing}\left(Q\right)$ need not be injective.\\
\\
%If a simplicial set $\widehat{S_*^{str}}\left(Q\right)\subset S_*\left(Q\right)$ of straight simplices is defined according
%to Lemma 11, we will denote the associated chain complex
%$$C_*^{str,inf}\left(Q\right):=C_*^{simp,inf}\left(\widehat{S_*^{str}}\left(Q\right)\right)\mbox{\ \ and\ \ }C_*^{str,inf}\left(Q,\partial Q\right):=C_*^{str,inf}\left(Q\right)/
%C_*^{str,inf}\left(\partial Q\right).$$
After having set up the necessary notations, we now start with the actual definition of the straightening.
We first mention that there is of course an analogue of the classical straightening (\cite{bp}, Lemma C.4.3.) in our setting.

\begin{obs}: Let $Q,\partial Q,\partial_0Q,\partial_1Q$ satisfy Assumption I and let $K_*^{str}\left(Q\right)\subset 
S_*\left(Q\right)$ satisfy conditions i)-viii) from Lemma 11. 
Then there exists a 'canonical straightening' map
$$str_{can}:C_*^{simp,inf}\left(K\left(Q\right)\right)\rightarrow
C_*^{simp,inf}\left(K^{str}\left(Q\right)\right),$$
mapping $C_*^{simp,inf}\left(K\left(\partial Q\right)\right)$
to $C_*^{simp,inf}\left(K^{str}\left(\partial Q\right)\right)$, with the following properties:\\
i) $str_{can}$ is a chain map,\\
%ii) if $\tau\in K^{str}\left(Q\right)$, then $str_{can}\left(\tau\right)=\tau$,\\
ii) if $\sum_{i\in I} a_i\tau_i\in C_*^{simp,inf}\left(K\left(Q\right)\right)$ and $\sum_{i\in I}a_i\sigma_i:=\sum_{i\in I}
a_i str_{can}\left(\tau_i\right)$, then
the maps $$
\tau,\sigma:\left(\mid\Upsilon\mid,\mid\partial\Upsilon\mid\right)\rightarrow \left(
Q,\partial Q\right)$$ 
(defined by Observation 7b) after fixing 
a set of cancellations $\mathcal{C}$ and a minimal representation of $\partial z$) are homotopic as maps of pairs. In particular, if $\sum_{i\in I} a_i\tau_i$ is a (relative) cycle, then 
%and $\mathcal{C}$ is a sufficient set of cancellations (Definition 8d), then 
$\sum_{i\in I} a_i str_{can}\left(\tau_i\right)$ is (relatively) homologous 
to $\sum_{i\in I} a_i\tau_i$,\\
iii) if $v\in S_0\left(Q\right)$, then the homotopy $H\left(v\right)$ between $v$ and $str_{can}\left(v\right)$
is either constant or $H\left(v\right)\in K_1\left(Q\right)$.
%iv)  if $v$ and $w$ belong to the same path-component $B$ of $\partial_0Q$, and $B$ is not
%a
%path-component of $\partial Q$, then there exists a path $\gamma:\left[0,1\right]\rightarrow B$ such that for
%$g:=\left\{\left[\gamma\right],\left[\overline{\gamma}\right]\right\}\in\Pi\left(K\left(\partial_0Q\right)\right)$ 
%we have $v=gw$ and $H\left(v\right)=gH\left(w\right)$. \\
\end{obs}
\begin{pf}
We define $str_{can}$ by induction on simplices. \\

{\em 0-simplices.}\\
We define $str_{can}\left(v\right)=v$ for each 0-simplex $v$ in $\partial_0Q$. The homotopy $H\left(v\right)$ is for each $v$ given by the constant map.

If $C$ is a path-component of $\partial Q-\partial_0Q$, then there is a unique straight 0-simplex $v_0$ in $C$. We define
$str_{can}\left(v\right)=v_0$ for all $v\in C$.
%If $B\subset C$ is a path-component of $\partial_0Q$, then for one $w\in B$ we choose some arbitrary 
%$H\left(w\right)\in K_1\left(\partial
%Q\right)$ with $\partial H\left(w\right)=v_0-w$ (this involves a choice because there may be different 1-simplices, corresponding to different
%homotopy classes rel.\ boundary, with the given vertices),
%and for each $v\in B$ with $v\not=w$ 
%we choose some path 
%$\gamma:\left[0,1\right]\rightarrow B$ with $\gamma\left(0\right)=v,\gamma\left(1\right)=w$, let $g:=
%\left\{\left[\gamma\right],\left[\overline{\gamma}\right]\right\}\in\Pi\left(K\left(\partial_0Q\right)\right)$
%(thus $v=gw$), and define $H\left(v\right)=g H\left(w\right)$ (this involves 
%a choice because $g$ is only determined up to multiplication with loops at $v$). Finally, if $v\in C$ but $v\not\in\partial_0Q$, then we 
We choose some arbitrary
$H\left(v\right)\in K_1\left(\partial Q\right)$ with $\partial H\left(v\right)=v_0-v$.

If $v\not\in\partial Q$, then we define $str_{can}\left(v\right)$ to be some straight 0-simplex in $\partial Q -\partial_0Q$ and we fix arbitrarily some $H\left(v\right)\in K_1\left(Q\right)$ with
$\partial H\left(v\right)=v-str_{can}\left(v\right)$.\\

{\em 1-simplices.}\\
For $e\in K_1\left(Q\right)$ we define $$str_{can}\left(e\right):=\left[\overline{H\left(\partial_1e\right)}*e*H\left(\partial_0e\right)\right],$$
where, as always, [.] denotes the unique 1-simplex in $K_1^{str}\left(Q\right)$, which is homotopic rel.\ boundary to the path in the brackets.

$e$ is homotopic to $str_{can}\left(e\right)$ 
by the canonical homotopy which is inverse to the homotopy moving $\overline{H\left(\partial_1e\right)}$ resp.\  
$H\left(\partial_0e\right)$ into constant maps. In particular, the restriction of this homotopy to $\partial_1e,\partial_0e$ gives $H\left(\partial_1e\right),\overline{H\left(\partial_0e\right)}$. Thus, for different
edges with common vertices, the homotopies are compatible. We thus have constructed a homotopy for the 1-skeleton
$\Upsilon_1$. We note that $H\left(v\right)\in K_1\left(\partial Q\right)$ or $H\left(v\right)$ constant whenever $v\in\partial Q$,
thus for $\tau\in K_1\left(\partial Q\right)$,
the homotopy between $\tau$ and $str_{can}\left(\tau\right)$ will take place in $\partial Q$. \\

{\em n-simplices.}\\
We assume inductively, that for some $n\ge 1$, we have 
defined $str_{can}$ on $K_{*\le n}\left(Q\right)$, 
mapping $K_{*\le n}\left(\partial Q\right)$ to $K_{*\le n}^{str}
\left(\partial Q\right)$, and satisfying i),ii),iii).

Let $\tau\in K\left(Q\right)$ be an n+1-simplex. Then we have by ii) a 
homotopy between $\partial\tau$ 
and $str_{can}\left(\partial\tau
\right)$. By Observation 1 this homotopy extends to $\tau$. The resulting simplex $\tau^\prime$ satisfies
$\partial\tau^\prime\in K_n^{str}\left(Q\right)$.
Condition v) from Lemma 11 means that
$\tau^\prime$ is homotopic rel.\ boundary to a unique simplex $str_{can}\left(\tau\right)\in 
K_{n+1}^{str}\left(Q\right)$. This proves the inductive step.\\
If $\tau\in K\left(\partial Q\right)$, then we can inductively assume that the homotopy of $\partial\tau$ has image 
in $\partial Q$. Then condition vii) from Lemma 11 implies $str_{can}\left(\tau\right)\in K_{n+1}^{str}\left(\partial Q\right)$ and that the homotopy of $\tau$ can be chosen to have image in $\partial Q$.\\

By construction, for any set of cancellations $\mathcal{C}$,
the induced maps $\tau$ and $\sigma$ 
are homotopic. In particular, if we chose a sufficient set of cancellations in 
the sense of Definition 8d), then Observation 6c) implies that $\sum_{i=1}^r a_i str_{can}\left(\tau_i\right)$ is (relatively) homologous
to $\sum_{i=1}^r a_i\tau_i$.

\end{pf}\\

However, we want to define a more refined straightening, 
which will be defined only on relative
cycles with some kind of additional information. 

\begin{df}: Let $Q,\partial Q,\partial_0Q,\partial_1Q$ 
satisfy Assumption I and let $K_*^{str}\left(Q\right)\subset 
S_*\left(Q\right)$ satisfy conditions i)-viii) from Lemma 11.
A set $D\subset K_1^{str}\left(Q\right)$ is called a set of distinguished 1-simplices if \\
- each $e\in D$ satisfies $\partial_0e,\partial_1e\in K_0^{str}\left(Q\right)= K_0^{str}\left(\partial Q\right)$,\\
- for each $$\left(x,y\right)\in
K_0^{str}\left( Q\right)\times K_0^{str}\left( Q\right)$$ 
there exists exactly one 1-simplex $e_{xy}\in D$ with
$$e_{xy}\left(0\right)=x,e_{xy}\left(1\right)=y,$$
%- for each $\left(x,y\right)\in
%K_0^{str}\left(\partial_1Q\right)\times K_0^{str}\left(\partial_1Q\right)$, $e_{xy}$ does not intersect $\partial_0Q$,\\
- for all $x\in\st$, $e_{xx}$ is the constant
loop,  \\
- for all $\left(x,y\right)\in
K_0^{str}\left( Q\right)\times K_0^{str}\left( Q\right)$ we have $e_{xy}=\overline{e_{yx}}$, i.e.\ $e_{xy}$ and $e_{yx}$ agree up to orientation,\\
- if $x_1$ and $x_2$ belong to the same path-component $C$ of $\partial Q$, and $y\in  K_0^{str}
\left(\partial Q\right)$, then $e_{x_1y}$ and $e_{x_2y}$ are homotopic by a 
homotopy moving $\partial_1e$ inside $\partial Q$,\\
if $C\cap\partial_0Q\not=\emptyset$, then the image of the homotopy of $\partial_1e$ is a path which is {\bf not} homotopic (rel.\ vertices) into $\partial_1Q$,\\
if $x_1$ and $x_2$ belong to the same path-component of $\partial_0 Q$, and $y\in  K_0^{str}
\left(\partial Q\right)$, then $e_{x_1y}$ and $e_{x_2y}$ are homotopic by a
homotopy moving $\partial_1e$ inside $\partial_0 Q$.\end{df}

Note that the last condition means exactly that $e_{x_2y}=ge_{x_1y}$ for some $g\in\Pi\left(K\left(\partial_0Q\right)\right)$ (cf.\ section 3.3).\\
The second-last condition may seem very artificial but it will be needed in the proof of Theorem 1, specifically in the construction of a $\pi_1$-injective cylinder.

\begin{obs}: Let the assumptions of Definition 9 be satisfied.
Then a set of distinguished 1-simplices exists.\end{obs}
\begin{pf}
(Recall that $K_0^{str}\left(\partial Q\right)=
K_0^{str}\left(\partial Q\right)$ by part iii) of Lemma 11.)

First, for each path-component $C$ of $\partial Q$ we fix some $x_C\in K_0^{str}\left(\partial Q\right)$, and for each path-component $E$ of $\partial_0Q$ we fix some $x_E\in K_0^{str}\left(\partial_0Q\right)$.

For each unordered pair $\left\{C,D\right\}$ of path-components of $\partial Q$ we choose
one straight 1-simplex $\sigma_{x_Cx_D}$ with
$$\sigma_{x_Cx_D}\left(0\right)=x_C, \sigma_{x_Cx_D}\left(1\right)=x_D.$$
In the case $C=D$ we choose $x_C=x_D$ and we choose $\sigma_{x_Cx_C}$
to be the
constant loop.

For each unordered pair $\left\{E,F\right\}$ of path-components of $\partial_0Q$ 
we choose 
one straight 1-simplex $\sigma_{x_Ex_F}$ with
$$\sigma_{x_Ex_F}\left(0\right)=x_E, \sigma_{x_Ex_F}\left(1\right)=x_F,$$
satisfying the condition that, for the unique path-components $C,D$ of $\partial Q$ with $E\subset C, F\subset D$, $$\sigma_{x_Ex_F}\sim \sigma_{x_Cx_D}$$
with a homotopy which keeps the endpoints in $\partial Q$ but such that the second-last condition is satisfied, i.e. even if $x_E, x_F$ belonged to $\partial_0Q\cap\partial_1Q$, their images under the homotopy are paths not homotopic (rel.\ vertices) into $\partial_1Q$. (This is possible because $\pi_1
\left(\partial_0Q\cap\partial_1Q\right)\rightarrow\pi_1\partial_0Q$ is not surjective by Assumption I.)\\
In the case $E=F$ we choose $x_E=x_F$ and we choose $\sigma_{x_Ex_E}$
to be the 
constant loop.

Moreover, for each path-component $E$ of $\partial_0Q$ and each $y\in K_0^{str}\left(Q\right)-K_0^{str}\left(\partial_0Q\right)$ 
we choose one straight 1-simplex 
$\sigma_{x_Ey}$ with
$$\sigma_{x_Ey}\left(0\right)=x_E, \sigma_{x_Ey}\left(1\right)=y,$$
satisfying the condition that, for the unique path-components $C,D$ of $\partial
Q$ with $E\subset C, y\in D$, $$\sigma_{x_Ey}\sim \sigma_{x_Cx_D}$$
with a homotopy keeping endpoints in $\partial Q$ but such that the second-last condition is satisfied.
 
(To avoid confusion, we note that in the above we have implicitly established an order on each pair of path-components, and this order will next be inherited by unordered pairs of points.)

Now the actual construction goes as follows.
First, for each unordered pair $$\left\{x,y\right\}\subset K_0^{str}\left(\partial_0Q
\right),$$
let $C$ resp.\ $D$ be the path-component of $\partial_0Q
$ which contains $x$ resp.\ $y$.
We choose one straight 1-simplex $\sigma_{xy}$ with 
$\sigma_{xy}\left(0\right)=x$ and $\sigma_{xy}\left(1\right)=y$, such that it is homotopic to $\sigma_{x_Cx_D}$ by a homotopy keeping endpoints in $\partial_0Q$. In the case $x=y$ we choose the constant loop.
 
Next, for each unordered pair $\left\{x,y\right\}$ with
$$x\in K_0^{str}\left(\partial_0Q
\right), y\in K_0^{str}\left(Q\right)-K_0^{str}\left(\partial_0Q\right)$$ we choose one straight 1-simplex $\sigma_{xy}$ with
$\sigma_{xy}\left(0\right)=x$ and $\sigma_{xy}\left(1\right)=y$, such that it is homotopic
to $\sigma_{x_Cy}$ by a homotopy keeping the first vertex in $\partial_0Q$.

Finally, for each unordered pair $$\left\{x,y\right\}\subset K_0^{str}\left(\partial
Q\right)-K_0^{str}\left(\partial_0Q\right)$$
we choose one straight 1-simplex $\sigma_{xy}$ with
$\sigma_{xy}\left(0\right)=x$ and $\sigma_{xy}\left(1\right)=y$, such that $\sigma_{xy}\sim \sigma_{x_Cx_D}$ (with a homotopy keeping vertices in $\partial Q$ but satisfying the second-last condition)
for the path-components $C,D$ of $\partial Q$ with $x\in C,y\in D$. In the case $x=y$ we choose the 
constant loop.

To finish the construction, for each ordered pair $\left(x,y\right)\in K_0^{str}\left(Q\right)\times K_0^{str}\left(Q\right)$
we define $e_{xy}=\sigma_{xy}$ and $e_{yx}=\overline{\sigma}_{xy}$ (that is, $\sigma_{xy}$ with the opposite orientation). 
%and call this edge a {\bf distinguished 1-simplex}. 
\end{pf}




\begin{df}: 
Let $Q,\partial Q,\partial_0Q,\partial_1Q$ satisfy assumption I.
Let $z=\sum_{i\in I} a_i\tau_i\in C_n^{inf}\left(Q\right)$
be a singular chain and $\Upsilon$ the associated simplicial set (for some set of cancellations $\mathcal{C}$). 

We say that a labeling of the elements of the 1-skeleton $\Upsilon_1$ by
0's and 1's is admissible, if \\
- each vertex of a 1-labeled $e\in \Upsilon_1$ belongs to $\partial_1 Q$,\\
- there does not exist a cycle of 1-labeled edges in $\mid\Upsilon\mid$, 
i.e.\ there does not exist a sequence of 1-labeled edges
$$e_1,e_2,\ldots,e_n$$ with $n\ge 2, e_i\in \Upsilon_1$ for $i=1,\ldots,n$ such that $\partial_0e_i=\partial_1e_{i+1}$ for $i=1,\ldots,n-1$ and $\partial_0e_n=\partial_1e_0$ in $\Upsilon_0$.\end{df}

We emphasize that we only exclude the existence of 1-labeled cycles in $\mid\Upsilon\mid$. It may be possible that the images of 1-labeled edges in $Q$ form a cycle.

\begin{lem}: Let $Q,\partial Q,\partial_0Q,\partial_1Q$ satisfy Assumption I.
Let $K_*^{str}\left(Q\right)\subset
S_*\left(Q\right)$ satisfy conditions i)-viii) from Lemma 11, and let $D\subset K_1^{str}\left(Q\right)$
be a set of distinguished 1-simplices.

Let $z=\sum_{i\in I}a_i\tau_i\in C_*^{inf,simp}\left(K\left(Q\right)\right)$ 
be a relative cycle, i.e.\ $ \partial z\in C_*^{inf,simp}\left( K\left(\partial Q\right)\right)$.

Let a set of cancellations $\mathcal{C}$ for $z$ and a minimal
presentation of $\partial z$ be given. Let $\Upsilon,\partial\Upsilon$ be the associated simplicial sets, $\tau:\left(\mid\Upsilon\mid,\mid\partial\Upsilon\mid\right)\rightarrow 
\left(Q,\partial Q\right)$ the associated continuous mapping.

Assume that we have an admissible 0-1-labeling
of $\Upsilon_1$. \\
Then there exists a relative cycle $$\sum_{i\in I}a_i\tau_i^\prime
\in C_*^{simp,inf}\left(K^{str}\left(Q\right)\right)$$ 
with $\partial
\sum_{i\in I}a_i\tau_i^\prime
\in C_*^{simp,inf}\left(K^{str}\left(\partial Q\right)\right)$, such that:\\
i) the associated continuous mappings $$\tau,\tau^\prime:
\left(\mid\Upsilon\mid,\mid\partial\Upsilon\mid\right)\rightarrow
\left(Q,\partial Q\right)$$ are homotopic as maps of pairs,\\ 
ii) if an edge of some $\tau_i$ is labeled by 1, then the corresponding edge of $\tau_i^\prime$ belongs to $D$.
\end{lem}

\begin{pf}
First we apply the 'canonical straightening' $str_{can}$ from Observation 7.
%to homotope $\sum_{i\in I} a_i\tau_i$ to another singular chain such that the 
%homotoped simplices all belong to $K_*^{str}\left(Q
%\right)$, and simplices in $K\left(\partial Q\right)$ are homotoped into simplices in 
%$K^{str}\left(Q\right)$.
%(As shown in the proof of Observation 8, we can assume that vertices of $\Upsilon$ 
%with image in $C$ remain fixed during the homotopy. Thus condition ii) will be satisfied.) 
The resulting chain $\sum_{i\in I} a_i str_{can}\left(\tau_i\right)$ satisfies i), but not necessarily ii). 

$\sum_{i\in I} a_i str_{can}\left(\tau_i\right)$ inherits the admissible
labeling
from $\sum_{i\in I} a_i \tau_i$. Thus we can w.l.o.g.\ restrict to the case that all $\tau_i$ belong to $K^{str}\left(Q\right)$.

Since the set of 1-labeled edges contains no cycle,
we can fix an order $<$
on the set of 1-labeled edges, such that for any 1-labeled edge $e$
there do not exist two 1-labeled edges $$d_1<e,d_2<e$$ with $\partial_1e
\subset\partial d_1 $ and $\partial_0e\subset\partial d_2$. 
We perform the straightening succesively, according to the order $<$.
So assume that 
$$e\in K_1^{str}\left(Q\right)$$ is a 1-labeled edge, and that we have 
a simplicial map $$\tau:\Upsilon\rightarrow S_*\left(Q\right)$$ such that for all 1-labeled 
edges $d$ with $d<e$ we have $$\tau\left(d\right)\in D.$$
We want to homotope $\tau$ into $\tau^\prime$ such that $\tau^\prime\left(e\right)\in D$ and $\tau^\prime\left(d\right)=
\tau\left(d\right)$ for all $d<e$.
By Definition 10, $x=\partial_0e$ and $y=\partial_1 e$ belong to $\partial_1 Q$.
By Definition 9, there exists a unique $e_{xy}\in D$ with $$\partial_1e_{xy}=x,
\partial_0e_{xy}=y.$$
%Next we want to homotope 1-labeled (straight) edges into distinguished 1-simplices. (For the following argument we can ignore the decomposition of straight 1-simplices.)
%Let $\hat{e}\in K_1^{str}\left(Q\right)$ be an edge labeled with 1. 
%Let $x=\partial_1 \hat{e},y=\partial_0 \hat{e}$ 
%be the vertices of $\hat{e}:=str_{can}\left(e\right)$.
By construction of the order $<$, at least one of $\partial_0e,\partial_1 e$ is not a vertex of any 1-labeled $d$ with $d<e$. Assume this is the case for $\partial_1e$, the other case being handled by analogy.
We homotope $e$ into $e_{xy}$ by the following sequence of homotopies
$$e\rightarrow e_{xy}*\overline{e}*e\rightarrow e_{xy},$$
where the first homotopy comes from the homotopy between the constant 
path $c_x$ and $e_{xy}*\overline{e}$, 
and the second homotopy comes from the homotopy between
$\overline{e}*e$ and the constant path $c_y$.
(Of course, we change 
the homotopy class relative boundary, so we can not keep
the endpoints fixed during the homotopy.) 
We notice that during the first homotopy $\partial_1e$ is moved along $e*\overline{e}_{xy}$ and $\partial_0e
$ is kept fixed, and during the second homotopy both $\partial_0e $ and $\partial_1e$ keep fixed.
In particular, these homotopies keep $\partial_0e$ fixed.
Thus, using Observation 1, this homotopy can be extended to a homotopy from $$\tau:\mid\Upsilon\mid\rightarrow Q$$
to some $$\hat{\tau}:\mid\Upsilon\mid\rightarrow Q,$$
such that $\hat{\tau}$ is
a simplicial map from $\Upsilon$ to $S_*\left(Q\right)$,
and the homotopy can be chosen to keep $\partial_0e$ and all 1-labeled edges $d$ with $d<e$ fixed. (The latter is possible
because no $d<e$ has $\partial_0e$ in its boundary $\partial d$.) This proves the inductive step.

Next we apply homotopies rel.\ boundary to (the already homotoped images of)
all 0-labeled edges $f\in K_1^{str}\left(Q\right)$, to homotope them to edges in $K_1^{str}\left(Q\right)$. 
Because this will be of importance in the proof of Corollary 3, we explicitly give the homotopy for (already canonically
straightened) 0-labeled edges. We say that $f$
is adjacent to a 1-labeled edge if $\partial_1e$ is a vertex of $f$. (We do not consider $\partial_0e$.) \\
Let $f\in K_1^{str}\left(Q\right)$ be a 0-labeled edge which is at both vertices adjacent
to 1-labeled edges $e_1,e_2$, i.e.\ $$\partial_0f=\partial_1e_1=x, \partial_1f=\partial_1e_2=v$$ and $$\partial_0e_1=y,\partial_0e_2=w.$$
Then we have seen that the homotopy from $e_1$ to $e_{xy}$ moves $\partial_1e_1$ along $e_1*\overline{e}_{xy}$, and 
the homotopy from $e_2$ to $e_{vw}$ moves $\partial_1e_2$ along $e_2*\overline{e}_{vw}$. Hence $$str\left(f\right)=
\left(e_{vw}*\overline{e}_2*f*e_1*\overline{e}_{xy}\right)_K.$$ Here, $\left(e_{vw}*\overline{e}_2*f*e_1*
\overline{e}_{xy}\right)_K$ means the unique 1-simplex in $K_1\left(Q\right)$ which is homotopic rel.\ boundary to $e_{vw}*\overline{e}_2*f*e_1*
\overline{e}_{xy}$.\\
Similarly if only one vertex of a straight 0-labeled edge $f$ is adjacent to a 1-labeled edge $e$, then we get either
$str\left(f\right)=
\left(e_{vw}*\overline{e}_2*f\right)_K$ or
$str\left(f\right)=
\left(f*e_1*\overline{e}_{xy}\right)_K$. If $f$ is not adjacent to a 1-labeled edge, then we get $str\left(f\right)=f$.

Now we have a simplicial map $\hat{\tau}:\Upsilon\rightarrow S_*\left(Q\right)$,
such that all 1-simplices are mapped to $K_1^{str}\left(Q\right)$,
and such that $$\hat{\tau}\left(e\right)\in D\subset K_1^{str}\left(Q\right)$$ holds for all 1-labeled edges $e$. Then we can,
as in the proof of Observation 7, by induction on $n$ apply homotopies rel.\ boundary to all  
n-simplices  to homotope them into $K_n^{str}\left(Q\right)$.
We obtain a homotopy,
which keeps all vertices and 1-labeled edges fixed, to 
a simplicial map $$\tau^\prime:\Upsilon\rightarrow K^{str}\left(Q\right),$$
mapping $\partial\Upsilon$ to $K^{str}\left(\partial Q\right)$ and satisfying i),ii).
\end{pf}\\
\\
A somewhat artificial 
formulation of the conclusion of Lemma 12 is that we have 
constructed a chain map $$str:C_*^{simp,inf}\left(\Upsilon,
\partial\Upsilon\right)\rightarrow 
C_*^{simp,inf}\left(K^{str}\left(Q\right),K^{str}\left(\partial Q\right)\right).$$
Unfortunately, this somewhat artificial formulation can 
not be simplified because $str$ depends on the chain $\sum_{i\in I}a_i\tau_i$. 
That is, we do not get a chain map $str:C_*^{simp,inf}\left(K\left(Q\right),K\left(\partial Q\right)\right)
\rightarrow C_*^{simp,inf}\left(K^{str}\left(Q\right),K^{str}\left(\partial Q\right)\right)$.

\subsection{Straightening of crushed cycles}




Here and elsewehere in the paper, $.\otimes_{{\Bbb Z}G}{\Bbb Z}$ will always mean the tensor product with the trivial
${\Bbb Z}G$-module ${\Bbb Z}$, that is, the quotient under the $G$-action.
We first state obvious generalizations of Observation 6 to the case of tensor products with a factor with trivial $G$-action.

\begin{obs}: Let $\left(Q,\partial Q\right)$ be a 
pair of topological spaces. Let $G$ be
a group acting
on a pair $\left(K,\partial K\right)$ with $K\subset S_*\left(Q\right)$ and $\partial K\subset S_*\left(\partial Q\right)$ both closed under face maps.\\
i) If $z=\sum_{i\in I}a_i\tau_i\otimes 1\in C_*^{simp,inf}
\left(K,\partial K\right)\otimes_{{\Bbb Z} G}{\Bbb Z}$
is a relative cycle, then $\hat{z}=\sum_{i\in I}\sum_{g\in G} a_i
\left(g\tau_i\right)\in C_*^{simp,inf}\left(K,\partial K\right)$ is a relative cycle. \\
If $\mathcal{C}$ is a sufficient set of cancellations for $z$, then 
there exists a set of cancellations $\widehat{\mathcal{C}}$ for $\hat{z}$ such
that $\left(\eta_1,\eta_2\right)\in
\widehat{\mathcal{C}}$ implies $\left(\eta_1\otimes 1,\eta_2\otimes 1\right)\in{\mathcal{C}}$.\\
If $\partial z=\sum_{a,i}c_{ai}\partial_a\tau_i\otimes 1$ is a minimal presentation for $\partial z$, then 
$\partial\hat{z}=\sum_{g\in G}\sum_{a,i}c_{ai}\partial_a\left(g\tau_i\right)$ is a 
minimal presentation for $\hat{z}$.\\
ii) Let $\widehat{\Upsilon},\partial\widehat{\Upsilon}$ be the simplicial sets associated to $\hat{z}$,
the sufficient set of cancellations 
$\widehat{\mathcal{C}}$ and the minimal presentation of 
$\partial\hat{z}$. They come with an obvious $G$-action.
Then we have an associated continuous mapping $\hat{\tau}:
\left(\mid \widehat{\Upsilon}\mid,\mid\partial\Upsilon\mid\right)
\rightarrow\left(Q,\partial Q\right)$. \end{obs}

\begin{cor}: Let $Q,\partial Q,\partial_0Q,\partial_1Q$ satisfy Assumption I.
Let $K_*^{str}\left(Q\right)\subset
S_*\left(Q\right)$ satisfy conditions i)-ix) from Lemma 11, 
and let $D\subset K_1^{str}\left(Q\right)$ be a set
of distinguished 1-simplices.

Let $G=\Pi\left(K\left(\partial_0Q\right)
\right)$ with its action on $K^{str}\left(Q\right)$ defined in 
Observation 5.

Let $\sum_{i\in I} a_i\tau_i\otimes 1\in C_n^{simp,inf}\left(K\left(Q\right),K\left(\partial Q\right)\right)
\otimes_{{\Bbb Z}G}{\Bbb Z}$ be a relative cycle. Fix a sufficient set of cancellations
$\mathcal{C}$ and a minimal presentation for $\partial z$. Let $\widehat{\Upsilon},\partial\widehat{\Upsilon}$ be defined by Observation 9.
Assume that we have a $G$-invariant admissible 0-1-labeling of the edges of $\widehat{\Upsilon}$.

Then there is a well-defined chain map $$str:C_*^{simp,inf}\left(\widehat{\Upsilon}
\right)
\otimes_{{\Bbb Z}G}{\Bbb Z}\rightarrow 
C_*^{simp,inf}\left(K^{str}\left(Q\right)\right)\otimes_{{\Bbb Z}G}{\Bbb Z},$$
mapping $C_*^{simp,inf}\left(\partial\widehat{
\Upsilon}\right)
\otimes_{{\Bbb Z}G}{\Bbb Z}$ to 
$C_*^{simp,inf}\left(K^{str}\left(\partial Q\right)\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$,
such that:\\
i) if $e\in\widehat{\Upsilon}_1$ is a 1-labeled edge, $str\left(e\otimes 1\right)=f\otimes 1$, then $f\in D$.\\
ii) if $Q$ is an orientable manifold with boundary $\partial Q$, and
if $\sum_{i\in I} a_i\tau_i\otimes 1\in C_*^{simp,inf}\left(K\left(Q\right),K\left(\partial Q\right)\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$ represents
the image of $\left[Q,\partial Q\right]\otimes 1$ under the canonical homomorphism, then 
$\sum_{i\in I} a_istr\left(\tau_i\right)\otimes 1\in C_*^{simp,inf}\left(K\left(Q\right),K\left(\partial Q\right)
\right)$
represents
the image of $\left[Q,\partial Q\right]\otimes 1$ under the canonical homomorphism.\end{cor}


\begin{pf} 

We can apply Lemma 12 to the infinite chain $\sum_{i\in I,g\in G}a_i \left(g\tau_i\right)$.
Thus Lemma 12 provides us with a chain map 
$str:C_*^{simp,inf}\left(\widehat{\Upsilon}\right)\rightarrow C_*^{simp,inf}\left(K^{str}\left(Q\right)
\right)$, given by $$str\left(g\tau_i\right):=\left(g\tau_i\right)^\prime.$$
We are going to 
define $str\left(\sigma\otimes z\right):=str\left(\sigma\right)\otimes z$ for each $\sigma\in \widehat{\Upsilon}, z\in{\Bbb Z}$. For this to be well-defined, we have to check the following claim: \\

{\em for each $\sigma\in
K,g\in G$, there exists $h\in G$ with $str\left(g\sigma\right)=h str\left(\sigma\right)$.}\\
\\
By condition ix) from Lemma 11 (asphericity of $K^{str}\left(Q\right)$), it suffices to check this for the 1-skeleton. 

It is straightforward to check the claim for the 0-skeleton. If $\sigma=v\in S_0\left(\partial_0Q\right)$ then also $gv\in S_0\left(\partial_0Q\right)$, thus
$str\left(v\right)=v,str\left(gv\right)=gv$, hence
$str\left(gv\right)=g str\left(v\right)$. 
If $\sigma=v\in S_0\left(Q\right)-S_0\left(\partial_0Q\right)$, then also $str\left(v\right)\not\in S_0\left(\partial_0Q\right)$, thus $gv=v,g str\left(v\right)=str\left(v\right)$, hence $str\left(gv\right)=g str\left(v\right)$.

The proof for 1-simplices consists of two steps. 
In the first step we prove that for $e\in K_1,g\in G$ we have 
$str_{can}\left(ge\right)=g str_{can}\left(e\right)$. 
In the second step we 
show that,
if
$e\in K_1^{str}\left(Q\right)$ and $g\in G$ , then there exists $h\in G$ with
$str\left(ge\right)=h str\left(e\right)$.
Hence altogether we will get $str\left(ge\right)=str\left(str_{can}\left(ge\right)\right)
= str\left(g\ str_{can}\left(e\right)\right)=h\ str\left(str_{can}\left(e\right)\right)=h\ str\left(e\right)$.\\
\\
First step: This is fairly obvious. If both vertices of $e$
belong to $\partial_0Q$, then $str_{can}\left(e\right)=e$ and $str_{can}\left(ge\right)=ge$, which 
implies the claim. If both vertices do not belong to $\partial_0 Q$, 
then this is also true for $str_{can}\left(e\right)$, and we have $ge=e, g str_{can}
\left(e\right)=str\left(e\right)$, which impies the claim. Finally 
we consider the case that one vertex, say $\partial_0e$ belongs to 
$\partial_0Q$, but $\partial_1 e$ does not belong. If $
g\partial_0e=\partial_0e$, we can apply the same argument as in the 
case $\partial_0e,\partial_1e\not\in\partial_0Q$. If $g\partial_0e
\not=\partial_0e$, then we have (possibly after renumbering) $g=\left\{
\left[\gamma_1\right],\ldots\right\}$ with $\partial_1\gamma_1=\partial_0e$. By definition we have that $ge$ is homotopic rel.\ boundary to $e*\gamma_1$, 
$str\left(ge\right)$ is homotopic rel.\ boundary to $\overline{H\left(\partial_1e\right)}*e*\gamma_1$, $str\left(e\right)$ is homotopic rel.\ boundary to
$\overline{H\left(\partial_1e\right)}*e$, $g str\left(e\right)$ 
is homotopic rel.\ boundary to $\overline{H\left(
\partial_1e\right)}*e*\gamma_1$. The claim follows.\\
\\
Second step: 
%In view of the first step we can restrict ourselves to show that 
%for (already canonically straightened)
%simplices 
Let $e\in K_1^{str}\left(Q\right)$.
% and $g\in G$ we have
%$str\left(ge\right)=g str\left(e\right)$.

First, if $e$ is a 1-labeled straight edge with $x=\partial_1e,y=\partial_0e$, and $g\in G$,
then $$str\left(e\right)=e_{xy}, str\left(ge\right)=e_{gx,gy}.$$ $x$ and $gx$ (resp.\ $y$ and $gy$) are either equal or belong to the same path-component of $\partial_0Q$. By the last assumption in Definition 9 this implies that there exists some $h\in G$ with 
$$e_{gx,gy}=he_{xy}.$$

%By the proof of Lemma 12, the 
%straightening of $e$ is realised by a homotopy which moves 
%only one vertex, w.l.o.g.\ $\partial_1e$. The homotopy is given 
%by the canonical homotopy $e\rightarrow e_{xy}*\overline{e}*e\rightarrow e_{xy}$ as described in the proof of Lemma 12. 

%If $\partial_1e$ does not belong to $\partial_0Q$, then clearly $str\left(ge\right)=str\left(e\right)=gstr\left(e\right)$.

%Thus assume $\partial_1e\in\partial_0Q$.
%In this case it follows from the last assumption in Definition 9 that there exists some $h\in G$ with $e_{gx,y}=he_{xy}$.

%If $y\not\in \partial_0Q$, then $gy=y$ and we get $str\left(ge)=h str\left(e\right)$. If $y\in\partial_0Q, y\not\in\partial_1Q$, then 
%Since $str\left(e\right)=e_{xy}$ and $str\left(ge\right)=e_{gx,gy}$, this yields the claimed conclusion.  

Second, let $f$ be a 0-labeled edge with one or two vertices 
adjacent to 1-labeled edges. 
%If these adjacent vertices do not belong to $\partial_0Q$, then $gf=f$ for all $g\in G$, and the claim trivially holds. Thus assume one or both vertices belong to $\partial_0Q$.
If $f$ is adjacent to one 1-labeled edge $e$ (with $\partial_1e=x,\partial_0e=y$), 
we have $$str\left(gf\right)= he_{xy}*\overline{ge}*gf=he_{xy}*\overline{e}*f=h\left(e_{xy}*\overline{e}*f\right)=h\ str\left(f\right).$$
Similarly if $f$ is adjacent to two 1-labeled edges.
Hence $str\left(gf\right)=h str\left(f\right)$ for 0-labeled edges $f$.\\
\\
Thus $str$ is well-defined and satisfies i) by Lemma 12. To prove ii), we first observe that, if $\sum_{i\in I} a_i\tau_i$ 
represents $\left[Q,\partial Q\right]$, then, by Observation 6c) and
condition i) from Lemma 12, $\sum_{i\in I} a_i str\left(\tau_i\right)=\sum_{i=1}^ra_i\tau_i^\prime$ 
represents $\left[Q,\partial
Q\right]$ and the claim follows. Thus 
it suffices to check: if $\sum_{i\in I} a_i\tau_i\otimes 1$ is (relatively)
homologous to $\sum_{j\in J} b_j\kappa_j\otimes 1$, then
$str\left(\sum_{i\in I} a_i\tau_i\otimes 1\right)$ is (relatively)
homologous to $str\left(\sum_{j\in J} b_j\kappa_j\otimes 1\right)$. 

So let $\sum_{i\in I} a_i\tau_i\otimes 1-
\sum_{j\in J} b_j\kappa_j\otimes 1=\partial \sum_{k\in K} c_k\eta_k\otimes 1\ mod\ C_*\left(\partial Q\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$
for some chain $\partial \sum_{k\in K} c_k\eta_k\otimes 1\in
C_*\left(Q\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$. 
In complete analogy with Lemma 12, we may extend $str$ to the simplicial set built by the $g\eta_k$'s, their faces and degenerations, and 
obtain a singular 
chain $str\left(\sum_{k\in K} c_k\eta_k\right)$
whose boundary is 
$\partial str\left(\sum_{k\in K} c_k\eta_k\right)=str\left(\sum_{i\in I} a_i\tau_i\otimes 1\right)-
str\left(\sum_{j\in J} b_j\kappa_j\otimes 1
\right)\ mod\ C_*\left(\partial Q\right)$.

%Since vertices contained in $\partial_0Q$ are fixed during the homotopy, we have $g str_{can}\left(e\right)=str_{can}\left(ge\right)$ for each 1-simplex $e$. 
%If $e$ is a 1-labelled edge, then its vertices belong to 
%$\partial_1Q$. If they belong to a closed component of $\partial_1Q$, then $ge=e$ and $g str\left(e\right)=str
%\left(e\right)$, from which the claim follows.
%If $e$ is adjacent to a 1-labelled edge, then $ge $ is adjacent to the same 1-labelled edge.

%The last claim holds because homotopic relative cycles (for a homotopy preserving $C_*\left(\partial Q\right)$) are relatively homologous.
\end{pf}





\subsection{Removal of 0-homologous chains}




\begin{df}: Let $Q$ be an n-dimensional compact manifold with boundary $\partial Q$.
We define 
$
rmv:
S_*\left(Q\right)\rightarrow S_*\left(Q
\right)$ by $$rmv\left(\sigma\right)=0$$ if $\sigma$ is weakly degenerate (Definition 7)
and $$rmv\left(\sigma\right)=\sigma$$ else.\end{df}

%It is easily checked (cf.\ the proof of Lemma 2.5.\ in \cite{k1}) that $rmv$ maps cycles to cycles and boundaries to boundaries.
\begin{lem}: Assume that $Q$ is a $n$-dimensional manifold with boundary $\partial Q$. Let $K_*^{str}\left(Q\right)\subset 
S_*\left(Q\right)$ satisfy the conditions i)-vii) from Lemma 11.
Then $$rmv:C_*^{simp,inf}\left(K^{str}\left(Q\right),K^{str}\left(\partial Q\right)
\right)\rightarrow C_*^{simp,inf}\left(
K^{str}\left(Q\right),K^{str}\left(\partial Q\right)\right),$$ 
defined by $$rmv\left(\left[\sigma\right]\right):=\left[rmv\left(\sigma\right)\right],$$
is a well-defined chain
map and induces the identity on $H_n^{lf}$, i.e.\ if $\sum_{j\in J} a_j\tau_j\in C_*^{simp,inf}\left(K^{str}\left(Q\right), K^{str}
\left(\partial Q\right)
\right)\subset C_*^{sing}\left(Q,\partial Q\right)
$ represents 
$\left[Q,\partial Q\right]$, then $\sum_{\in J} a_j rmv\left(\tau_j\otimes 1\right)$ represents
$\left[Q,\partial Q\right]$.
\end{lem}
\begin{pf}
If $\sigma\in C_*^{str}\left(\partial Q\right)$, then $rmv\left(\sigma\right)=0\in C_*^{str}\left(\partial Q\right)$, thus $rmv$ is well-defined.\\
In a first step, we prove that $rmv$ is a chain map.

Assume that $rmv\left(\sigma\right)=0$.\\
If $\sigma$ has image in $\partial Q$, then $rmv\left(\sigma\right)=0$ and $rmv\left(\partial\sigma\right)=0$, thus
$\partial rmv\left(\sigma\right)=rmv\left(\partial\sigma\right)$.\\
If some edge $e$ of $\sigma$, say connecting the $i$-th and $j$-th vertex,
is a constant loop, then all faces of $\sigma$ except possibly $\partial_i\sigma$ and $\partial_j\sigma$
have a constant edge. 
Thus $rmv\left(\partial_k\sigma\right)=0$ if $k\not\in\left\{i,j\right\}$. Moreover, since $e$ is constant, corresponding edges of $\partial_i\sigma$ and $\partial_j\sigma$
are homotopic rel.\ boundary and thus agree (possibly up to orientation) by condition v) from
Lemma 11. By induction on the dimension of subsimplices we get, again 
using condition v) from Lemma 11, that $\partial_i\sigma=\left(-1\right)^{i-j}\partial_j\sigma$. Altogether we get $rmv\left(\partial\sigma\right)=0$, thus
$\partial rmv\left(\sigma\right)=rmv\left(\partial\sigma\right)$.

Assume that $rmv\left(\sigma\right)=\sigma$. Since no edge of $\sigma$ is a constant loop, of course also no edge of
a face $\partial_i\sigma$ is a constant loop. If the image of
$\partial_i\sigma$ is not contained in $\partial Q$, this implies $rmv\left(\partial_i\sigma\right)=\partial_i\sigma=\partial_i rmv\left(\sigma\right)$. If $\partial_i\sigma$ has image in $\partial Q$, then of course $\left[\partial_i\sigma\right]=\left[0\right]=\left[\partial_i rmv\left(\sigma\right)\right]$, which implies $rmv\left(\partial_i\sigma\right)=\partial_i rmv\left(\sigma\right)$.\\
Now we prove that $rmv$ induces the identity on $H_n$.

Let $\sum_{j=1}^r a_j\tau_j$ be a straight relative cycle, representing the relative homology class $h$.

We denote by $J_1\subset \left\{1,\ldots,r\right\}$ the indices of those $\tau_j$ which have a constant edge.
The sum $\sum_{j\in J_1} a_j\tau_j$ is a relatively 0-homologous relative cycle. Indeed, each face of $\partial_i\tau_k$ 
not contained in $\partial Q$ has to cancel against some face of some $\tau_l$, because
$\sum_{j=1}^r a_j\tau_j$ is a relative cycle. If $\partial_i\tau_k$
is degenerate, then necessarily $l\in J_1$. 
Moreover, if $\tau_k$ is degenerate and
$\partial_i\tau_k$ is nondegenerate, then we have proved in the first step that $\partial_i\tau_k$ cancels against some $\partial_j\tau_k$. 

Thus $\sum_{j\in J_1} a_j\tau_j$ represents some relative homology class. The isomorphism 
$H_n\left(C_*^{str}\left(Q,\partial Q\right)
\right)\rightarrow {\Bbb R}$ is given by pairing with the volume
form of an arbitrary Riemannian metric. After smoothing the relative cycle, we
can apply Sard's lemma, and conclude that degenerate simplices have volume 0. Thus 
$\sum_{j\in J_1} a_j\tau_j$ is 0-homologous.

We denote by $J_2\subset \left\{1,\ldots,r\right\}$ the indices of those $\tau_j$ which are contained in
$\partial Q$. For $j\in J_2$ we have $\left[\tau_j\right]=\left[0\right]\in
C_*^{str}\left(Q,\partial Q\right)=C_*^{str}\left(Q\right)/C_*^{str}\left(\partial Q\right)$.

Thus $\sum_{j\not\in J_1\cup J_2}a_j\tau_j$ is another representative of the homology class $h$.
By Definition 11, it represents $\left(rmv\right)_*\left(h\right)$.
\end{pf} \\

\begin{lem}: Assume that $Q$ is a compact $n$-dimensional manifold with boundary $\partial Q$.
Let the assumptions of Corollary 3 be satisfied. 
Then we can extend $rmv$ to a well-defined chain map $$rmv:C_*^{simp,inf}
\left(K^{str}\left(Q\right), K^{str}\left(\partial Q\right)\right)\otimes_{{\Bbb Z}G}{\Bbb Z}
\rightarrow C_*^{simp,inf}\left(K^{str}\left(Q\right), K^{str}\left(\partial Q\right)
\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$$ by defining $$rmv\left(\sigma
\otimes z\right)=\left\{\begin{array}{cc} 0:& rmv\left(g\sigma\right)=0 \mbox{\ for\ some\ } g\in G\\
\sigma\otimes z:&\mbox{\ else\ }\end{array}\right\}.$$
Moreover, if $\sum_{j\in J} a_j\tau_j\otimes 1\in C_*^{simp,inf}\left(K^{str}\left(Q\right), K^{str}\left(\partial Q\right)
\right)\otimes_{{\Bbb Z}G}{\Bbb Z}\subset C_*^{sing}\left(Q,\partial Q\right)
\otimes_{{\Bbb Z}G}{\Bbb Z}$ represents the image 
of $\left[Q,\partial Q\right]\otimes 1$, then $\sum_{\in J} a_j rmv\left(\tau_j\otimes 1\right)$ represents the image of $\left[Q,\partial Q\right]\otimes 1$.
\end{lem}
\begin{pf} Well-definedness of $rmv$ is clear from the definition. The same proof as for Lemma 13 shows that
$rmv$ is a chain map. (The only non-obvious point is the case that 
$rmv\left( g\sigma \right)=0$ and $g\sigma$ has a constant edge, with 
vertices $v_i$ and $v_j$. In this case all $g\partial_k\sigma$ with $k\not\in\left\{
i,j\right\}$ have a constant edge. By the definition of the $G$-action in Observation 9, 
we have that the $G$-action
fixes constant edges. Thus all $\partial_k\sigma$ with $k\not\in\left\{
i,j\right\}$ have a constant edge. Moreover, $g\partial_i\sigma=\left(-1\right)^{i-j}g\partial_j\sigma$, which implies
$\partial_i\sigma=\left(-1\right)^{i-j}\partial_j\sigma$,
thus $rmv\left(\partial\sigma\right)=0$.)

If $\sum_{j=1}^r a_j\tau_j$ represents $\left[Q,\partial Q\right]$, then the second claim follows from Lemma 13.
If $\sum_{j\in J} a_j\tau_j\otimes 1$ is homologous to $\sum_{i=1}^s b_i\kappa_i\otimes 1$ and 
$\sum_{i=1}^s b_i\kappa_i$ represents $\left[Q,\partial Q\right]$, then (because $rmv$ is a
chain map) 
$rmv\left(\sum_{j\in J} a_j\tau_j\otimes 1\right)$ is homologous to 
$rmv\left(\sum_{i=1}^s b_i\kappa_i\otimes 1\right)$, which implies the second claim.\end{pf}\\
\\
The proof of Theorem 1 will pursue the idea of straightening a given cycle such that many simplices either become 
weakly degenerate or will have an edge in $\partial_0Q$. In the first case, they will disappear after application of $rmv$. In the second case, they disappear in view of the following observation, which is a variant of an argument used in \cite{gro}.

\begin{lem}: Let all assumptions of Corollary 3 be satisfied for a topological space $Q$ and
a relative cycle $z=\sum_{i\in I} a_i\tau_i\otimes 1\in C_n^{simp,inf}\left(K\left(Q
\right),K\left(\partial Q\right)\right)\otimes_{{\Bbb Z}G}
{\Bbb Z}$. Let $\sigma\subset\tau_i$ be a subsimplex of some $\tau_i, i\in I$.

If $str\left(\sigma\right)$ has an edge in $\partial_0Q$, then $str\left(\sigma\otimes 1\right)=0$. \end{lem}
\begin{pf} Let $\gamma$ be the edge of $str\left(\sigma\right)$ with image in $\partial_0Q$,
then $g=\left\{\left[\gamma,\overline{\gamma}\right]\right\}$ is an element of 
$G=\Pi\left(K\left(\partial_0Q\right)\right)$ and $g str\left(\sigma\right)=\overline{str\left(\sigma\right)}$.
In the simplicial chain complex $C_*^{simp,inf}
\left(K^{str}\left(Q\right),K^{str}\left(\partial Q\right)\right)\subset
C_*^{simp,inf}\left(K\left(Q\right),K\left(\partial Q\right)\right)$, one has $\overline{str\left(\sigma\right)}=-str\left(\sigma\right)$. Thus $g str\left(\sigma\right)=-str\left(\sigma\right)$, which implies $str\left(\sigma\otimes 1\right)=str\left(\sigma\right)\otimes 1=0$.  \end{pf}




\section{Proof of Theorem 1}

\subsection{Motivating examples }

{\bf Example 1}: {\em Let $M$ be a connected, orientable, hyperbolic $n$-manifold, $F$ an orientable, geodesic n-1-submanifold, $Q=\overline{M-F}$. For simplicity we assume
that $M$ and $F$ are closed, thus $Q$ is a hyperbolic manifold with geodesic boundary $\partial_1Q\not=\emptyset$,
and $\partial_0Q=\emptyset$.}

{\bf Outline of proof of $\parallel M\parallel_F^{norm}\ge\frac{1}{n+1}\parallel \partial Q\parallel$}:
Start with a fundamental cycle 
$\sum_{i=1}^r a_i\sigma_i$ of $M$, 
such that $\sigma_1,\ldots, \sigma_r$ are
normal to $F$. 
Since we want to consider laminations without isolated leaves, we replace $F$ by a trivially foliated product $\mathcal{F}$.
We can assume after a suitable homotopy that each component of $\sigma_i^{-1}\left(\partial Q\right)$ either contains no vertex of $\Delta^n$ or 
consists of exactly one vertex, and that each vertex of $\Delta^n$ belongs to $\sigma_i^{-1}\left({\mathcal{F}}\right)$, for $i=1,\ldots,r$.

Each $\sigma_i^{-1}\left(Q\right)$
consists of polytopes, which can each be further triangulated (without introducing new vertices)
in a coherent way (i.e., such that boundary cancellations between different $\sigma_i$'s will remain)
into $\tau_{i1},\ldots,\tau_{is\left(i\right)}$. 

$\sum_{i=1}^r a_i\left(\tau_{i1}
+\ldots+\tau_{is\left(i\right)}\right)$ is a relative fundamental cycle for $Q$.

For each $\sigma_i$, preimages of
the boundary leaves of $\mathcal{F}$ cut $\Delta^n$ into regions which we colour 
with black (components of $\sigma_i^{-1}\left(\mathcal{F}\right)$) and white 
(components of $\sigma_i^{-1}\left(Q\right)$). Moreover, if $\sigma_i^{-1}\left(
\partial Q\right)$ contains vertices, these vertices are coloured black. This is a canonical colouring (Definition 4). 

The edges of the simplices $\tau_{i,j}$ fall into two classes:
'old edges', i.e.\ subarcs of edges of 
$\sigma_i$, and 'new edges', which are contained in the interior of some subsimplex of $\sigma_i$ of dimension $\ge 2
$.

We define the labeling of the edges of $\tau_{ij}$ such that 'old edges' 
are labelled 1 and 'new edges' are labelled 0. This is an admissible labeling (Definition 10). With this labeling, we
apply the straightening procedure from section 5.1 to get a straight cycle $\sum_{i=1}^r a_i\left(str\left(\tau_{i1}\right)
+\ldots+str\left(\tau_{is\left(i\right)}\right)\right)$. 
(Thus 'old edges' are straightened to distinguished 1-simplices.)

After straightening we remove all weakly degenerate simplices (simplices contained in $\partial Q$ or having a constant edge), i.e.\ we apply 
the map $rmv$ from section 5.4. By Lemma 13, this does not change the homology 
class.
In particular, the boundary of the relative cycle,
$\partial \sum_{i,j} rmv\left(str\left(\tau_{ij}\right)\right)$ still represents the fundamental class $\left[\partial Q\right]$ of $\partial Q$.

{\em Claim: for each $\sigma_i$, after straightening there remain at most $n+1$ faces of
nondegenerate simplices $str\left(\tau_{ij}\right)$ contributing to $\partial 
\sum_{i,j} rmv\left(str\left(\tau_{ij}\right)\right)$.}

In view of Lemma 10, it suffices 
to show the following subclaim: if, for a fixed $i$, $T_1=\partial_{k_1}\tau_{ij_1},T_2=\partial_{k_2}\tau_{ij_2}$ are faces of some $\tau_{ij_1}$ resp.\ $\tau_{ij_2}$ such 
that $T_1,T_2$ have a white-parallel arc (Definition 6), then $rmv\left(str\left(\tau_{ij_1}\right)\right)=0, 
rmv\left(str\left(\tau_{ij_2}\right)\right)=0$ and in particular the corresponding straightened faces $str\left(T_1\right),
str\left(T_2\right)$ do not occur (with nonzero coefficient)
in $\partial \sum_{i,j} rmv\left(str\left(\tau_{ij}\right)\right)$. 
(Notational remark: for a subsimplex $T$ of an affine subset $S\subset \Delta^n$ we get a singular simplex $\sigma_i\mid_T$ by restricting $\sigma_i$ to $T$. We denote by $str\left(T\right)$ the straightening of $\sigma_i\mid_T$.)

To prove the subclaim, 
%assume that $T_1,T_2$ have a white-parallel arc. 
let $W$ be the white region of $\Delta^n$
containing $T_1$ and $T_2$ in its boundary. 
By assumption of the subclaim, there is a white square bounded by two arcs $e_1\subset T_1, e_2\subset T_2$ and two arcs
$f_1,f_2$ which are subarcs of edges of $\Delta^n$. (The square is a formal sum of two triangles, 
$U_1+U_2$, which are 2-dimensional faces of some $\tau_{ij}$'s.) 

\psset{unit=0.1\hsize}
$$\pspicture(0,-1)(15,4)
\pspolygon[](0,0)(4,0)(2,4)(0,0)
\psline(0.5,1)(3.5,1)


\psline(1.5,3)(2.5,3)

\uput[0](1.5,0.8){$e_2$}
\uput[0](0.3,1.6){$f_1$}
\uput[0](1.8,3.1){$e_1$}
\uput[0](3.2,1.6){$f_2$}
\uput[0](1.5,2){square}




\endpspicture$$

We want to show that all edges of $str\left(\tau_{ij_1}\right)$ belong to $S_1^{str}\left(\partial
Q\right)$.
Note that $T_1,T_2\subset \partial W$ are mapped to $\partial Q$. Let $x_1\in S_0^{str}\left(Q\right)$
resp.\ $x_2\in S_0^{str}\left(Q\right)$ be the unique elements of
$S_0^{str}\left(Q\right)$ in the same connected component 
$C_1$ resp.\ $C_2$ of $\partial Q$ as $\sigma_i\left(
T_1\right)$ resp.\ $\sigma_i\left(T_2\right)$. 
In particular $\partial_0str\left(e_1\right)=x_1=\partial_1str\left(e_1\right)$ and $\partial_0str\left(e_2\right)=x_2=
\partial_1str\left(e_2\right)$.
Thus $e_1$ and $e_2$ are straightened to loops $str\left(e_1\right)$ resp.\ $str\left(e_2\right)$ based at $x_1$ resp.\ $x_2$. 
The straightenings of
the other two arcs, $str\left(f_1\right),str\left(f_2\right)$
connect $x_1$ to $x_2$, and they are distinguished 1-simplices because they arise as straightenings of 'old edges'.
Thus $str\left(f_1\right)=str\left(f_2\right)$, by uniqueness of distinguished 1-simplices with given vertices. 
This means that the square is straightened to a cylinder. 

But $\left(Q,\partial Q\right)$ is acylindrical, thus either $str\left(e_1\right),str\left(e_2\right)$ are
constant (in which case $rmv\left(str\left(\tau_{ij_1}\right)\right)=rmv\left(str\left(\tau_{ij_2}\right)\right)=0$),
or the cylinder 
must be homotopic into $\partial Q$. In the latter case, $str\left(f_1\right)$ must (be
homotopic into and therefore) be contained in $\partial Q$. In particular, $\partial_0 str\left(f_1\right)$ and $\partial_1 str\left(f_1\right)$ belong to the same component of $\partial Q$. This implies
$\partial_0 str\left(f_1\right)=\partial_1 str\left(f_1\right)$. Since $str\left(f_1\right)$ is a distinguished 1-simplex, this implies that $str\left(f_1\right)$ is constant.

Let $P_1,P_2$ be the affine planes whose intersections with $\Delta^n$ contain $T_1$ resp.\ $T_2$.
We have now that there is an arc $f_1$ connecting 
$P_1\cap\Delta^n$ to $P_2\cap\Delta^n$ such that $str\left(f_1\right)$ is contained in $\partial Q$. This
implies that for each other arc $f$ connecting
$P_1\cap\Delta^n$ to $P_2\cap\Delta^n$ its straightening $str\left(f\right)$
must (be homotopic into and therefore) be contained in $\partial Q$.

If $P_1$ and $P_2$ are of the same type, then all edges of $str\left(\tau_{ij_1}\right)$ connect
$P_1\cap\Delta^n$ to $P_2\cap\Delta^n$, hence all
edges of $str\left(\tau_{ij_1}\right)$ belong to $S_1^{str}\left(\partial Q\right)$.
If $P_1$ and $P_2$ are not of the same type, then
existence of a parallel arc implies that
at least one of them, say $P_1$, must be of type $\left\{0a_1\ldots a_k\right\}$ with $k\not\in\left\{0,n-1\right\}$. Then, if $P_3$ 
is any other plane bounding $W$, it follows from Corollary 2 that $P_3$ has a white-parallel arc with $P_1$. 
Thus, repeating the argument in the last paragraph with $P_1$ and $P_3$ in place of $P_1$ and $P_2$, we 
obtain that for each arc
$f$ connecting
$P_1\cap\Delta^n$ to $P_3\cap\Delta^n$ its straightening $str\left(f\right)$
must (be homotopic into and therefore) be 
contained in $\partial Q$. Hence, for each $\tau_{ij_1}$ in the chosen triangulation of $W$, its 1-skeleton is straightened into $\partial Q$. 

Since straight simplices $\sigma$ (of dimension $\ge 2$) with $\partial\sigma$ 
in the geodesic boundary $\partial Q$, must be in $\partial Q$,
this implies by induction that the $k$-skeleton of $str\left(\tau_{ij_1}\right)$ is in $\partial Q$, for each $k$.
In particular, $str\left(\tau_{ij_1}\right)\in S_n^{str}\left(\partial Q\right)$. Hence $rmv\left(
str\left(\tau_{ij_1}\right)\right)=0$.
We have proved the subclaim.\\
\\
By Lemma 10, the {\em subclaim} implies the {\em claim}. Since $\sum_{i=1}^r a_i \partial \sum_j rmv\left(str\left(\tau_{ij}\right)\right)$ represents the fundamental class $\left[\partial Q\right]$,
we conclude 
$\parallel \partial Q\parallel\le \left(n+1\right)\sum_{i=1}^r\mid a_i\mid$.\\
\\
\\
The simplifications of Example 1 in comparison to the proof in section 6.2 are essentially all due to the fact that $\partial_0Q=\emptyset$. We remark that in Example 2, if $F$ is not geodesic, then $Q\not=N$ and thus $\partial_0Q\not=\emptyset$ (even though 
$\partial M=\emptyset, \partial F=\emptyset$). Thus the generalization to $\partial_0Q\not=\emptyset$ would be necessary even if one only wanted to consider closed manfifolds $M$ and $F$. \\
\\
{\bf Example 2}: {\em Let $M$ be a connected, closed, hyperbolic 3-manifold, $F\subset M$ a closed, incompressible surface, $N=\overline{M-F}, Q=Guts\left(N\right)$. }

{\bf Outline of proof of $\parallel M\parallel_F^{norm}\ge\frac{1}{4}\parallel \partial Q\parallel$}:
Start with a fundamental cycle
$\sum_{i=1}^r a_i\sigma_i$ of $M$,
such that $\sigma_1,\ldots, \sigma_r$ are
normal to $F$.
As in Example 1 we get a relative fundamental cycle $\sum_{i=1}^r a_i\left(\tau_{i1}                                                                             +\ldots+\tau_{is\left(i\right)}\right)$
of $N$. We can not apply the argument from Example 1 to $N$ because $N$ is not acylindrical. Therefore we 
would like to work with
a relative fundamental cycle for the acylindrical manifold $Q$.

Using Lemma 2, we can assume that all $\tau_{ij}$ belong to $K\left(N\right)$. Then we can apply the retraction $r$ from Lemma 5. Since $r$ is only defined after tensoring $.\otimes_{{\Bbb Z}G}{\Bbb Z}$,
we get $r\left(\tau_{ij}\otimes 1\right)=\kappa_{ij}\otimes 1$ with $\kappa_{ij}$ only determined up to choosing one $\kappa_{ij}$ in its $G$-orbit. The rest of the proof then basically boils down to
copying the proof of Example 1 (with $\tau_{ij}$ replaced by $\kappa_{ij}$), but taking care of the ambiguity in the choice of $\kappa_{ij}$. The details can be found in the next section.


\subsection{Proof}

\begin{df}: Let $Q$ be a manifold with 
a given decomposition $$\partial Q=\partial_0Q\cup\partial_1Q.$$
The pair $\left(Q,\partial_1Q\right)$ is called a pared
acylindrical manifold, if any continuous mapping of pairs
$f:\left({\Bbb S}^1\times \left[0,1\right],{\Bbb S}^1\times\left\{0,1\right\}\right)
\rightarrow \left(Q,\partial_1Q\right)$, which is $\pi_1$-injective as a map of pairs, must be homotopic, as a map of pairs 
$$\left({\Bbb S}^1\times \left[0,1\right],{\Bbb S}^1\times\left\{0,1\right\}\right)\rightarrow \left(Q,\partial_1Q\right),$$
into $\partial Q.$
\end{df}
%Remark: In 3-manifolds theory some authors only require 
%that $f$ is homotopic into $\partial Q$. For the applications 
%in chapter 7, $\partial_0Q$ will consist of tori and annuli, hence the two conditions are equivalent.

\begin{df}: Let $\left(N,N^\prime\right)$ be a topological space, such that $N=Q\cup R$ for two subspaces
$Q,R$. (In particular, $N^\prime=Q^\prime\cup R^\prime$ for $Q^\prime=Q\cap N^\prime,R^\prime=N^\prime\cap R$).
Let $Q_0^\prime:=Q\cap R$. 
We say that the
decomposition $N=R\cup Q$ is an essential decomposition of $\left(N,N^\prime\right)$
if $$Q\cap R=Q^\prime\cap R^\prime$$ and if the inclusions $$Q_0^\prime\rightarrow Q\rightarrow N, Q_0^\prime
\rightarrow R\rightarrow N,
Q_0^\prime\rightarrow Q^\prime\rightarrow N^\prime,
Q_0^\prime\rightarrow R^\prime\rightarrow N^\prime$$
are each $\pi_1$-injective (for each connected component).
\end{df}


\begin{thm}:  
Let $M$ be a compact, orientable, connected n-manifold and $\mathcal{F}$ a lamination (of codimension one)
of
$M$. Assume that $$N:=\overline{M - {\mathcal{F}}}$$
has a decomposition $N=Q\cup R$ into orientable n-manifolds with boundary $Q,R$ such that, if we
let $$
\partial_0Q=Q\cap R,
\partial_1Q=\partial
Q\cap\partial N, N^\prime=\partial N\cup\partial_0Q,$$
then\\
i) each connected component of
$\partial_0Q$ has
amenable fundamental group, \\
ii) $\left(Q,\partial_1Q\right)$ is pared acylindrical, $\partial_1Q\rightarrow Q$ is $\pi_1$-injective, $\partial_0Q\cap\partial_1Q\rightarrow\partial_1Q$ is not $\pi_1$-surjective\\
iii) $Q,\partial Q,\partial_1Q$ are aspherical,\\
iv) the decomposition $N=Q\cup R$
is an essential decomposition of $\left(N,N^\prime\right)$.

Then 
$$\parallel M,\partial M\parallel^{norm}_{\mathcal{F}}
\ge\frac{1}{n+1}\parallel \partial Q\parallel.$$\end{thm}
%$$\parallel M
%\parallel_{\mathcal{F}}\ge
%\frac{1}{\left[\frac{n}{2}\right]+1}\parallel \partial Q\parallel.$$\end{thm}

{\em Proof:}
%We prove the claimed inequality for the normal Gromov norm. Afterwards, at the end, we will give the necessary modifications for the proof of the stronger inequality for the (transverse) Gromov norm.\\

Consider a relative cycle $z=\sum_{i=1}^r a_i\sigma_i$, representing $\left[M,\partial M\right]$,
such that $\sigma_1,\ldots, \sigma_r$ are
normal to $\mathcal{F}$. 
%(This is in particular the case if
%$\sigma_1,\ldots, \sigma_r$ are
%transverse to $\mathcal{F}$.)
Our aim is to show: 
$\sum_{i=1}^r\mid a_i\mid\ge \frac{1}{n+1}\parallel\partial Q\parallel$.\\
% (resp.\ $\sum_{i=1}^r\mid a_i\mid\ge \frac{1}{\left[\frac{n}{2}\right]+1}\parallel\partial Q\parallel$ if $\sigma_1,\ldots,\sigma_r$ are even transverse to $\mathcal{F}$).\\

Denote $$N=\overline{M-{\mathcal{F}}}.$$ 
Since each $\sigma_i$ is normal to $\mathcal{F}$,
we have for each $i=1,\ldots,r$ that, 
after application of a simplicial homeomorphism $h_i:\Delta^n
\rightarrow\Delta^n$,
the image of
$\sigma_i^{-1}\left(N\right)$
consists of polytopes, which can each be further triangulated in a coherent way (i.e., such that boundary cancellations between different $\sigma_i$'s will remain)
into simplices $\theta_{ij},j\in \hat{J}_i$. (It is possible that
$\mid \hat{J}_i\mid=\infty$, because $N$ may be noncompact.) 
We choose these triangulations of the $\sigma_i^{-1}\left(N\right)$ 
to be minimal (Definition 6), that is, we do not introduce new vertices. (Indeed, 
compatible minimal triangulations of the $\sigma_i^{-1}\left(N\right)$ do exist: one starts with common minimal triangulations of 
the common faces and extends them to minimal triangulations of each polytope.)
 
Because boundary cancellations are preserved, we have that $$\sum_{i=1}^r a_i\sum_{j\in \hat{J}_i}\theta_{ij}
$$ is a countable (possibly infinite) relative cycle 
representing the fundamental class
$\left[N,\partial N\right]$ in the sense of section 3.2.\\
We fix a sufficient set of cancellations ${\mathcal{C}}^M$ for the 
relative cycle $\sum_{i=1}^r a_i\sigma_i$, in the sense of Definition 8. 
This induces a sufficient set of cancellations ${\mathcal{C}}^N$ for the relative cycle $\sum_{i=1}^r \sum_{j\in \hat{J}_i}a_i\theta_{ij}$.\\


%Recall that we can assume that $\mathcal{F}$ has no isolated leaves, upon replacing isolated leaves with canonically foliated $I$-bundles.
If $\partial M$ is a leaf of $\mathcal{F}$, then all faces of $z$ contributing to $\partial z$ are contained in $\partial N$. We call these faces exterior faces. We can assume that, for each $i$, \\
- each component of $\sigma_i^{-1}\left(\partial N\right)$ either contains no vertex of $\Delta^n$, or
consists of exactly one vertex, or consists of an exterior face,\\
- and each vertex of $\Delta^n$ belongs to $\sigma_i^{-1}\left(\mathcal{F}\right)$.\\
Indeed, by a small homotopy of the relative 
fundamental cycle $\sum_{i=1}^r a_i\sigma_i$, preserving normality, we can obtain that no component of $\sigma_i^{-1}\left(\partial N\right)$ contains 
a vertex of $\Delta^n$, except for exterior faces. Afterwards, if some vertices of $\sum_{i=1}^r a_i\sigma_i$ do not belong to $\mathcal{F}$, 
we may homotope a small neighborhood of the vertex,
until the vertex (and no other point of the neighborhood) meets $\partial N$. This, of course, 
preserves normality to
%of $M$. The homotopy can be chosen to preserve normality of the $\sigma_i$ to 
$\mathcal{F}$. 
%Namely, if some vertices belong to $N$, then we may homotope these vertices until
%$\partial N$, without affecting the intersection of the $\sigma_i$ with $\mathcal{F}$, and afterwards we may slightly homotope the 
%vertices into the interior of the 
%laminated part, such that near to the vertices the intersections of leaves 
%$with the simplex are normal.

Since each $\sigma_i$ is normal to $\mathcal{F}$, in particular each $\sigma_i$ is normal to the union of boundary leaves 
$$\partial_1 N:=\overline{\partial N - 
\left(\partial M\cap \partial N\right)}.$$
Thus for each $\sigma_i$, after application of a simplicial
homeomorphism $h_i:\Delta^n\rightarrow\Delta^n$,
the image of $\sigma_i^{-1}\left(\partial_1 N\right)$ consists of a (possibly infinite) 
set $$Q_1,Q_2,\ldots\subset\Delta^n,$$
such that $$Q_i=P_i\cap\Delta^n$$ for some
affine hyperplanes $P_1,P_2,\ldots$. 
We define a colouring by declaring that (images under $h_i$ of)
components of $$\sigma_i^{-1}
\left(int\left(N\right)\right):=\sigma_i^{-1}\left(N-\partial_1N\right)$$ are coloured
white and (images under $h_i$ of)
components of $\sigma_i^{-1}\left({\mathcal{F}}\right)$ 
are coloured black. (In particular, all $Q_i$ are coloured black.)
Since we assume that all vertices of $\Delta^n$ belong to 
$\sigma_i^{-1}\left({\mathcal{F}}\right)$, and since each boundary leaf is adjacent to at least one component of $\sigma_i^{-1}\left(int\left(N\right)\right)$, this is a canonical colouring (Definition 4).\\
\\
%We denote $N^\prime:=\partial N\cup\partial_{0}Q$. 
Since $\partial N\subset N^\prime$, we may of course also consider 
$\sum_{i=1}^r a_i\sum_{j\in \hat{J}_i}\theta_{ij}
$ as a relative 
cycle in $C_*^{inf}\left(N,N^\prime\right)$.\\

By Lemma 2a), 
we can homotope $$z=\sum_{i=1}^r a_i\sum_{j\in \hat{J}_i}\theta_{ij}$$
to a chain $$\sum_{i=1}^r a_i\sum_{j\in \hat{J}_i}\hat{\theta}_{ij}$$
such that each $\hat{\theta}_{ij}$ is a simplex of $\widetilde{K}\left(N\right)$, as defined
in section 3.2, and such that $\partial z$ is homotoped into $\widetilde{K}\left(N^\prime\right)$. Then we apply Lemma 2c) and obtain that
$$\sum_{i=1}^r\sum_{j\in\hat{J}_i}a_i\tau_{ij}:=
\sum_{i=1}^r\sum_{j\in\hat{J}_i}a_ip\left(\hat{\theta}_{ij}\right)\in
C_n^{simp,inf}\left(K\left(N\right)\right)$$
represents $\left[N,\partial N\right]$. (Remark: to apply Lemma 2c), 
we need $\parallel N,\partial N\parallel\not=0$. If we had $\parallel N,\partial N\parallel=0$, 
then Lemma 6 would easily imply $\parallel
\partial Q\parallel=0$ and the conclusion of Theorem 1 would be trivially true. Hence we can restrict to the case that
the assumption of Lemma 2c) holds true.) \\

Consider $Q\subset N$ as in the assumptions of Theorem 1.
We denote $$G:=\Pi\left(K\left(\partial_{0} Q\right)\right).$$

We have by assumption that $N=Q\cup R$ is an essential decomposition of $\left(N,N^\prime\right)$ (Definition 13), which means exactly that
the assumptions of Lemma 5 are satisfied.
%with $R^\prime=N^\prime\cap R, Q^\prime=N^\prime\cap Q=\partial Q, Q_0^\prime=\partial_{0}Q$. 
Thus, according to Lemma 5, there exists a retraction 
$$r:C_n^{simp,inf}\left(K\left(N\right)
\right)\otimes_{{\Bbb Z}G}{\Bbb Z}
\rightarrow C_n^{simp,inf}\left(K\left(Q\right)\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$$ 
for $n\ge 2$, mapping $C_n^{simp,inf}\left(K\left(N^\prime\right)
\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$ to $C_n^{simp,inf}\left( K\left(\partial Q\right)\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$, 
such that, for each simplex $\tau_{ij}\in K\left(N\right)$, we either have $r\left(\tau_{ij}\otimes 1\right)=0$ or
$$r\left(\tau_{ij}\otimes 1\right)=\kappa_{ij}\otimes 1$$ for some simplex $\kappa_{ij}\in K\left(Q\right).$ (Remark: if $n=1$, then Theorem 1 is trivially true. Hence we can restrict to the case $n\ge 2$.)
%(According to Lemma 5, the image of $r$ is actually
%$C^{simp,lf}_*\left(K\left(Q\right),K\left(\partial Q\right)\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$. 
%Compactness of $Q$ implies that $C^{simp,lf}_*\left(K\left(Q\right),K\left(\partial Q\right)\right)=C_*^{simp}
%\left(K\left(Q\right),K\left(\partial Q\right)\right)$.)

Thus $$r\left(\sum_{i=1}^r a_i\sum_{j\in \hat{J}_i}\tau_{ij}\otimes 1\right)=
\sum_{i=1}^r a_i\sum_{j\in J_i}\kappa_{ij}\otimes 1$$ with $J_i\subset\hat{J}_i$ for all $i$. (It may still be possible that $\mid J_i\mid=\infty$.)\\
We remark that $\kappa_{ij}$ is only determined up to choosing one $\kappa_{ij}$ in its G-orbit.

Since $r$ is a chain map, we get a sufficient set of cancellations for \\
$\sum_{i=1}^r a_i\sum_{j\in J_i}\kappa_{ij}\otimes 1$
by $${\mathcal{C}}^Q:=\left\{\left(\partial_{a_1}\kappa_{i_1j_1}\otimes 1, \partial_{a_2}\kappa_{i_2j_2}\otimes 1\right): \left(\partial_{a_1}\tau_{i_1j_1},
\partial_{a_2}\tau_{i_2j_2}\right)\in{\mathcal{C}}^N\right\}.$$

By part b) of Lemma 6 we have that 
$$\partial\left(\sum_{i=1}^r a_i \sum_{j\in J_i} \kappa_{i,j}\otimes 1\right)\in C_*^{simp,inf}
\left(K\left(\partial Q\right)\right)\otimes_{{\Bbb Z}G}
{\Bbb Z}$$
represents (the image of) $\left[\partial Q\right]\otimes 1$.

Lemma 4a) gives that $G$ is amenable. Together with Lemma 7 this implies $$\parallel\partial Q\parallel
\le \sum_{i=1}^r \mid a_i\mid
\left(n+1\right)\mid J_i\mid.$$

In the remainder of the proof, we will use Lemma 14 to improve this inequality and, in 
particular, get rid of the unspecified (possibly infinite)
numbers $\mid J_i\mid$.\\

%We can homotope $\sum_{i=1}^r a_i \sum_{j=1}^{s\left(i\right)} \kappa_{i,j}\otimes 1$ such each vertex 
%is contained in $\partial_0Q$ or in a closed component of $\partial_1Q$ (because this is a $G$-invariant condition).
If $\partial_1 Q$ were empty, then $\partial
Q=\partial_0Q$ and amenability of $\pi_1\partial_0Q$ would imply
$\parallel\partial Q\parallel=0$, in particular Theorem 1 would be trivially true. Hence we can restrict to the case
$\partial_1Q\not=\emptyset$. This means that $$Q,\partial Q,\partial_0Q,\partial_1Q$$
satisfy Assumption I from section 5. Thus there exists a simplicial
set $$K_*^{str}\left(Q\right)\subset S_*\left(Q\right)$$ 
satisfying conditions i)-viii) from Lemma 11, and a set $$D\subset K_1^{str}\left(Q\right)$$
of distinguished 1-simplices (Definition 9).

Recall that, for each $i$, $$\sum_{j\in\hat{J}_i} \theta_{i,j}$$ was defined by choosing a triangulation
of $\sigma_i^{-1}\left(N\right)$.
%, where the triangulations for different $\sigma_i$'s had to be chosen 
%compatibly at common boundary faces. 
The simplices $\theta_{i,j}$ thus have 'old edges', i.e.\ subarcs of edges of $\sigma_i$, and 'new edges', whose interior
is contained in the interior of some subsimplex of $\sigma_i$ of dimension $\ge 2$. 

Associated to $z=\sum_{j=1}^{t\left(i\right)} \theta_{i,j}$ and ${\mathcal{C}}^N$ (and 
an arbitrary minimal presentation of $\partial z$) are, by Definition 8, simplicial sets $\Upsilon^N,\partial\Upsilon^N$. 
%Let $\Upsilon^r:=\left\{\tau\in\Upsilon^N:r\left(\tau\otimes 1\right)\not=0\right\}\subset\Upsilon$. 

The only possibility that two 'old edges' have a vertex in $\Upsilon^N$ in common is that this vertex is a vertex of $\sigma_i$. 
%(Even though they may have common vertices in $Q$.)
Hence there does not exist a cycle of 'old edges' in $\Upsilon^N$. 

So the labeling of edges of $$\sum_{i=1}^r a_i\sum_{j\in\hat{J}_i}\theta_{i,j}$$ by labeling 'old edges' with
label 1 and 'new edges' with label 0 is an admissible labeling (Definition 10).\\

Associated to $$w=\sum_{j\in J_i} \kappa_{i,j}\otimes 1$$ and ${\mathcal{C}}^Q$ 
(and
an arbitrary minimal presentation of $\partial w$) there are simplicial sets $\Upsilon,\partial\Upsilon$.
By our definition of ${\mathcal{C}}^Q$, $\Upsilon$ is isomorphic to a 
simplicial subset of $\Upsilon^N$, namely to the subset generated by the set 
$$\left\{\tau\in\Upsilon^N:r\left(\tau\otimes 1\right)\not=0\right\}$$ together with all iterated faces and degenerations.
In particular, the admissible 0-1-labeling of $\Upsilon^N$ induces an admissible 
0-1-labeling of $\Upsilon$ (even though possibly the $\kappa_{ij}$'s may yield a 1-labeled cycle in $Q$).
%For each $\left(i,j\right)$, we give the edges of $\kappa_{i,j}$ the same labelling as the 
%corresponding edges of $\tau_{i,j}$. 
%Then $\sum_{i=1}^r a_i\sum_{j=1}^{s\left(i\right)}\kappa_{ij}\otimes 1\in 
%C_*\left(K,\partial K\right)$ is a relative cycle, with an admissible 0-1-labelling
%of its edges.

Let $\widehat{\Upsilon},\partial\widehat{\Upsilon}$ be defined by Observation 10.
By Corollary 3, there is a chain map 
$$str:C_*^{simp,inf}\left(\widehat{\Upsilon}
\right)\otimes_{{\Bbb Z}G}{\Bbb Z}
\rightarrow
C_*^{simp,inf}\left(K^{str}\left(Q\right)
\right)\otimes_{{\Bbb Z}G}{\Bbb Z},$$
mapping $C_*^{simp,inf}\left(\partial\widehat{\Upsilon}
\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$ to
$C_*^{simp,inf}\left(
K^{str}\left(\partial Q\right)
\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$, such that 
$$\sum_{i=1}^ra_{i} \sum_{j\in J_i}str\left(\kappa_{ij}\right)\otimes 1$$
represents (the image of) $\left[Q,\partial Q\right]\otimes 1$
and such that 1-labeled edges 
are mapped to distinguished 1-simplices. (We keep in mind that $\kappa_{ij}$ and thus $str\left(\kappa_{ij}\right)$
are determined only up to choosing one simplex in its $G$-orbit.)

We then apply Lemma 14 to get the relative cycle
$$\sum_{i=1}^r a_i \sum_{j\in J_i}
rmv\left(str\left(\kappa_{ij}\right)\otimes 1\right)\in C_*^{simp,inf}\left(K^{str}\left(Q\right),K^{str}\left(\partial Q\right)\right)\otimes_{{\Bbb Z}G}{\Bbb Z}$$
representing (the image of) $\left[Q,\partial Q\right]\otimes 1$. Thus
$$\sum_{i=1}^r a_i \partial \sum_{j\in J_i}
rmv\left(str\left(\kappa_{ij}\right)\otimes 1\right)\in C_*^{simp,inf}\left(K^{str}\left(\partial Q\right)\right)
\otimes_{{\Bbb Z}G}{\Bbb Z}$$ represents (the image of) $\left[\partial Q\right]\otimes 1$. 
We want to show that it is actually 
a finite chain of $l^1$-norm at most $\left(n+1\right)\sum_{i=1}^r\mid a_i\mid$.\\
\\
{\em Claim}: For each $i$, $$\partial\sum_{j\in J_i}rmv\left( 
str\left(\kappa_{i,j}\right)\otimes 1\right)$$
is the formal sum of at most $n+1$ n-1-simplices $L\otimes 1$ with coefficient 1. \\

%This will imply the wanted inequality
%$$\parallel\partial Q\parallel\le\sum_{i=1}^r\mid a_i\mid \left(n+1\right).$$


%To prove the claim, our aim is to show that, under the assumption of pared acylindricity of $\left(Q,\partial_1Q\right)$, for 
%each $i$, the set of n-1-simplices $L$ such that $L\otimes 1$
%occurs with coefficient 1 in 
%$\partial\sum_{j\in J_i}rmv\left(
%str\left(\kappa_{i,j}\right)\otimes 1\right)$
%can 
%not contain any n-1-simplices $L_1\otimes 1=str\left(r\left(T_1\otimes 
%1\right)\right)$ and
%$L_2\otimes 1=str\left(r\left(T_2\otimes
%1\right)\right)$
%such that $T_1$ 
%does have a 
%white-parallel arc with with some $T_2$ (Definition 5).
The claim will be a consequence of the following subclaim and Lemma 10.\\
%This will imply, with Lemma 10,
%that there can be at most $n+1$ n-1-simplices 
%occuring with coefficient 1, yielding the wanted inequality.\\
\\
{\em Subclaim}: 
Assume that for some fixed $i\in I$, for the 
chosen triangulation 
$$\sigma_i^{-1}\left(N\right)=\bigcup_{j\in \hat{J}_i}\theta_{ij},$$ 
and the associated canonical colouring, there exist 
$j_1,j_2\in \hat{J}_i, k_1,k_2\in \left\{0,\ldots,n\right\}$ such that the
faces $$T_1=\partial_{k_1}\theta_{ij_1}\in S_{n-1}\left(\partial N\right), T_2=\partial_{k_2}\theta_{ij_2}\in S_{n-1}\left(\partial N\right)$$ 
have a white-parallel arc (Definition 6). Then $$rmv\left(str\left(\kappa_{ij_1}\right)\otimes 1\right)=0,
rmv\left(str\left(\kappa_{ij_2}\right)\otimes 1\right)=0.$$

We are going to prove the subclaim. $$\partial_{k_l}\theta_{ij_l}\in S_{n-1}\left(\partial N\right)\subset
S_{n-1}\left(N^\prime\right)$$ implies $$\partial_{k_l} str\left(\kappa_{ij_l}\right)\in K_*^{str}\left(\partial Q\right)$$ for $l=1,2$.
Argueing by contradiction, we assume that $$rmv\left(str\left(\kappa_{ij_1}\right)\otimes 1\right)\not=0.$$

By assumption of the subclaim, there are white-parallel arcs $e_1,e_2$ of $T_1$ resp.\ $T_2$.
This means that there are arcs
$e_1,e_2$ in a 2-dimensional subsimplex $\tau^2  \subset\Delta^n$ of the standard simplex, and that there are arcs $f_1,f_2$, which are subarcs of some edge of $\tau^2$, such that
$$\partial_0e_1=\partial_1f_2,\partial_0f_2=\partial_0e_2,\partial_1e_2=\partial_0f_1,\partial_1f_1=\partial_1e_1$$ 
and such that $$e_1,f_2,e_2,f_1$$ bound a square in the boundary of 
a white component. (Cf.\ the picture in section 6.1. We will use the same letter for an affine subset of $\Delta^n$ and 
for the singular simplex obtained by restricting $\sigma_i$ to this subset.) The square is of the form $U_1+U_2$, where $U_1,U_2$ are n-2-fold iterated faces of some $\theta_{ij}$'s.
Hence $$\partial U_1=e_1+f_2+\partial_2U_1$$ and $$\partial U_2=-e_2-f_1-\partial_2U_1,$$ i.e. 
$$\partial \left(U_1+U_2\right)=e_1+f_2-e_2-f_1.$$ 
We emphasize that $f_1,f_2$ need not be edges of $\theta_{ij_1}$ or $\theta_{ij_2}$.\\
\\
Notational convention: for each iterated face $f=\partial_{k_1}\ldots\partial_{k_l}\theta_{ij}$ with $i\in I,j\in J_i$, we
will denote $f^\prime$ the $n-l$-simplex with
$$f^\prime\otimes 1=\partial_{k_1}\ldots\partial_{k_l}\kappa_{ij}\otimes 1=
r\left(\partial_{k_1}\ldots\partial_{k_l}\tau_{ij}\otimes 1\right)=r\left(
\partial_{k_1}\ldots\partial_{k_l} p\left(\hat{\theta}_{ij}\right)\otimes 1\right).$$
(The last two equations are 
true because $r,p$ and the homotopy from $\sum_{i,j}a_i\theta_{ij}$ to $\sum_{i,j}\hat{\theta}_{ij}$ are chain maps.) In other words, if $f$ is an iterated face of some $\tau_{ij}$, then $f^\prime$ is, up to the ambiguity by the $G$-action, the corresponding iterated face of $\kappa_{ij}$.\\

Since $r,p$ and the homotopy are chain maps, we have 
$$\partial_0e_1^\prime\otimes 1=\partial_1f_2^\prime\otimes 1,\partial_0f_2^\prime\otimes 1=\partial_0e_2^\prime\otimes 1,
\partial_1e_2^\prime\otimes 1=\partial_0f_1^\prime\otimes 1,\partial_1f_1^\prime\otimes 1=\partial_1e_1^\prime\otimes 1$$
and 
$$\partial \left(U_1^\prime\otimes 1\right)= e_1^\prime\otimes 1+f_2^\prime\otimes 1 +\partial_2U_1^\prime\otimes 1,
\partial \left(U_2^\prime\otimes 1\right)=-e_2^\prime\otimes 1-f_1^\prime\otimes 1-\partial_2U_1^\prime\otimes 1.$$
(One should keep in mind that $x\otimes 1=y\otimes 1$ means that $y=gx$ for some $g\in G$.)

We claim that, upon choosing the $\kappa_{ij}$'s 
in their respective $G$-orbits, we can achieve that
\begin{itemize}
\item $\partial_0e_1^\prime=\partial_1f_2^\prime\in \partial_1Q,\partial_0f_2^\prime=\partial_0e_2^\prime\in 
\partial_1Q, \\
\partial_1e_2^\prime=\partial_0f_1^\prime\in\partial_1Q,
\partial_1f_1^\prime=\partial_1e_1^\prime\in\partial_1Q$
\item $\partial\left(str\left(U_1^\prime\right)+str\left(U_2^\prime\right)\right)=str\left(e_1^\prime\right)+str\left(f_2^\prime\right)
-str\left(e_2^\prime\right)-str\left(f_1^\prime\right)$
%\item $str\left(e_1^\prime\right)$ and $str\left(e_2^\prime\right)$ are loops.
\end{itemize}
Indeed, since by Lemma 5b) no vertex belongs to a path-component $C$ of $\partial Q$ contained in
$\partial_0Q$, we can clearly use the $G$-action to move all vertices to $\partial_1Q$. Moreover, if $f_2^\prime$ belongs to $\kappa_{ij_k}$ with $k\not=1$, then $\partial_1f_2^\prime=g\partial_0e_1^\prime$ 
implies that after replacing $\kappa_{ij_k}$ by 
$g^{-1}\kappa_{ij_k}$ (which does not affect $e_1^\prime$) we get $\partial_1f_2^\prime=\partial_0e_1^\prime$. 
On the other hand, 
if $f_2^\prime$ does belong to $\kappa_{ij_1}$, then of course $\partial_0e_1^\prime$ and $\partial_1f_2^\prime$ are the same vertex of $\kappa_{ij_1}$. \\
The other three equations of the first item are handled analogously. (In each case $g$ can be chosen to affect only the vertex in question. This is possible because $f_1^\prime$ and $f_2^\prime$ belong, by construction, to distinct simplices, and because, by Lemma 5c), $f_1^\prime$ and $f_2^\prime$ are not loops.)

For the second item: let $\overline{\partial_2U_1^\prime}=g\partial_2U_2^\prime$. 
If $\overline{\partial_2U_1^\prime}$ and $\partial_2U_2^\prime$ 
belong to the same $\kappa_{ij}$, then they agree. If not, then 
we can apply $g$ to one $\kappa_{ij}$ without affecting the other and get $\overline{\partial_2U_1^\prime}=
g\partial_2U_2^\prime$, which implies the claim of the second item. 


%The 1-skeleton of $\partial_{k_1}str\left(\kappa_{ij_1}\right)$ belongs to $K_1^{str}\left(\partial Q\right)$.
%Moreover, since $\partial_{k_1}\tau_{ij_1}\in K\left(\partial N\right)$, 
%Lemma 5b) implies that no vertex of $\partial_{k_1}\kappa_{ij_1}$
%is contained in a path-component $C$ of $\partial Q$ with $C\subset\partial_0Q$.
%By condition i) of Lemma 11, this implies that each vertex of $\partial_{k_1}str\left(\kappa_{ij_1}\right)$ belongs to $K_0^{str}\left(\partial_1Q\right)$.
%In particular,
%$$\partial_0str\left(e_1^\prime\right),\partial_1str\left(
%e_1^\prime\right),\partial_0str\left(e_2^\prime\right),\partial_1
%str\left(e_2^\prime\right)\in K_0^{str}\left(\partial_1Q\right).$$

%Since $K_0^{str}\left(\partial_0Q\right)\cap K_0^{str}\left(\partial_1Q\right)=\emptyset$ by condition iii) from Lemma 11,
%this implies that none of $
%str\left(U_1^\prime\right),str\left(U_2^\prime\right),
%str\left(e_1^\prime\right),str\left(f_2^\prime\right),str\left(e_2^\prime\right),str\left(f_1^\prime\right)$ has a vertex
%in $\partial_0Q$. In particular,
%$
%str\left(U_1^\prime\right),str\left(U_2^\prime\right),
%str\left(e_1^\prime\right),str\left(f_2^\prime\right),str\left(e_2^\prime\right),str\left(f_1^\prime\right)$
%are fixed by each $g\in G=\Pi\left(K\left(\partial_{0}Q\right)\right)$.
%Thus, $\partial\left(U_1^\prime\otimes 1+U_2^\prime\otimes 1\right)=e_1^\prime\otimes 1+f_2^\prime\otimes 1-e_2^\prime\otimes 1-f_1^\prime\otimes 1$ and  $\partial str=str\partial$
%implies
%$$\partial\left(str\left(U_1^\prime\right)+str\left(U_2^\prime\right)\right)=str\left(e_1^\prime\right)+str\left(f_2^\prime\right)
%-str\left(e_2^\prime\right)-str\left(f_1^\prime\right).$$

%It follows from the construction of $K_*^{str}\left(Q\right)$ that
%$str\left(e_1^\prime\right),str\left(e_2^\prime\right)$ are {\em loops}. 
%Indeed, for $i=1,2$, $\partial_0str\left(e_i^\prime\right)$ and $\partial_1str\left(e_i^\prime\right)$ belong 
%to $\partial_1Q$ and moreover belong to the same path-component $C$ of $\partial Q$, because 
%$str\left(e_i^\prime\right)\in K_1^{str}\left(\partial Q\right)$. But 
%from condition (i) of Lemma 11, it follows that $C$ contains only one straight 0-simplex, hence $\partial_0
%str\left(e_i^\prime\right)=\partial_1str\left(e_i^\prime\right)$.

%So, $str\left(e_1^\prime\right)$ is a loop 
%based at some $x_1\in K_0^{str}\left(\partial_1Q\right)$, $str\left(e_2^\prime\right)$ 
%is a loop based at some $x_2\in K_0^{str}\left(\partial_1 Q\right)$.
%Hence $\partial_1str\left(f_1^\prime\right)=\partial_1str\left(f_2^\prime\right)=x_2$
%and $\partial_0str\left(f_2^\prime\right)=\partial_0str\left(f_1^\prime\right)=x_1$. 

After these modifications we have that $str\left(e_1^\prime\right), str\left(f_1^\prime\right),str\left(e_2^\prime\right),
str\left(f_2^\prime\right)$ bound a square with vertices in $\partial_1Q$. 
%Moreover, both vertices of $str
%\left(e_1^\prime\right)$ belong to the same path-component 
%of $\partial_0Q$ and both vertices of 
%$str\left(e_2^\prime\right)$
%belong to the same path-component of $\partial_0Q$.

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We note that $f_1^\prime$ and $f_2^\prime$
are edges with label 1. By condition (i) of Corollary 3,
this implies that $str\left(f_1^\prime\right)$ and $str\left(f_2^\prime\right)$ are distinguished
1-simplices.

Moreover, $str\left(f_1^\prime\right)$ and 
$str\left(f_2^\prime\right)$ have their respective endpoints in the same path-components 
of $\partial Q$.
By Definition 9, this implies that {\bf $str\left(f_1^\prime\right)$
can be homotoped into $str\left(f_2^\prime\right)$ by a homotopy keeping
endpoints in $\partial Q$}. 
%(Resp., if both vertices are not contained in $\partial_0Q$, then $str\left(f_1^\prime\right)=str\left(f_2^\prime\right)$; resp., if only one vertex is contained in $\partial_0Q$, then the other is fixed during the homotopy.) 
This homotopy can be followed by a homotopy of $$str\left(U_1^\prime\right)\cup str\left(U_2^\prime\right)$$ 
{\em into straight simplices}.
We denote by $$str^\prime\left(e_1^\prime\right),str^\prime\left(e_2^\prime\right), str^\prime\left(f_1^\prime\right)=str^\prime\left(f_2^\prime\right), str^\prime\left(U_1^\prime\right), str^\prime\left(U_2^\prime\right)\in K_*^{str}\left(Q\right)$$ 
the images of 
$str\left(e_1^\prime\right),str\left(e_2^\prime\right), str
\left(f_1^\prime\right),str\left(f_2^\prime\right),str\left(U_1^\prime\right), str\left(U_2^\prime\right)$ after this homotopy.

%must
%{\bf agree (independently what the original homotopy class of $f_1,f_2$ was)}, i.e.
$$str^\prime\left(f_1^\prime\right)=str^\prime\left(f_2^\prime\right)$$
means that, after the homotopy, 
the square $str\left(U_1^\prime\right)+str\left(U_2^\prime\right)$
is mapped to 
a {\bf cylinder} $str^\prime\left(U_1^\prime\right)+str^\prime\left(U_2^\prime\right)$. 
(This is why we have performed the straightening construction 
in section 5 such that up certain homotopies
there should be only one distinguished 1-simplex 
for any given pair of connected components.)\\
\\
The assumption $rmv\left(str\left(\kappa_{ij_1}\right)\otimes 1\right)\not=0$ implies that
the loops $str^\prime\left(e_1^\prime\right)$ and $str^\prime\left(e_2^\prime\right)$ are not 0-homotopic.\\
Indeed,
if one of them, say $str^\prime\left(e_1^\prime\right)$, were a 0-homotopic (thus constant) loop, then also $str^\prime\left(e_2^\prime\right)$ would be a
0-homotopic (thus constant) loop, because $str^\prime\left(f_1^\prime\right)=str^\prime\left(
f_2^\prime\right)$ and $str^\prime\left(e_1^\prime\right)*str^\prime\left(f_1^\prime\right)*str^\prime
\left(e_2^\prime\right)^{-1}* str^\prime\left(f_2^\prime\right)^{-1}$ 
is 0-homotopic. \\
If $str\left(e_1^\prime\right)\not=str^\prime\left(e_1^\prime\right)$, then 
we have by the second-last condition of Definition 9 that the homotopy from 
$str\left(e_1^\prime\right)$ to $str^\prime\left(e_1^\prime\right)$ moves the vertex along a path not homotopic (rel.\ boundary) into $\partial_1Q$. But
it follows from Lemma 5c) that $str\left(e_1^\prime\right)$ is contained in
$\partial_1Q$. Hence $str^\prime\left(e_1^\prime\right)$ can not be 0-homotopic in this case. Similarly for $e_2^\prime$.\\
If $str\left(e_1^\prime\right)=str^\prime\left(e_1^\prime\right)$ and 
$str\left(e_2^\prime\right)=str^\prime\left(e_2^\prime\right)$, we 
observe that at least 
one of $str\left(e_1^\prime\right), str\left(e_2^\prime\right)$ is an edge of $str\left(\kappa_{ij_1}\right)$. 
%Moreover, the homotopy between $str\left(e_1^\prime\right)$ and 
%$str^\prime\left(e_1^\prime\right)$ keeps endpoints in $\partial_0Q$, thus is realised by some $g_1\in G$. The same way, 
%the homotopy between $str\left(e_2^\prime\right)$ and
%$str^\prime\left(e_2^\prime\right)$ is realised by some $g_2\in G$.
In particular, either $\kappa_{ij_1}$ or $\kappa_{ij_2}$ would have
one of the constant loops $str\left(e_1^\prime\right)$ or $str\left(e_2^\prime\right)$ as an edge. 
By Lemma 14 and Definition 7, this would contradict the assumption
$rmv\left(str\left(\kappa_{ij_1}\right)\otimes 1\right)\not=0$.

Because $str^\prime\left(e_1^\prime\right)$ and $str^\prime\left(e_2^\prime\right)$ 
are not 0-homotopic,
the cylinder $$str^\prime\left(U_1^\prime\right)+str^\prime\left(U_2^\prime\right)$$ is $\pi_1$-injective as a 
map of pairs. Since $\left(Q,\partial_1Q\right)$ is a 
pared acylindrical manifold, the cylinder must then 
be homotopic into $\partial Q$, as a map of
pairs 
$$\left({\Bbb S}^1\times \left[0,1\right],{\Bbb S}^1\times\left\{0,1\right\}\right)                              \rightarrow \left(Q,\partial_1Q\right).$$
In particular, $str^\prime\left(f_1^\prime\right)$ is homotopic into $\partial Q$, by a homotopy keeping its vertices in $\partial_1Q$. 

Thus $$x_1=\partial_0 str^\prime\left(f_1^\prime\right),x_2=\partial_1 str^\prime\left(f_1^\prime\right)$$
belong to the same connected component $C$ of $\partial Q$. We know that $x_1,x_2$ belong to $\partial_1Q$. This implies
$$str^\prime\left(f_1^\prime\right)\in K_1^{str}\left(\partial_1Q\right).$$
Indeed, if not, then we had $str^\prime\left(f_1^\prime\right)\in K_1^{str}\left(\partial_1Q\right)$, hence
$x_1,x_2\in\partial_0Q\cap \partial_1Q$, and then Lemma 5c) implies that
$x_1,x_2$ belong to distinct path-components of $\partial_0Q$. (For the other two alternatives of Lemma 5c) $str\left(f^\prime\right)$ would automatically be in the $G$-orbit of an edge in $\partial_1Q$, because both vertices of $f$ belonged to $\partial_1N$.)
%$C\not
%\subset\partial_0Q$. 
%Thus condition (i) from Lemma 11 implies that $x_1=x_2$. Hence
%$str\left(f_1^\prime\right)$ is a distinguished 
%1-simplex with $\partial_0str\left(f_1^\prime\right)=\partial_1str\left(f_1^\prime\right)$. By Definition 9 this implies that $str\left(f_1^\prime\right)$ is a constant loop.\\
Hence $str^\prime\left(f_1^\prime\right)\in K_1^{str}\left(\partial_1Q\right)$.\\
\\
Let $P_1,P_2$ be the affine planes whose intersections with $\Delta^n$ contain $T_1$ resp.\ $T_2$. Let $W$ be the white component whose boundary contains the 
white-parallel arcs of $T_1,T_2$.
We have seen that there is are arcs $f_1,f_2$ connecting
$P_1\cap\Delta^n$ to $P_2\cap\Delta^n$ such that $$str^\prime\left(f_1^\prime\right)=str^\prime\left(f_2^\prime\right)$$ is contained in $\partial_1Q$.
This
implies that for each other arc $f$ connecting
$P_1\cap\Delta^n$ to $P_2\cap\Delta^n$ the straightening $str\left(f^\prime\right)$
must be (homotopic into and therefore) contained in $\partial_1 Q$. 
%Since $\partial_i str\left(f^\prime\right)$ belongs to the same component of $\partial Q$ as $\partial_istr\left(f_1^\prime\right)$, for $i=1,2$,
%the same argument as for $str\left(f_1^\prime\right)$ implies that $str\left(f^\prime\right)$ must be constant.

If $P_1$ and $P_2$
are of the same type (Definition 2), then this shows that for all arcs $f\subset W$:
$$str\left(f^\prime\right)\in K_1^{str}\left(\partial_1Q\right)$$

If $P_1$ and $P_2$ are not of the same type, then the existence of a parallel arc
implies that at least one 
of them, say $P_1$, must be of type $\left\{0a_1\ldots
a_k\right\}$ with $k\not\in\left\{0,n-1\right\}$. Then, for each plane $P_3\not=P_1$ with $P_3\cap\Delta^n\subset \partial W$, it follows
from Corollary 2 that $P_3\cap\Delta^n$ has a white-parallel arc with $P_1\cap\Delta^n$.
Thus, repeating the
argument with $P_1$ and $P_3$ in place of $P_1$ and $P_2$, we prove that there are arcs in $\partial_1Q$
connecting
$P_1\cap\Delta^n$ to $P_3\cap\Delta^n$, and consequently
for each arc
$f\subset W$ connecting
$P_1\cap\Delta^n$ to $P_3\cap\Delta^n$, the straightening $str\left(f^\prime\right)$
must be (homotopic into and therefore) contained in $\partial_1 Q$.

Consequently, also  for arcs 
connecting
$P_2\cap\Delta^n$ to $P_3\cap\Delta^n$, we have that
$str\left(f^\prime\right)$
must be (homotopic into and therefore) contained in $\partial_1 Q$. This finally shows that the 
1-skeleta of 
$str\left(\kappa_{ij_1}\right)$ and $str\left(\kappa_{ij_2}\right)$ belong to $K_1^{str}\left(\partial_1 Q\right)$.
By $\pi_1$-injectivity of $\partial_1Q\rightarrow Q$, asphericity of $\partial_1Q$, and condition vii) from Lemma 11, this implies
that the
2-skeleta of
$str\left(\kappa_{ij_1}\right)$ and $str\left(\kappa_{ij_2}\right)$ belong to $K_1^{str}\left(\partial_1 Q\right)
$. Inductively, if the $k$-skeleta of
$str\left(\kappa_{ij_1}\right)$ and $str\left(\kappa_{ij_2}\right)$ belong to 
$K_k^{str}\left(\partial_1 Q\right)
$, then by asphericity of $Q$, asphericity of $\partial_1Q$, and 
condition vii) from Lemma 11 we obtain
that the $k+1$-skeleta of
$str\left(\kappa_{ij_1}\right)$ and $str\left(\kappa_{ij_2}\right)$ belong to 
$K_{k+1}^{str}\left(\partial_1 Q\right)
$.

By Definition 7, Definition 11 and Lemma 14 this implies $$rmv\left(
str\left(\kappa_{ij_1}\right)\otimes 1\right)=0,
rmv\left(
str\left(\kappa_{ij_2}\right)\otimes 1\right)=0.$$

So we have shown the subclaim: if $T_1=\partial_{k_1}\theta_{ij_1}, T_2=\partial_{k_2}\theta_{ij_2}$
have a white-parallel arc, then $rmv\left(
str\left(\kappa_{ij_1}\right)\otimes 1\right)=0,
rmv\left(
str\left(\kappa_{ij_2}\right)\otimes 1\right)=0$. In particular, $$str\left(T_1^\prime\right),
str\left(T_2^\prime\right)$$ do not occur (with non-zero coefficient)
in $$\partial \sum_{j\in J_i} rmv\left(str\left(\kappa_{ij}\right)\otimes 1
\right).$$
\\
By Lemma 10, for a canonical colouring associated to a
set of affine planes $P_1,P_2,\ldots$, and a fixed triangulation of each $Q_i=P_i\cap\Delta^n$,
we have at most $n+1$ n-1-simplices whose 1-skeleton does not contain a white-parallel arc.
Therefore the {\em subclaim} implies the {\em claim}. \\
Thus we have presented $\left[\partial Q\right]\otimes 1 $ as a finite chain of
$l^1$-norm $\le \left(n+1\right)\sum_{i=1}^r\mid a_i\mid$. By Lemma 4a) we know that $G=\Pi\left(K\left(\partial_{0}Q\right)\right)$ is amenable. Thus Lemma 7, applied to $\widehat{K}$ with its $G$-action, implies
$$\parallel \partial Q\parallel\le \left(n+1\right)\sum_{i=1}^r\mid a_i\mid.$$
\hfill$\Box$
\\
\\
We remark that Theorem 1 is not true without assuming amenability of $\pi_1\partial_{0}Q$.
Counterexamples can be found, for example,
using \cite{jun} or \cite{k1}, Theorem 6.3.


\section{Specialization to 3-manifolds}

{\bf Guts of essential laminations.} We start with recalling the guts-terminology.
 Let $M$ be a compact 3-manifold with boundary and $\F$ an essential lamination
transverse or tangential to the boundary. $N=\ol$ is a, possibly noncompact,
irreducible 3-manifold with boundary $\partial N$. We denote $\partial_0N=\partial
N\cap\partial M$ and $\partial_1N=\overline{\partial N - \partial_0N}$.
(Thus $\partial_1N$ is the union of boundary leaves of the lamination.)
By the proof of \cite{gk}, Lemma 1.3., the noncompact ends of $N$ are essential
$I$-bundles
over noncompact subsurfaces of $\partial_1N$. After cutting off
each of these ends along an essential, properly fibered annulus, one obtains a compact 3-manifold to which one can 
apply the JSJ-decomposition of \cite{js}, \cite{joh}. 
Hence we have a decomposition of $N$ into the characteristic submanifold 
$Char\left(N\right)$ (which consists of $I$-bundles 
and Seifert fibered 
solid tori, where the fibrations have to respect boundary patterns 
as defined in \cite{joh}, p.83) and the guts of $N$, $Guts\left(N\right)$. The $I$-fibered ends of $N$ will be added to the characteristic submanifold, which thus may become noncompact.
(We mention that there are
different notions of guts in the literature. Our notion is compatible with
\cite{ag}, \cite{ag2}, but differs from the definition in \cite{gk} or
\cite{cd} by taking the Seifert fibered solid tori into the 
characteristic submanifold and not into the guts. Thus, solid torus guts in the
paper of Calegari-Dunfield is the same as empty guts in our setting.)

For $Q=Guts\left(N\right)$ we denote $\partial_1 Q=\partial_1N\cap\partial Q$ and $\partial_0Q=\overline{\partial Q-\partial_1 Q}$.
For $R=Char\left(N\right)$ we denote $\partial_1 R=\partial_1N\cap\partial R$ 
and $\partial_0R=\overline{\partial R-\partial_1 R}$.
The guts of $N$ has the following properties: the pair $\left(Q,\partial_1Q\right)$ is a pared acylindrical manifold as defined in Definition 3, and the inclusions $\partial_0Q\rightarrow Q, \partial_1Q\rightarrow Q, Q\rightarrow N$,
$\partial_0R\rightarrow R,\partial_1R\rightarrow R,R\rightarrow N$
are $\pi_1$-injective (see \cite{js},\cite{joh}).
It follows from Thurston's hyperbolization theorem for Haken manifolds that $Q$ admits a hyperbolic metric with geodesic boundary $\partial_1Q$ and 
(if $\partial_0Q$ consists of tori and annuli) cusps 
corresponding to $\partial_0Q$.


\begin{thm}: Let $M$ be a compact 3-manifold with incompressible boundary and $\mathcal{F}$ an essential lamination of
$M$. Then $$\parallel
M,\partial M\parallel^{norm}_{\mathcal{F}}\ge-\chi\left(Guts\left({\mathcal{F}}\right)
\right).$$
%, \parallel M\parallel_{\mathcal{F}} \ge -2\chi\left(Guts\left({\mathcal{F}}\right)\right).$$
More generally, if $P$ is a polyhedron with $f$ faces, then
 $$ \parallel
M,\partial M\parallel^{norm}_{{\mathcal{F}},P}\ge-\frac{2}{f-2}\chi\left(Guts\left({\mathcal
{F}}\right)\right).$$\end{thm}

\begin{pf} 

%For an essential lamination $\mathcal{F}$, it follows from \cite{goe}, Theorem 6.1., that 
%the complement 
Let $N=\overline{M-{\mathcal{F}}}$. Since $\mathcal{F}$ is essential, $N$ is irreducible.
%From incompressibility of $\partial M$ and of boundary leaves one obtains that $\partial N\rightarrow N$ is $\pi_1$-injective. 
% is aspherical. Thus one can apply Theorem 1 to essential laminations of 
%3-manifolds.\\
Let $R=Char\left(N\right)$ be the characteristic submanifold and
$Q=Guts\left(N\right)$ be the complement of the characteristic submanifold of $N$. 
We denote $\partial_1 Q=\partial
N\cap\partial Q,
\partial_0Q=\overline{\partial Q-\partial_1 Q}$.
% and
%$\partial_1 R=\partial_1N\cap\partial R,
%\partial_0R=\overline{\partial R-\partial_1 R}$. 
It follows from the construction of the
characteristic submanifold (see \cite{js},\cite{joh}) that\\
- $Q$ is compact and aspherical,\\
- $\left(Q,\partial_1Q\right)$ is pared acylindrical, $\partial_1Q$ is aspherical, $\partial_1Q\rightarrow Q$ is $\pi_1$-injective, and\\
- $\partial_0Q$ consists of essential annuli and tori, in particular $\pi_1\partial_0Q$ is amenable and injects into $\pi_1 N$.

(Even stronger, Thurston has proven that $Q$ is a hyperbolic manifold with geodesic boundary $\partial_1Q$ and cusps corresponding to $\partial_0Q$. In particular, $\chi\left(\partial Q\right)\le 0$, thus $\partial Q$ is aspherical.)
Moreover, $\pi_1$-injectivity of $Q\cap R=\partial_0Q\rightarrow N$ implies that the maps
$\partial_0Q\rightarrow Q\rightarrow N$ and $\partial_0Q\rightarrow R\rightarrow N$ are each $\pi_1$-injective for each connected component. (This is in both cases immediate for the first inclusion, and it must be true for the inclusions $Q\rightarrow N$ resp.\ $R\rightarrow N$ because a disk $D^2\rightarrow N$ with boundary in $Q$ resp.\ $R$, 
but not homotopic into $Q$ resp.\ $R$ would intersect $\partial_0Q$ in an essential circle.) 
$\pi_1$-injectivity of $\partial_0Q   \rightarrow Q$ resp.\ $\partial_0Q\rightarrow R$ implies
in particular $\pi_1$-injectivity of $\partial_0Q\rightarrow\partial Q$ resp.\ $\partial_0Q\rightarrow
\partial R$. If we denote $N^\prime=\partial N\cup\partial_0Q$, 
then we have that $N^\prime=\partial Q\cup\partial R$ and the Seifert-van Kampen 
theorem implies that $\partial Q\rightarrow N^\prime$ and $\partial R\rightarrow N^\prime$ are 
$\pi_1$-injective. Thus $N=Q\cup R$ is an essential decomposition as in Definition 13.

In conclusion, the decomposition $N=R\cup Q$ satisfies the assumptions of Theorem 1.

From the computation of the simplicial volume for surfaces
(\cite{gro}, section 0.2.) and
$\chi\left(Q\right)=\frac{1}{2}\chi\left(\partial Q\right)$ (which is a consequence of
Poincare duality for the closed 3-manifold $Q\cup_{\partial Q}Q$), it follows that
$$-\chi\left(Guts\left({\mathcal{F}}\right)\right)=
-\frac{1}{2}\chi\left(\partial Guts\left({\mathcal{F}}\right)\right)=\frac{1}{4}\parallel \partial
Guts\left({\mathcal{F}}\right)\parallel.$$
Thus, the first claim is obtained as application of Theorem 1 to
$Q=Guts\left({\mathcal{F}}\right)$. 

The second claim, that is the generalisation to arbitrary polyhedra, is
obtained as in \cite{ag}. 
Namely, one
uses the same straightening as above, and asks again how many nondegenerate 
2-simplices may, after straightening, occur in the intersection of 
$\partial Q$ with some polyhedron $P_i$. In \cite{ag}, p.\ 11, it is shown that this
number is at most $2f-4$, where $f$ is the number of faces of $P_i$. The same argument as above shows then $\sum_{i=1}^r\mid a_i\mid\ge\frac{1}{2f-4}\parallel
\partial Guts\left({\mathcal{F}}\right)\parallel$, giving the wanted inequality.
\end{pf}\\


\noindent The following corollary applies, for example, to all hyperbolic manifolds
obtained by Dehn-filling the complement of the figure-eight knot in ${\Bbb S}^3
$. (Note that Hatcher has proved in \cite{hat} that each hyperbolic manifold
obtained by Dehn-filling the complement of the figure-eight knot in ${\Bbb S}^3
$ carries essential laminations.)

\begin{cor}: If $M$ is a closed hyperbolic manifold with $Vol\left(M\right)< 2V
_3 =2.02...$, then $M$ carries no essential lamination $\mathcal{F}$
with $\parallel M\parallel^{norm}_{{\mathcal{F}},P}=\parallel M\parallel_P$ for
 all polyhedra P, and nonempty guts. In particular, there is no tight essential
lamination with nonempty guts.\end{cor}

\begin{pf} The derivation of corollary 1 from Theorem 2 is exactly
the same as in \cite{ag} for the usual (non-laminated)
Gromovnorm. Namely, by \cite{stt} (or \cite{ag}, end of section 6) there exists a sequence $P_n$ of straight polyhedra in ${\Bbb H}^3$ with $\lim_{n\rightarrow\infty} \frac{Vol\left(P_n\right)}{f_n-2}=V_3$, with $f_n$ denoting
the number of faces of $P_n$. Assuming
that $M$ carries a lamination $\mathcal{F}$ with $ \parallel M\parallel^{norm}_{{\mathcal{F}},P_n}=\parallel M\parallel_{P_n}$
for all $n$, one gets $$-\chi\left(Guts\left({\mathcal{F}}\right)\right)\le
\frac{f_n-2}{2}\parallel  M\parallel_{{\mathcal{F}},P_n}=
\frac{f_n-2}{2}\parallel  M\parallel_{P_n}
\le \frac{f_n-2}{2}\frac{Vol\left(M\right)}{Vol\left(P_n\right)} 
\rightarrow \frac{Vol\left(M\right)}{2V_3} < 1.$$
On the other hand, if $Guts\left({\mathcal{F}}\right)$ is not empty, then it is
a hyperbolic manifold with nonempty geodesic boundary, hence $$
\chi\left(Guts\left({\mathcal{F}}\right)\right)\le -1,$$
giving a contradiction.\end{pf}\\

\begin{df}: The Weeks manifold is the closed 3-manifold obtained by $\left(-\frac{5}{1},-\frac{5}{2}\right)$-surgery at the Whitehead link (\cite{ro}, p.68).\end{df}

It is known that the Weeks manifold is hyperbolic and that its hyperbolic volume is
approximately $0.94..$. (It is actually the hyperbolic 3-manifold of smallest volume.)

\begin{cor} (\cite{cd}, Conjecture 9.7.): The Weeks manifold admits no tight
lamination $\mathcal{F}$.\end{cor}
\begin{pf} According to \cite{cd}, the Weeks manifold can not carry
a tight lamination with empty guts. Since tight laminations satisfy
$\parallel M\parallel^{norm}_{{\mathcal{F}},P}=\parallel M\parallel$ for each 
polyhedron (see lemma 1 in section 2),
and since the
Weeks manifold has volume smaller than $2V_3$, it follows from corollary 1 that
it can not carry a tight lamination with nonempty guts neither.\end{pf}\\

The same argument shows that a hyperbolic 3-manifold $M$ with \\
- $Vol\left(M\right) < 2V_3$, and\\
- no injective homomorphism $\pi_1M\rightarrow Homeo^+\left({\Bbb S}^1\right)$\\
can not carry a tight lamination, because it was shown by Calegari-Dunfield
in \cite{cd} that the existence of a tight lamination with empty guts implies the existence of an injective homomorphism $\pi_1M\rightarrow Homeo^+\left({\Bbb S}^1\right)$. Some methods for excluding the existence 
of injective homomorphisms $\pi_1M\rightarrow Homeo^+\left({\Bbb S}^1\right)$
have been developed in \cite{cd} (which yielded in particular the nonexistence
of such homomorphisms for the Weeks manifold, used in the corollary above), but
in general it is still hard to apply this criterion to other hyperbolic 3-manifolds of volume $< 2V_3$.

As indicated in \cite{cal2}, an approach to
a generalization of some of the above arguments 
to essential, non-tight laminations, yielding possibly a proof for nonexistence of essential laminations on the Weeks manifold, could consist in trying to define a straightening of cycles (as in the proof of lemma 1) upon possibly changing the essential lamination. 

As a consequence of a recent paper of Tao Li, one can at least exclude the 
existence of transversely orientable essential laminations on the Weeks manifold.

\begin{cor}: The Weeks manifold admits no transversely orientable essential 
lamination $\mathcal{F}$.\end{cor}
\begin{pf} According to \cite{li}, Theorem 1.1, the following statement is true: if a closed, orientable, atoroidal 3-manifold $M$ contains a transversely orientable essential lamination, then it contains a transversely orientable tight essential lamination. Hence corollary 3 is a direct consequence of corollary 2.\end{pf}





\section{Higher dimensions}

We want to finish this paper with showing that Theorem 1 is interesting also in higher dimensions. Meanwhile, 
in dimension 3, the assumptions of Lemma 1 hold for each essential lamination, it is likely that this will not be the case for many laminations 
in higher dimensions. However, the most straightforward, but already interesting  
application of the inequality is the following Corollary 7 which
means that, for a given negatively curved manifold $M$, we can give an explicit bound on the topological complexity of geodesic hypersurfaces. Such a bound seems to be new except, of course, in the 3-dimensional case where it is due to Agol (\cite{ag}) and (with nonexplicit constants) to Hass (\cite{ha}).

\begin{cor}: Let $M$ be a compact Riemannian $n$-manifold of negative sectional curvature and finite volume. Let $F\subset M$ be a geodesic $n-1$-dimensional
hypersurface of finite volume. Then $\parallel
F\parallel\le \frac{n+1}{2}\parallel M\parallel$.\end{cor}

\begin{pf} Consider $N=\overline{M-F}$. $\left(N,\partial N\right)$
is acylindrical. This is well-known and
can be seen as follows: assume that $N$ contained an essential
cylinder, then the double $DN=N\cup_{\partial_1 N}N$ would contain an essential
2-torus.
% and, in particular, its fundamental group contained
%a non-peripheral subgroup isomorphic to ${\Bbb Z}^2$.
But, since $N$ is a negatively curved manifold with geodesic boundary,
we can glue the Riemannian metrics to get a complete negatively curved Riemannian metric
on $DN$. In particular, $DN$ contains no essential 2-torus, giving a contradiction.
%$\pi_1\left(DN\right)$ must be (relatively) word-hyperbolic (relative to $\pi_1\left(\partial DN\right)$)
%and can thus not contain any non-peripheral subgroup isomorphic to ${\Bbb Z}^2$.

Moreover, the geodesic boundary $\partial N$ is $\pi_1$-injective and negatively curved, thus aspherical. 
Therefore we can choose $Q=N$, in which case the other assumptions of Theorem 1
are
trivially satisfied. From Theorem 1 we conclude \\
$\parallel M\parallel_{F}^{norm}\ge\frac{1}{n+1}\parallel \partial N\parallel$. The boundary of $N$ consists of two copies of $F$, hence $\parallel \partial N\parallel=2\parallel F\parallel$. The leaf space of $\widetilde{F}\subset\widetilde{M}$ is a Hausdorff tree, thus Lemma 1b) implies
$\parallel M\parallel_{F}^{norm}=\parallel M\parallel$. The claim follows.\end{pf}\\
\\
This statement should be read as follows: for a given manifold $M$ (with given volume) one has an upper bound on the topological complexity of compact geodesic hypersurfaces.

We finish this section by a reformulation of Corollary 7 for hyperbolic manifolds.


\begin{cor}: Let $M$ be a closed hyperbolic n-manifold and $F$ a closed n-1-dimensional
geodesic hypersurface. Then $$Vol\left(M\right)\ge
C_n\chi\left(F\right)$$ for a constant $C_n$ depending only on $n$.\end{cor}

\begin{pf}
Assume $n$ is odd. (If $n$ is even, then $\chi\left(F\right)=0$ and the claim is trivially true.) Then, $F$ is an even-dimensional hyperbolic manifold and, by the Chern-Gau\ss -Bonnet-Theorem we have $Vol\left(F\right)=G_{n-1}\chi\left(F\right)$
for a constant $G_{n-1}$. Together with $Vol\left(M\right)=V_n\parallel M\parallel,
Vol\left(F\right)=V_{n-1}\parallel F\parallel$ and Corollary 7,
we conclude $Vol\left(M\right)\ge \frac{2}{n+1}\frac{V_n}{V_{n-1}}G_{n-1}\chi\left(F\right)$.\end{pf}





\begin{thebibliography}{29}
\bibitem{ag}I.\ Agol, {\em Lower bounds on volumes of hyperbolic Haken 3-manifolds}, Preprint.  http://front.math.ucdavis.edu/math.GT/9906182.
\bibitem{ag2}I.\ Agol, P.\ Storm, W.\ Thurston, with an appendix by N.\ Dunfield, {\em Lower bounds on volumes of 
hyperbolic Haken 3-manifolds}, J.\ Amer.\ Math.\ Soc.\ {\bf 20}, pp.1053-1077 (2007).
\bibitem{bp}R.\ Benedetti, C.\ Petronio, {\em Lectures on
Hyperbolic Geometry}. Universitext, Springer-Verlag, Berlin (1992).
\bibitem{bri}M.\ Brittenham, {\em Essential laminations and Haken normal form}, 
Pac.\ J.\ Math.\ {\bf 168}, 217-234 (1995).
\bibitem{cal}D.\ Calegari, {\em The Gromov norm and foliations}.
GAFA 11, pp.1423-1447 (2001).
\bibitem{cal2}D.\ Calegari, {\em Problems in foliations and laminations on 3-manifolds}.
 Proc.\ Sympos.\ Pure Math.\ {\bf 71}, 297-335 (2003).
\bibitem{cd}D.\ Calegari, N.\ Dunfield, {\em Laminations and groups of homeomorphisms
of the circle}, Invent.\ Math.\ {\bf 152}, 149-207 (2003).
\bibitem{cs}M.\ Culler, P.\ Shalen, {\em Volumes of hyperbolic Haken manifolds}, 
Invent.\ Math.\ {\bf 118}, pp.285-329 (1994).
\bibitem{fe}S.\ Fenley, {\em Laminarfree hyperbolic manifolds}, Comm.\ Math.\ Helv.\ {\bf 82}, pp.247-321 (2007).
\bibitem{fr}S.\ Francaviglia, {\em Hyperbolic volume of representations of 
fundamental groups of cusped 3-manifolds}, Int.\ Math.\ Res.\ Not.\ {\bf 9}, 425-459 (2004).
\bibitem{gab}D.\ Gabai, {\em Essential laminations and Kneser normal form}, J.\ 
Diff.\ Geom.\ {\bf 53}, 517-574 (1999).
\bibitem{gk}D.\ Gabai, W.\ Kazez, {\em Group negative curvature for 3-manifolds with genuine laminations}, Geom.\ Topol.\ {\bf 2}, 65-77 (1998).
\bibitem{goe}D.\ Gabai, U.\ Oertel, {\em Essential laminations in 3-manifolds}, 
Ann.\ of Math.\ {\bf 130}, 41-73 (1989).
\bibitem{gro}M.\ Gromov, {\em Volume and Bounded
Cohomology}, Public.\ Math.\ IHES {\bf 56}, 5-100 (1982).
\bibitem{ha}J.\ Hass, {\em Acylindrical surfaces in 3-manifolds}, Michigan Math.\ J.\ 42, pp.357-365 (1995).
\bibitem{hat}A.\ Hatcher, {\em Some examples of essential laminations in 
3-manifolds}, Ann.\ Inst.\ Fourier {\bf 42}, 313-325 (1992).
\bibitem{iva}N.\ Ivanov, {\em Foundations of the theory of
bounded cohomology}, J.\ Sov.\ Math.\ {\bf 37}, 1090-1114 (1987).
\bibitem{js}W.\ Jaco, P.\ Shalen, {\em Seifert fibered spaces in 3-manifolds}, 
Mem.\ Amer.\ Math.\ Soc., {\bf 21}, (1979).
\bibitem{joh}K.\ Johannson, {\em Homotopy Equivalences of 3-Manifolds with Boundaries},
Lecture Notes in Mathematics 761, Springer Verlag, (1979).
\bibitem{jun}D.\ Jungreis, {\em Chains that realize the Gromov
invariant of hyperbolic manifolds},  Ergodic Theory and Dynamical Systems {\bf 17},
643-648 (1997).
\bibitem{k1}T.\ Kuessner, {\em Efficient fundamental cycles for
cusped hyperbolic manifolds}, Pac.\ J.\ Math.\ {\bf 211}, 283-314 (2003).
\bibitem{k2}T.\ Kuessner, {\em Multicomplexes, 
bounded cohomology and additivity 
of simplicial volume}, Preprint, http://www.math.uni-siegen.de/~kuessner/preprints/bc.pdf.
\bibitem{k3}T.\ Kuessner, {\em Foliated norms on fundamental group and homology}, Top.\ Appl.\ {\bf 144}, 229-254 (2004).
\bibitem{li}T.\ Li, {\em Compression branched surfaces and tight essential laminations}, Preprint.
\bibitem{rs}R.\ Roberts, M.\ Stein, {\em Group actions on order trees}, Top.\ Appl.\ {\bf 115}, 175-201 (2001).
\bibitem{rss}R.\ Roberts, J.\ Shareshian, M.\ Stein, {\em Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation}, J.\ Amer.\ Math.\ Soc.\ {\bf 16}, 639-679 (2003).
\bibitem{ro}D.\ Rolfsen, {\em Knots and Links}, AMS Chelsea Publishing (2003).
\bibitem{stt}D.\ Sleator, R.\ Tarjan, W.\ Thurston, {\em Rotation distance, triangulations, and hyperbolic geometry}, J.\ Amer.\ Math.\ Soc.\ {\bf 1}, 647-681 (1988).
\bibitem{thu}W.\ Thurston, {\em The Geometry and Topology of 3-Manifolds
}. Lecture Notes, Princeton.
\bibitem{thua}W.\ Thurston, {\em Hyperbolic geometry and 3-manifolds}.
Low-dimensional topology, Proc.Conf.Bangor, Lond.Math.Soc.Lect.NoteSer.48, pp.9-
25 (1982).
\end{thebibliography}


Thilo Kuessner\\
Mathematisches Institut, Universit\"at M\"unster\\
Einsteinstra\ss e 62\\
D-48149 M\"unster\\
Germany\\
e-mail: kuessner@math.uni-muenster.de


\end{document}

math0506528.extracted_bib
\begin{thebibliography}{29}
\bibitem{ag}I.\ Agol, {\em Lower bounds on volumes of hyperbolic Haken 3-manifolds}, Preprint.  http://front.math.ucdavis.edu/math.GT/9906182.
\bibitem{ag2}I.\ Agol, P.\ Storm, W.\ Thurston, with an appendix by N.\ Dunfield, {\em Lower bounds on volumes of 
hyperbolic Haken 3-manifolds}, J.\ Amer.\ Math.\ Soc.\ {\bf 20}, pp.1053-1077 (2007).
\bibitem{bp}R.\ Benedetti, C.\ Petronio, {\em Lectures on
Hyperbolic Geometry}. Universitext, Springer-Verlag, Berlin (1992).
\bibitem{bri}M.\ Brittenham, {\em Essential laminations and Haken normal form}, 
Pac.\ J.\ Math.\ {\bf 168}, 217-234 (1995).
\bibitem{cal}D.\ Calegari, {\em The Gromov norm and foliations}.
GAFA 11, pp.1423-1447 (2001).
\bibitem{cal2}D.\ Calegari, {\em Problems in foliations and laminations on 3-manifolds}.
 Proc.\ Sympos.\ Pure Math.\ {\bf 71}, 297-335 (2003).
\bibitem{cd}D.\ Calegari, N.\ Dunfield, {\em Laminations and groups of homeomorphisms
of the circle}, Invent.\ Math.\ {\bf 152}, 149-207 (2003).
\bibitem{cs}M.\ Culler, P.\ Shalen, {\em Volumes of hyperbolic Haken manifolds}, 
Invent.\ Math.\ {\bf 118}, pp.285-329 (1994).
\bibitem{fe}S.\ Fenley, {\em Laminarfree hyperbolic manifolds}, Comm.\ Math.\ Helv.\ {\bf 82}, pp.247-321 (2007).
\bibitem{fr}S.\ Francaviglia, {\em Hyperbolic volume of representations of 
fundamental groups of cusped 3-manifolds}, Int.\ Math.\ Res.\ Not.\ {\bf 9}, 425-459 (2004).
\bibitem{gab}D.\ Gabai, {\em Essential laminations and Kneser normal form}, J.\ 
Diff.\ Geom.\ {\bf 53}, 517-574 (1999).
\bibitem{gk}D.\ Gabai, W.\ Kazez, {\em Group negative curvature for 3-manifolds with genuine laminations}, Geom.\ Topol.\ {\bf 2}, 65-77 (1998).
\bibitem{goe}D.\ Gabai, U.\ Oertel, {\em Essential laminations in 3-manifolds}, 
Ann.\ of Math.\ {\bf 130}, 41-73 (1989).
\bibitem{gro}M.\ Gromov, {\em Volume and Bounded
Cohomology}, Public.\ Math.\ IHES {\bf 56}, 5-100 (1982).
\bibitem{ha}J.\ Hass, {\em Acylindrical surfaces in 3-manifolds}, Michigan Math.\ J.\ 42, pp.357-365 (1995).
\bibitem{hat}A.\ Hatcher, {\em Some examples of essential laminations in 
3-manifolds}, Ann.\ Inst.\ Fourier {\bf 42}, 313-325 (1992).
\bibitem{iva}N.\ Ivanov, {\em Foundations of the theory of
bounded cohomology}, J.\ Sov.\ Math.\ {\bf 37}, 1090-1114 (1987).
\bibitem{js}W.\ Jaco, P.\ Shalen, {\em Seifert fibered spaces in 3-manifolds}, 
Mem.\ Amer.\ Math.\ Soc., {\bf 21}, (1979).
\bibitem{joh}K.\ Johannson, {\em Homotopy Equivalences of 3-Manifolds with Boundaries},
Lecture Notes in Mathematics 761, Springer Verlag, (1979).
\bibitem{jun}D.\ Jungreis, {\em Chains that realize the Gromov
invariant of hyperbolic manifolds},  Ergodic Theory and Dynamical Systems {\bf 17},
643-648 (1997).
\bibitem{k1}T.\ Kuessner, {\em Efficient fundamental cycles for
cusped hyperbolic manifolds}, Pac.\ J.\ Math.\ {\bf 211}, 283-314 (2003).
\bibitem{k2}T.\ Kuessner, {\em Multicomplexes, 
bounded cohomology and additivity 
of simplicial volume}, Preprint, http://www.math.uni-siegen.de/~kuessner/preprints/bc.pdf.
\bibitem{k3}T.\ Kuessner, {\em Foliated norms on fundamental group and homology}, Top.\ Appl.\ {\bf 144}, 229-254 (2004).
\bibitem{li}T.\ Li, {\em Compression branched surfaces and tight essential laminations}, Preprint.
\bibitem{rs}R.\ Roberts, M.\ Stein, {\em Group actions on order trees}, Top.\ Appl.\ {\bf 115}, 175-201 (2001).
\bibitem{rss}R.\ Roberts, J.\ Shareshian, M.\ Stein, {\em Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation}, J.\ Amer.\ Math.\ Soc.\ {\bf 16}, 639-679 (2003).
\bibitem{ro}D.\ Rolfsen, {\em Knots and Links}, AMS Chelsea Publishing (2003).
\bibitem{stt}D.\ Sleator, R.\ Tarjan, W.\ Thurston, {\em Rotation distance, triangulations, and hyperbolic geometry}, J.\ Amer.\ Math.\ Soc.\ {\bf 1}, 647-681 (1988).
\bibitem{thu}W.\ Thurston, {\em The Geometry and Topology of 3-Manifolds
}. Lecture Notes, Princeton.
\bibitem{thua}W.\ Thurston, {\em Hyperbolic geometry and 3-manifolds}.
Low-dimensional topology, Proc.Conf.Bangor, Lond.Math.Soc.Lect.NoteSer.48, pp.9-
25 (1982).
\end{thebibliography}