Multicomplexes, bounded cohomology and \\ additivity of simplicial volume
Thilo Kuessner
We discuss some additivity properties of the simplicial volume for manifolds with boundary: we give proofs of additivity for glueing amenable boundary components and of superadditivity for glueing amenable submanifolds of the boundary, and we discuss doubling of 3-manifolds.\footnotetext{2000 Mathematics Subject Classification 57N65}
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\begin{document}
 
\title{Multicomplexes, bounded cohomology and \\
additivity of simplicial volume }
\author{Thilo Kuessner}
\date{}
\maketitle
\begin{abstract}
We discuss some additivity properties of the simplicial volume for manifolds with boundary: we give
proofs of
additivity for glueing amenable boundary components and of superadditivity for glueing
amenable submanifolds of the boundary,
and we discuss doubling of 3-manifolds.\footnotetext{2000 Mathematics Subject Classification 57N65}
\end{abstract}
 
\noindent                                                

The simplicial volume is a homotopy invariant of manifolds. It was defined in \cite{gro}, using the $l^1$-norm on singular chains, as follows. \\
Let $\left(X,Y\right)$ be a pair of spaces. For a relative homology class $h\in
H_*\left(X,Y;{\Bbb R}\right)$ define its Gromov norm $\parallel h\parallel:=
inf\left\{\sum_{i=1}^r\mid a_i\mid :
\sum_{i=1}^r a_i\sigma_i\mbox { represents }h\right\}$.\\
To get an invariant of a compact, connected, orientable n-manifold $M$, consider its fundamental class $\left[M,\partial M\right]$, that is the image of a generator of $H_n\left(M,\partial M;{\Bbb Z}\right)$ under the canonical 
homomorphism $H_n\left(M,\partial M;{\Bbb Z}\right)\rightarrow H_n\left(M,\partial M;{\Bbb R}\right)$, and 
define the simplicial volume of $M$, to be denoted $\parallel M,\partial M\parallel$, as the Gromov norm
of $\left[M,\partial M\right]$.
Extend this definition to non-connected manifolds by summing the simplicial volumina 
of the connected
components.
 
The simplicial volume remains an invariant about which relatively little is known. This article is devoted to the study of the behaviour of simplicial volume with
respect to cut and paste.
That means, we are given an (n-1)-submanifold $F\subset M$ with $\partial F\subset\partial M$, and we wish to compare $\parallel M_F,\partial M_F\parallel$
to $\parallel M,\partial M\parallel$.
Here, $M_F$ denotes the manifold obtained by cutting off $F$, that is $M_F:=\overline{M - N\left(F\right)}$ for a regular neighborhood $N\left(F\right)$ of $F$.
 
$\parallel M,\partial M\parallel$ may be strictly smaller than 
$\parallel M_F,\partial M_F\parallel$,
as there may be fundamental cycles of $M$ which are not the images of fundamental cycles of $M_F$. For example, we showed in \cite{k} that $\parallel M_F,\partial M_F\parallel > \parallel M,\partial M\parallel$
if $int\left(M\right)$ is a hyperbolic n-manifold of finite volume, $n\ge3$, 
and $F$ is a closed geodesic hypersurface.                      
On the other hand, somewhat counter-intuitively, $\parallel M,\partial M\parallel$ may be strictly larger
than $\parallel M_F,\partial M_F\parallel$, as fundamental cycles for $M_F$ need not fit
together at the copies of $F$.

For theorem 1, we consider the case that $F$ is amenable, and prove:\\
{\bf Theorem 1:} {\em Let $M_1,M_2$ be two compact n-manifolds,
$A_1\subset \partial M_1$ resp.$A_2\subset\partial M_2$ (n-1)-dimensional submanifolds,
$f:A_1\rightarrow A_2$ a homeomorphism,
$M=\glu$ the glued manifold.
If $\pi_1A_1$, $\pi_1A_2$ are amenable and $f_*$ restricts to an isomorphism
$f_*:ker\left(\pi_1A_1\rightarrow\pi_1M_1\right)\rightarrow
ker\left(\pi_1A_2\rightarrow\pi_1M_2\right)$, \\
then $\parallel M,\partial M\parallel\ge\sima+\simb$.\\
If moreover $A_1,A_2$ are connected components of $\partial M_1$ resp.\ $\partial
M_2$,\\
then $\parallel M,\partial M\parallel=\sima+\simb$.}\\
We prove analogous facts if $A_1,A_2$ are in the boundary of the same manifold $M_1$.\\        
Applied to 3-manifolds, theorem 1 means that simplicial volume is additive with respect to glueing along incompressible tori and superadditive with respect to glueing along incompressible annuli. In the special case that the boundary of the 3-manifolds consists of tori,  Soma proved in
\cite{som1} that simplicial volume is
additive with respect to glueing incompressible tori or annuli. We think that our proof, apart from being a generalisation to manifolds with arbitrary boundary,
should be of interest because the proof in \cite{som1} heavily relies on theorem 6.5.5. from Thurston's lecture notes, of which no published proof is available so far.\\

In chapter 4, we consider the special case of doubling a manifold, that is of glueing two (differently oriented) copies of $M$ by the identity of $\partial M$.
Here, two fundamental cycles of $M$, corresponding to opposite orientations, fit together at $\partial M$ to give a fundamental cycle of $DM$, hence $\parallel DM\parallel\le 2\parallel M,\partial M\parallel$ trivially holds. We give precise conditions for 3-manifolds to satisfy the strict inequality.
{\bf Theorem 2:} {\em Let $M$ be a manifold of dimension $n\le3$. Then\\
$\parallel DM\parallel
< 2\parallel M,\partial M\parallel$ if and only if $\parallel \partial M\parallel >0$.}\\
Theorem 2 will, using geometrization of 3-manifolds, follow from our result in \cite{k} together
with application of theorem 1 to 3-manifolds with boundary. \\
 
The article is organized as follows. Chapter 1 gives the necessary facts about multicomplexes. We usually refer to
\cite{gro} where it contains complete proofs and just fix notations in a way useful for later chapters. Chapter 2 discusses treelike multicomplexes. The proved results are the same which are needed in \cite{gro} to prove results about glueing manifolds without boundary, our contribution consisting in writing complete proofs for the ideas indicated in section 3.5. of \cite{gro}.
Theorem 1 is proved in chapter 3, and theorem 2 is proved in chapter 4.\\

It might be helpful for the reader that we give some non-rigorous motivation for the proof of theorem 1.
Let us consider a toy example. We glue two manifolds $M_1$ and $M_2$ to get 
$M_1\vee M_2$. (This is not a manifold but one may define a fundamental class in the obvious way.) We want to show $\parallel M_1\vee M_2\parallel\ge\parallel M_1\parallel +\parallel
M_2\parallel$, thus we have to find an efficient way to map representatives of $\left[M_1\vee M_2\right]$
to a sum of representatives of $\left[M_1\right]$ and of $\left[M_2\right]$.
That means,
we look
for a
chain map $r$, leftinverse to the inclusions, which maps
simplices in $M_1\vee M_2$
to simplices either in $M_1$ or in $M_2$.
 
The universal cover of $M_1\vee M_2$ is a tree-like complex made from copies of $\tilde{M_1}$ and $\tilde{M_2}$. In a tree, any nondegenerate
tripel of vertices has a unique central point, belonging to all three geodesics between the vertices.
More generally, in a tree-like complex, one might try to construct a central simplex associated to any nondegenerate tupel of at least three points.
For example, if $M_1$ and $M_2$ admitted hyperbolic metrics, one would have unique geodesics between two vertices in $\tilde{M_1}$ or $\tilde{M_2}$, hence also in the tree-like complex, and one can actually show that, for a nondegenerate tupel of vertices, the associated set of geodesics intersects the full 1-skeleton of exactly one top-dimensional simplex. This 'central' simplex belongs to a copy of $\tilde{M_1}$ or $\tilde{M_2}$. It is then easy to define $r$.
 
There is clearly no such construction for arbitrary manifolds. However, we show in 
section 2 that this construction can be done if $M_1$ and $M_2$ are
aspherical,
minimally complete multicomplexes. Such multicomplexes have in fact several features in common with hyperbolic spaces.
The theorem that we actually prove in section 2 is a generalization of the above. We consider not only multicomplexes glued at one vertex, but glueing multicomplexes
along an arbitrary submulticomplex with the additional condition that a 
suitable group $G$ acts with certain transitivity properties on the submulticomplex along which the glueing is performed, and we do the above construction not for simplices but for $G$-orbits of simplices. (It might be tempting to consider the quotients with respect to the $G$-action to reduce the glueing to a generalized wedge. However, this would raise technical problems related to the fact that these quotients are not multicomplexes.)

It should be noted that in the case of glueing two closed manifolds, \cite{gro},3.5.\ avoids the use of multicomplexes by using the classifying spaces of the fundamental groups, where the corresponding constructions are easier. This construction generalizes to manifolds with boundary only if one were to consider manifolds with exactly one boundary component.

Technically, the line of argument is as follows.
To any space $X$,
one associates an aspherical, minimally complete multicomplex $K\left(X\right)$. Its simplicial bounded cohomology $H_b^*\left(K\left(X\right)\right)$ is isometrically isomorphic
to the singular bounded cohomology $H_b^*\left(X\right)$. Bounded cohomology is a device that admits to dualize problems about the simplicial volume. We show in 3.1., using the isometric isomorphisms between bounded cohomology groups, that the glueing problem for manifolds $M_1, M_2$ can be translated into an analogous problem for the aspherical, minimally complete multicomplexes $K\left(M_1\right),K\left(M_2\right)$.

If $M$ is a manifold obtained by glueing along $A$, one 
lets act a                                             
large amenable group $\Pi_M\left(A\right)$, which consists of all homotopy classes in $M$
of pathes in $A$. This group action satisfies the transitivity properties needed to
apply the results of chapter 2, i.e.\ to
solve the glueing problem for the associated multicomplexes, and hence also for the original manifolds. In the proofs we will always have to distinguish two cases: the "amalgamated" case, where {\em two} manifolds are glued along parts of their boundary, and
the "HNN"-case, where the glueing is performed by identifying two boundary subsets of {\em one} manifold.\\
Convention: 
For simplicity, we assume all manifolds to be oriented.\\
I would like to thank Michel Boileau and Elmar Vogt for discussions about the content of this paper.

\section{Multicomplexes}

\subsection{Definitions}
 
\begin{df}: A multicomplex $K$ consists of the following data:\\
- {\em Simplices}: a set $V$ and, for any finite ordered subset $F=\left\{v_0,\ldots,v_n\right\}\subset V$ 

\hspace*{0.7in} with $\mid F\mid \ge 2$, 
a (possibly empty) set $I_F$, 

such that for any permutation $\pi:F_1\rightarrow F_2$ there is a bijection $I_\pi:I_{F_1}\rightarrow I_{F_2}$,\\
- {\em Face maps}:
for any finite ordered set $F=\left\{v_0,\ldots,v_n\right\}\subset V$ a family of maps 

$\left\{d_j:I_F\rightarrow I_{F-\left\{v_j\right\}}\right\}_{0\le j\le n}$.\end{df}
The elements of $V$ are the 0-simplices or vertices of $K$.
The pairs
$\sigma=\left(F,i\right)$ with $\mid F\mid=n$ and $i\in I_F$ are the $n$-simplices of $K$. The $j$-th face of an $n$-simplex $\left(F,i\right)$ is given by $\partial_j\left(F,i\right):=\left(F-\left\{v_j\right\},d_j\left(F\right)\right)$.\\
Let $K_j$ be the union of $j$-simplices and $K^n$ the $n$-skeleton of $K$, that is the union $\cup_{0\le j\le n}K_j$.\\

The {\bf geometric realization} $\mid K\mid$ of $K$ is defined as follows:\\
$\mid K\mid$ is the set of pairs $\left\{\left(\lambda,i\right):i\in I_{F_\lambda}\right\}$, where $\lambda:V\rightarrow\left[0,1\right]$ are maps such that $\sum_{v\in V}\lambda\left(v\right)=1$ and $F_\lambda=\left\{v\in V:\lambda\left(v\right) > 0\right\}$ is finite. \\
The set of all $\lambda$ with a given $F_\lambda$ and a given $i\in I_{F_\lambda}$ is canonically identified with a standard simplex
and herits a topology via this identification. We consider the topology on $\mid K\mid$ defined such that a set is closed if its intersection with each simplex is closed.

\begin{df} We call a multicomplex minimally complete, or m.c.m.\ for short, if the following holds:
whenever $\sigma:\Delta\rightarrow \mid K\mid$ is a singular simplex, such that
$\partial\sigma$ is a simplex of $K$, then $\sigma$ is homotopic relative $\partial\Delta$ to
a {\em unique} simplex in $K$.\end{df}
If $\sigma$ is an $n$-simplex, its n-1-skeleton is the set $\left\{\partial_0\sigma,\ldots,\partial_n\sigma\right\}$. By recursion, we define that the n-k-skeleton $\sigma_{n-k}$ of an $n$-simplex $\sigma$ is the union of the n-k-skeleta of the simplices belonging to the n-k+1-skeleton of $\sigma$.
\begin{df} We call a multicomplex $K$ aspherical if all simplices $\sigma\not =\tau$ in $K$ 
satisfy $\sigma_1\not=\tau_1$.        \end{df}
Orientation: Let $\pi:F_1\rightarrow F_2$ be a permutation of finite sets and $i\in I_{F_1}$. We say that $\left(F_1,i\right)$ and $\left(F_2,I_\pi\left(i\right)\right)$ have the same orientation if $\pi$ is even, and that they have different orientation if $\pi$ is odd.\\
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.

\subsection{Bounded Cohomology} 
For a multicomplex $K$, let $F_j\left(K\right)$ be the $R$-vector space with basis 
the set of $j$-simplices of $K$. Let $O_j\left(K\right)$ be the subspace generated by the set 
$\left\{\sigma-sign\left(\pi\right)\pi\left(\sigma\right)\right\}$ where $\sigma$ runs over all $j$-simplices,
$\pi$ runs over all permutations, and $\pi\left(F,i\right):=\left(\pi\left(F\right),I_\pi\left(i\right)\right)$.
Define the $j$-th chain group $C_j\left(K\right)=F_j\left(K\right)/O_j\left(K\right)$.\\
Let $$C^j_a\left(K\right):=Hom_R\left(C_j\left(K\right),{\Bbb R}\right),$$
i.e., the group of antisymmmetric cochains.\\
If $\left(K,L\right)$ is a pair of multicomplexes, we get an inclusion $C_j\left(L\right)\rightarrow
C_j\left(K\right)$ and we define 
$$C^j_a\left(K,L\right):=\left\{\omega\in C^j_a\left(K\right): \omega\left(c\right)=0 \mbox{ for all } c\in C_j\left(L\right)\right\}$$ with the norm 
$$\parallel \omega\parallel_\infty:=sup\left\{\omega\left(\sigma
\right) : \sigma \mbox{ j-simplex}\right\}.$$
Let
$C_b^j\left(K,L\right):=\left\{\omega\in C^j_a\left(K,L\right):
\parallel\omega\parallel_\infty < \infty\right\}$. The coboundary
operator preserves $C_b^*\left(K,L\right)$, hence induces
maps $\delta_b^i:C_b^i\left(K,L\right)\rightarrow
C_b^{i+1}\left(X,Y\right)$. Define the bounded cohomology
of $\left(K,L\right)$ by $H_b^i\left(K,L\right):=
ker\delta_b^i/im\delta_b^{i-1}$. $\parallel .\parallel_\infty$
induces a pseudo-norm $\parallel .\parallel$ on
$H_b^*\left(K,L\right)$.\\
For the bounded cohomology of topological spaces, we refer to \cite{iva}. 
When dealing with a pair of multicomplexes $\left(K,L\right)$, we will distinguish between $H_b^*\left(K,L\right)$ and 
$H_b^*\left(\mid K\mid,\mid L\mid\right)$. Since any bounded singular cochain is in particular a bounded simplicial cochain,
we have an inclusion $h:C_b^*\left(\mid K\mid\right)\rightarrow C_b^*\left(K\right)$.
 
\begin{pro} (Isometry lemma, \cite{gro}, S.43): If $K$ is a connected minimally complete multicomplex with infinitely
many vertices, then
$h^*:H^*_b\left(\mid K\mid\right)\rightarrow
H^*_b\left(K\right)$ is an isometric isomorphism. \end{pro}
 
For an n-dimensional compact, connected, orientable manifold let $\beta_M$ be the unique
class in $H^n\left(M,\partial M\right)$
such that $<\beta_M,\left[M,\partial M\right]>=1$.\\
By \cite{gro}, see \cite{kue}, Thm.2.1., $\parallel M,\partial M\parallel =
\frac{1}{\parallel \beta_M\parallel}$. In particular,  $\parallel M,\partial M\parallel=0$ if and only if $ im\left(H^n_b
\left(M,\partial M\right)\rightarrow H^n\left(M,\partial M\right)\right)=0$.
 
It can be shown (\cite{gro},\cite{iva}) that the bounded cohomology and its pseudonorm depend only on the
fundamental group. This works also for pairs $\left(X,Y\right)$, if $\pi_1Y\rightarrow \pi_1X$ is
injective. In particular, one has:                        
\begin{lem}:
If $M$ and $N$ are compact manifolds with incompressible boundary of the same dimension and there exists a map
$f:\left(M,\partial M\right)\rightarrow\left(N,\partial N\right)$ inducing an
isomorphism of pairs of fundamental groups, then $\parallel M,\partial M\parallel=deg\left(f\right)\parallel N,\partial N\parallel$.\end{lem}
\subsection{Aspherical multicomplexes}
\begin{pro} (\cite{gro}, p.46): Let $X$ be a topological space. Then there is an aspherical m.c.m.
$K\left(X\right)$ such that
there is an isometric isomorphism $I: H_b^*\left( K\left(X\right)\right)\rightarrow H^*_b\left(X\right)$. \end{pro}
Warning: It will be convenient for us to use notation different from Gromov's, since we will make use of a certain functoriality of $K\left(X\right)$. So one should be aware that our $K\left(X\right)$ corresponds to $K/ \Gamma_1$ in Gromov's notation, as well as that our $\tilde{K}\left(X\right)$ will correspond to Gromov's $K$ or $K\left(X\right)$. According to \cite{gro}, $\mid K\left(X\right)\mid$ is actually an
aspherical topological space. We will only need that $K\left(X\right)$ is
aspherical in the sense of definition 3, what will follow from the
geometric description of $K\left(X\right)$.\\
Because it will be of importance in the proof of lemma 8 and 9, we mention that $I$ is constructed as the following composition:$$ \begin{diagram}H_b^*\left(K\left(X\right)\right)&\rTo^{p^*}& H_b^*\left(\tilde{K}\left(X\right)\right)&\rTo^{\left(h^*\right)^{-1}}& H_b^*\left(\mid\tilde{K}\left(X\right)\mid\right)&\rTo^{\left(S^*\right)^{-1}}& H_b^*\left(X\right).\end{diagram}$$ 
Here, $h^*$ is the isomorphism from proposition 1, $p$ and $S$ are described in the geometrical description below.\\
A proof of proposition 2 is given in \cite{gro}. Because it will be crucial for the proof of theorem 1, 
we recall the geometrical description of $K\left(X\right)$ as it can be read off the constructions in \cite{gro}.

{\bf Geometrical description of $\tilde{K}\left(X\right)$:} For a topological space $X$,
$\tilde{K}\left(X\right)$ is the multicomplex defined as follows. Its 0-skeleton is $\left\{x: x\in X\right\}$. Its 1-skeleton is $\left\{\left(\left\{x,y\right\},i\right):x\not= y\in X, i\in I_{\left\{x,y\right\}}\right\}$ where $I_{\left\{x,y\right\}}$ is the set of homotopy classes relative $\left\{0,1\right\}$
of maps from $\left[0,1\right]$ to $X$ mapping $0$ to $x$ and $1$ to $y$. Having defined the n-1-skeleton of $\tilde{K}\left(X\right)$, the n-simplices of $\tilde{K}\left(X\right)$ are the
pairs $\left(\left\{x_0,\ldots,x_n\right\},i\right)$ with $x_0,\ldots,x_n\in X$ and $i$ a homotopy class relative $\partial\Delta^n$
of mappings from the standard simplex $\Delta^n$ to $X$, taking the i-th vertex of $\Delta^n$ into $x_i$ for $i=0,\ldots,n$ and the n-1-skeleton of $\Delta^n$ into the n-1-skeleton of $\tilde{K}\left(X\right)$.\\
We have a canonical continuous mapping $S:\widetilde{K}\left(X\right)\rightarrow X$ which maps each singular simplex to its image in $X$.\\
\\
{\bf Geometrical description of $K\left(X\right)$:} The multicomplex $K\left(X\right)$ is obtained from $\tilde{K}\left(X\right)$ by identifying, via simplicial maps, all n-simplices with a common n-1-skeleton, for all $n\ge 2$, succesively in order of increasing dimension. \\
In particular, we have a canonical projection $p:\tilde{K}\left(X\right)\rightarrow K\left(X\right)$.
\begin{pro}: Let $Y\subset X$ be a subspace, such that $\pi_1\left(Y,y\right)\rightarrow\pi_1\left(X,y\right)$
is injective for all $y\in Y$. Then
$K\left(Y\right)$ is a submulticomplex of $K\left(X\right)$.\\
$I: H^*_b\left(K\left(X\right),K\left(Y\right)\right)\rightarrow
H^*_b\left(X,Y\right)$ is an isometric isomorphism.\end{pro}  
\begin{pf} If two distinct 1-simplices in $Y$ mapped to the same 1-simplex in $X$, the corresponding pathes in $Y$ would be in $ker\left(\pi_1Y\rightarrow\pi_1X\right)$. By asphericity, simplices are determined by their 1-skeleton and the first claim follows. From the five lemma, $I$ is an isomorphism. Therefore, it must be an isometry, since $I$ and $I^{-1}$ are composed by maps of norm $\le1$.\end{pf}
%The vertices of $K\left(X\right)$ are the points
%of $X$. Given two vertices $v_0,v_1$, the
%1-simplices with vertices $v_0$ and $v_1$ are the homotopy classes relative $\left\{0,1\right\}$
%of singular 1-simplices $\sigma:\left[0,1\right]\rightarrow X$ with $\sigma\left(0\right)=v_0,\sigma\left(1\right)=v_1$.
%Then define the set of i-simplices as follows: given i+1 vertices $v_0,\ldots,v_i$
%and a set $E$ of $\frac{1}{2}\left(i+1\right)\left(i+2\right)$ 1-simplices such that to any pair $\left\{v_i,v_j\right\}$ there exists a unique
%1-simplex in $E$ with vertices $v_i,v_j$, there exists {\em one} i-simplex with $E$ as the set of 1-subsimplices if in $X$ exists
%a singular i-simplex with vertices $v_0,\ldots,v_n$ whose 1-subsimplices are in the homotopy classes of the elements of $E$. Otherwise there is no
%i-simplex with $E$ as set of 1-subsimplices.
%The subsimplex structure on $K\left(X\right)$ is defined in the obvious way.
\subsection{Amenable Groups and Averaging}
\begin{df}: For a group $\Gamma$ let $B\left(\Gamma\right)$ be the space of bounded real-valued functions on $\Gamma$. $\Gamma$ is called amenable if there is a $\Gamma$-invariant linear functional
$Av:B\left(\Gamma\right)\rightarrow R$
such that
$inf\left(f\right)\le Av\left(f\right)\le sup\left(f\right)$ holds for all $f\in B\left(\Gamma\right)$.\end{df}
If a group $G$ acts simplicially on a pair of multicomplexes $\left(K,L\right)$, we denote 
$C_b^*\left(K^G,L^G\right):=\left\{c\in C^*_b\left(K,L\right): gc=c\mbox{ for all }g\in G\right\}$,\\
$\delta_i^G:= \delta\mid_{C_b^i\left(K^G,L^G\right)}$ and \\
$H_b^*\left(K^G,L^G\right):=ker\delta_i^G/im\delta_{i-1}^G$.
\begin{lem}: (i)
If an amenable group $\Gamma$ acts on a pair of multicomplexes $\left(K,L\right)$, and $p:C_b^*\left(K^\Gamma,L^\Gamma\right)\rightarrow
C_b^*\left(K,L\right)$ is the
inclusion, then there is a homomorphism $Av:H_b^*\left(K,L\right)\rightarrow H_b^*\left(K^\Gamma,L^\Gamma\right)$ such that $Av\circ p^*=id$ and $\parallel Av\parallel =1$.\\
(ii) If, moreover, all elements of $\Gamma$ are homotopic to the identity in the category of
continuous maps of pairs of
spaces $\left(\mid K\mid, \mid L\mid\right)$, then $p^*\circ Av=id$.\end{lem}
\begin{pf}: The proof of (i) works, for $L=\emptyset$, the same way as for singular bounded cohomology in \cite{gro},p.39. In the relative setting, if $L\not=\emptyset$, $Av$ still is an isometry because
of $\parallel Av\parallel\le1,\parallel p^*\parallel\le1$ and $Av p^*=id$, and it is an isomorphism as a consequence of the five lemma. Part
(ii) follows from the homotopy lemma, \cite{gro},p.42 and is implicit in \cite{gro},p.46,cor.D.\end{pf}

  \subsection{Group actions on multicomplexes}     
In the following, $\left(X,Y\right)$ will be a pair of topological spaces. For a path $\gamma:\left[0,1\right]\rightarrow X$, we denote by $\left[\gamma\right]$ its homotopy class {\em in X} relative $\left\{0,1\right\}$.
\begin{df}: Define $\Pi_X\left(Y\right):=\\
\left\{\left(\left[\gamma_1\right],\ldots,\left[\gamma_n\right]\right):n\in N,
\gamma_1,\ldots,\gamma_n:\left[0,1\right]\rightarrow Y,\left\{\gamma_1\left(
0\right),\ldots,\gamma_n\left(0\right)\right\}=\left\{
\gamma_1\left(1\right),\ldots,\gamma_n\left(1\right)\right\}\right\}$.\end{df}
$\Pi_X\left(Y\right)$ is a group with respect to the following product: \\
$\left\{\left[\gamma_1\right],\ldots,\left[\gamma_m\right]\right\}
\left\{\left[\gamma_1^\prime\right],\ldots,\left[\gamma_n^\prime\right]\right\}:=
\left\{\left[\gamma_1*\gamma_1^\prime\right],\ldots,\left[\gamma_i*\gamma_i^\prime\right],\left[\gamma_{i+1}\right],\ldots,
\left[\gamma_m\right],\left[\gamma_1^\prime\right],
\ldots,\left[\gamma_n^\prime\right]\right\}$, where $*$ denotes the concatenation of pathes
and $i\ge 0$ is chosen 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 $j\ge i+1, k\ge i+1$. (Such an $i$ exists for a unique reindexing of the elements in the unordered sets $\left\{\left[\gamma_1\right],\ldots,\left[\gamma_m\right]\right\}$ and $
\left\{\left[\gamma_1^\prime\right],\ldots,\left[\gamma_n^\prime\right]\right\}$.)\\
\\

{\bf Action of $\Pi_X\left(Y\right)$ on $K\left(X\right)$:}\\
To define an action of $\Pi_X\left(Y\right)$ on the 0-skeleton of $K\left(X\right)$, we recall that \\
$i:X\rightarrow K\left(X\right)$ maps $X$ bijectively to $K\left(X\right)_0$.
For $g=\left\{\left[\gamma_1\right],\ldots,\left[\gamma_n\right]\right\}\in
\Pi_X\left(Y\right)$, we define
$gi\left(\gamma_1\left(0\right)\right)=i\left(\gamma_1\left(1\right)\right),
\ldots,gi\left(\gamma_n\left(0\right)\right)=i\left(\gamma_n\left(1\right)\right)$ and $gi\left(v\right)=i\left(v\right)$ if $v\not\in\left\{\gamma_1\left(0\right),\ldots,\gamma_n\left(0\right)\right\}$.\\
As a next step, we extend this to an action on the 1-skeleton of $K\left(X\right)$.
Recall that $1$-simplices $\left[\sigma\right]$ in $K\left(X\right)$ correspond to homotopy 
classes $\left[\sigma\right]$  of pathes $\sigma:\left[0,1\right]\rightarrow X$.
Let $g=\left\{\left[\gamma_1\right],\ldots,\left[\gamma_n\right]\right\}\in
\Pi_X\left(Y\right)$.
For a 1-simplex $\left[\sigma\right]$ define $g\left[\sigma\right]=\left[\sigma\right]$ if $
\sigma\left(0\right)\not\in\left\{
\gamma_1\left(0\right),\ldots,\gamma_n\left(0\right)\right\}$ and
$\sigma\left(1\right)\not\in\left\{\gamma_1\left(0\right),\ldots,
\gamma_n\left(0\right)\right\}$.
If $\sigma\left(0\right)
=\gamma_i\left(0\right)$ and $\sigma\left(1\right)$ differs from all
$\gamma_j\left(0\right)$, define $g\left[\sigma\right]$ to be the 1-simplex of $K\left(X\right)$
corresponding to the homotopy class of
the concatenation $\sigma * i\left(\overline{\gamma_i}\right)$, where
$\overline{\gamma_i}\left(t\right):=\gamma_i\left(1-t\right)$ for
$t\in\left[0,1\right]$. If $\sigma\left(1\right)
=\gamma_i\left(0\right)$ and $\sigma\left(0\right)$ differs
from all $\gamma_j\left(0\right)$, define $g\left[\sigma\right]$ as the 1-simplex 
corresponding to the homotopy class of 
$i\left(\gamma_i\right) * \sigma$. If $\sigma\left(0\right)=\gamma_i\left(0\right)$ and $\sigma\left(1\right)=\gamma_j\left(0\right)$, define
$g\sigma$ to be the 1-simplex corresponding to the homotopy class of
$i\left(\gamma_j\right)*\sigma*i\left(\overline{\gamma_i}\right)$. All these definitions were independent of the choice of $\sigma$ in its homotopy class relative $\left\{0,1\right\}$.
\\
To define the action of $\Pi_X\left(Y\right)$ on all of $K\left(X\right)$,
we claim that for a simplex
$\sigma\in K\left(X\right)$ with 1-skeleton $\sigma_1$, and $g\in\Pi_X\left(Y\right)$,
there exists {\em some} simplex in $K\left(X\right)$ with 1-skeleton $g\sigma_1$. Since $K\left(X\right)$ is aspherical, this will allow a unique extension of the group action from $K\left(X\right)_1$ to $K\left(X\right)$.
To prove the claim, observe the following:
if $g$ is a path in
$X$ connecting $v_0$ to $v_0^\prime$ and if $\sigma$ is a simplex in $K\left(X\right)$
represented by a singular simplex $\hat{\sigma}$ in $X$ with 0-th vertex $v_0^\prime$,
then $\hat{\sigma}$
can
clearly be homotoped so that one gets a singular simplex in $X$ with 0-th vertex $v_0$, leaving the other vertices fixed, so that we get a simplex whose 1-skeleton is $g\sigma_1$. Argueing succesively, we get the claim for general $g\in\Pi_X\left(Y\right)$.
 
\begin{df}: Let $\left(K,L\right)$ be a pair of multicomplexes. Let $\left\{e_1,\ldots,e_n\right\}$ and $\left\{e_1^\prime,\ldots,e_n^\prime\right\}$ be two n-tuples of 1-simplices in $K$ with vertices in $L$. We say that $\left\{e_1,\ldots,e_n\right\}$ and $\left\{e_1^\prime,\ldots,e_n^\prime\right\}$ are $L$-related, if there 
are 1-simplices $f_1,\ldots,f_m$ in  $L$ such that \\
- the vertices $f_1\left(0\right),f_1\left(1\right),\ldots,f_m\left(0\right),f_m\left(1\right)$ of $f_1,\ldots,f_m$ are all distinct and are in bijection with the set of vertices 
of $e_1,\ldots,e_n,e_1^\prime,\ldots,e_n^\prime$, (note that $m\le 2n$, we do not assume the vertices of the $e_i$'s and $e_i^\prime$'s to be distinct)\\
- the concatenations $f_ke_if_le_i^{\prime-1}$ (with $f_k,f_l$ uniquely selected such that the vertices match) represent the identity in $\pi_1K$.\end{df}

The following observation is obvious from the construction.
\begin{lem}: Let $\left(X,Y\right)$ be a pair 
of spaces such that $\pi_1Y\rightarrow\pi_1X$ is injective. Then the action of $\Pi_X\left(Y\right)$ on $K\left(X\right)$ is
transitive on $K\left(Y\right)$-related $1$-simplices with vertices in $K\left(Y\right)$. That is, if $e_1$ and $e_2$ are $K\left(Y\right)$-related 1-simplices in $K\left(X\right)$, 
then exists $g\in\Pi_X\left(Y\right)$ with $ge_1=e_2$.\end{lem}
 
%{\bf Description of $K_Y\left(X\right)$:} For a subspace $Y\subset X$ denote $K_Y\left(X\right):=
%\Pi_X\left(Y\right)\backslash K\left(X\right)$.\\
%$\Pi_X\left(Y\right)\backslash K\left(X\right)_0=X/_\sim$, where
%$x_1\sim x_2$ if $x_1$ and $x_2$ belong to the same path-connected component of $Y\subset X$.
%\\
%Denote $Y_1,Y_2,\ldots$ the path-connected components of $Y\subset X$.
%For $x_1,x_2\in
%X - Y$ orbits of 1-simplices with vertices $x_1,x_2$ are
%in 1-to-1 correspondence with homotopy classes of pathes between $x_1$ and $x_2$ in $X$. 
%For $x\in X - Y$ there
%may be several 1-simplices with vertices $x$ and $\left[Y_i\right]$,
%since $\pi_1Y_i\rightarrow\pi_1X$ needn't be surjective. However, there is again 
%at most one i-simplex with a given 1-skeleton. \\
%Remark: An analogous description holds, whenever a group $G$ acts simplicially on a pair of aspherical,
%minimally complete multicomplexes $\left(M,M^\prime\right)$ satisfying that $G$ acts transitively on the set of $j$-simplices of $M^\prime$, for all $j\ge 0$.

%\begin{pro}: Let $Y\subset X$ be a subspace, such that $\pi_1Y$ injects into
%$\pi_1X$. Then
%$K\left(Y\right)$ may be realized as a submulticomplex of $K\left(X\right)$.\\
%$h^*: H^*_b\left(K\left(X\right),K\left(Y\right)\right)\rightarrow
%H^*_b\left(X,Y\right)$ is an isometric isomorphism.\end{pro}
%\begin{pf} Since higher-dimensional simplices are determined by the 1-skeleton,
%it suffices to show that the 1-skeleton of $K\left(Y\right)$
%embeds into the 1-skeleton of $K\left(X\right)$. If this were not
%the case, we would have singular 1-simplices
%$\sigma:\left[0,1\right]\rightarrow Y$ belonging to different
%homotopy classes relative $\left\{0,1\right\}$ in $Y$ but to the same homotopy class
%relative $\left\{0,1\right\}$ in $X$, hence, nontrivial elements in
%$ker\left(\pi_1Y\rightarrow\pi_1X\right)$.\end{pf} 


\subsection{An application of averaging}
\begin{lem}: If $A\subset X$ is a subspace such that $im\left(\pi_1\left(A,x\right)\rightarrow\pi_1\left(X,x\right)\right)$ is amenable for all $x\in A$, then $\Pi_X\left(A\right)$ is
amenable.\end{lem}
\begin{pf} There is an exact sequence \\
$1\rightarrow \bigoplus_{y\in A}im\left(\pi_1\left(A,y\right)\rightarrow\pi_1\left(X,y\right)\right)\rightarrow
\Pi_X\left(A\right)\rightarrow Perm_{fin}\left(A\right)\rightarrow 1$,
where $Perm_{fin}$ are the permutations with finite support.\\
It is well known that a group is amenable if any finitely generated subgroup
is amenable. All finitely supported permutations have finite order. It follows that
any finitely generated subgroup of $Perm_{fin}\left(A\right)$ is finite and
therefore amenable. Also any finitely generated subgroup of
$\bigoplus_{y\in Y}im\left(\pi_1\left(A,y\right)\rightarrow\pi_1\left(X,y\right)\right)$
is contained in a finite sum
of amenable groups and is therefore amenable. Thus
$\Pi_X\left(A\right)$ is an amenable extension of an amenable group
and, hence, is amenable.\end{pf}\\
\\
\\
For $\epsilon\in R$ define a norm on $C_*\left(X,Y\right)$ by
$\parallel z\parallel_\epsilon=\parallel z\parallel + \epsilon\parallel\partial z\parallel$.
We get an induced pseudonorm $\parallel .\parallel_\epsilon$ on $H_*\left(X,Y\right)$. \\
More generally, if $A$ is a union of connected components of $Y$,
we define a norm on relative cycles of $C_*\left(X,Y\right)$ by $\parallel z\parallel^A_\epsilon=\parallel z\parallel +\epsilon\parallel \partial z\mid_A\parallel$ and
consider the induced pseudonorm $\parallel .\parallel^A_\epsilon$
on $H_*\left(X,Y\right)$.
\begin{pro}: If $im\left(\pi_1\left(A,y\right)\rightarrow\pi_1\left(X,y\right)\right)$ is
amenable for all $y\in A$, then $\parallel h\parallel=
\parallel h\parallel^A_\epsilon$ for all $h\in H_*\left(X,Y\right)$.\end{pro}
\begin{pf}: With the additional assumption $A=Y$, proposition 4 becomes the equivalence theorem in \cite{gro}, p.57.
To get the general claim, we give a straigtforward modification of Gromov's proof.
 
We consider the dual norm on bounded cohomology, which, by the
Hahn-Banach theorem, is induced from the dual norm on the relative
cocycles. We will show that $\parallel c\parallel^A_\epsilon
=\parallel c\parallel$ for relative cocycles $c$.
 
By propositions 1 and 2, we may assume that we are working with the complex of antisymmetric simplicial cochains of $K\left(X\right)$. By lemma
2, we may assume the cochains to be invariant under the action
of the amenable group $\Pi_X\left(A\right)$. Hence, we may assume
that the relative cocycle $c$ factors over $Q$, where $Q$ is the quotient
of $F_*\left(K\left(X\right)\right)/F_*\left(K\left(Y\right)\right)$ under the relations $\overline{\sigma}=-\sigma$ and $a\sigma=\sigma$ for all $a\in \Pi_X\left(A\right)$
and all simplices $\sigma$, where $\overline{\sigma}$ is $\sigma$ with
the opposite orientation. We can define in an obvious way analogs of
our norms on the dual of $Q$ and we get then $\parallel c\parallel=
\parallel c^Q\parallel$ and $\parallel c\parallel^A_\epsilon=
\parallel c^Q\parallel^A_\epsilon$, where $c^Q$ is $c$ considered as a
map from $Q$ to $R$.
 
But in $Q$, any simplex $\sigma$ with an edge in $A$
becomes trivial, because there is some element of $\Pi_X\left(A\right)$
mapping $\sigma$ to $\overline{\sigma}$. Hence, for any
relative cycle $z\in C_*\left(X,Y\right)$, the image of $\partial z\mid_A$
in $Q$ is trivial. Hence, $\parallel c^Q\parallel$ and
$\parallel c^Q\parallel^A_\epsilon$ agree.\end{pf}
 
 
\begin{cor}: If $M$ is a compact manifold, $A$ a union of
connected components of $\partial M$, and $im\left(\pi_1\left(A,x\right)\rightarrow\pi_1\left(M,x\right)\right)$
is amenable for all $x\in A$, then for any $\epsilon
>0$ exists a representative $z$ of $\left[M,\partial M\right]$
with $\parallel z\parallel\le\parallel M,\partial M\parallel+\epsilon$
and $\parallel \partial z\mid_A\parallel\le\epsilon$.\end{cor}

\section{Retraction in aspherical treelike complexes}
%We denote by $A_\infty$ the multicomplex which has exactly one i-simplex
%$\sigma_i$ for any $i\in N$, such that $\partial \sigma_i=\left(i+1\right)\sigma_{i-1}$. Note that $pi_1A_\infty=Z/3Z$. 
If a group $G$ acts simplicially on a multicomplex $M$, then $C_*\left(M\right)/G$ are abelian groups with well-defined boundary operator, even though $M/G$ may not be a multicomplex. (An instructive example for the latter phenomenon is the action 
of $G=\Pi_X\left(X\right)$ on $K\left(X\right)$, for a topological space $X$.)

\subsection{The 'amalgamated' case}
\begin{lem}: Assume that $\left(M,M^\prime\right), \left(K,K^\prime\right),
\left(L,L^\prime\right)$ are pairs of path-connected, minimally complete submulticomplexes,
such that\\
(i) $K,L$ are submulticomplexes of $M$, with inclusions $i_K:K\rightarrow M, i_L:L\rightarrow M$,\\
(ii) $M_0=K_0\cup L_0$,\\ 
(iii) $A:=K\cap L$ is a submulticomplex, $\pi_1A\rightarrow \pi_1K$ and $\pi_1A\rightarrow\pi_1L$ are injective,\\
(iv) the inclusion $K\cup L\rightarrow M$ induces an isomorphism $\pi_1\left(K\cup L\right)\rightarrow\pi_1\left(M\right)$,\\
(v) $K$ and $L$ are aspherical in the sense of definition 3,\\
(vi) $K^\prime =M^\prime\cap K, L^\prime =M^\prime\cap L$.\\
Assume moreover that a group $G$ acts simplicially on $\left(M,M^\prime\right)$ such that \\
(vii) $G$ maps $\left(K,K^\prime\right)$ to $\left(K,K^\prime\right)$ and $\left(L,L^\prime\right)$ to $\left(L,L^\prime\right)$,\\
(viii) $G$ acts transitively on $A$-related tuples of 1-simplices.\\
Then there is a relative chain map $r:G\backslash C_*\left(M,M^\prime\right)\rightarrow
G\backslash C_*\left(K,K^\prime\right)\oplus G\backslash C_*\left(L,L^\prime\right)$ in degrees $*\ge2$ such that\\
- if $G\sigma$ is the orbit of a simplex in $M$, then $r\left(G\sigma\right)$ either is the $G$-orbit of a simplex in $K$ or the $G$-orbit of a simplex in $L$,\\
- $r{i_K}_*=id_{G\backslash C_*\left(K,K^\prime\right)},\ r{i_L}_*=id_{G\backslash C_*\left(L,L^\prime\right)}$.\end{lem}
 
\begin{pf} We consider first the case $M^\prime=\emptyset$.

The plan of the proof is as follows: let $\sigma$ be a simplex in $M$,
let $\tilde{\sigma}$ be a lift to the universal cover $\tilde{M}$, and let
$v_0,\ldots,v_n\in\tilde{M}_0$ be the vertices of $\tilde{\sigma}$. To 
each pair $\left\{v_i,v_j\right\}$ we associate a family of 'minimizing' pathes $\left\{p\left(\left\{a_l^{ij}\right\};\left\{h_k^{ij}\right\}\right)\right\}$ parametrised 
by vertices $a_0^{ij},\ldots,a_{m_{ij}}^{ij}\in A_0$ and by elements $h_k^{ij}$ of $\pi_1K$ or $\pi_1L$ satisfying conditions described below. Associated to $\left\{v_0,\ldots,v_n\right\}$ and these families of 'minimizing' pathes,
we construct a family of 'central' simplices \\
$\left\{\tilde{\tau}\left(\left\{a_l^{ij}\right\};\left\{h_k^{ij}\right\}\right)\subset\tilde{M}\right\}$ and 
their projections $\left\{\tau\left(\left\{a_l^{ij}\right\};\left\{h_k^{ij}\right\}\right)\subset M\right\}$, which actually lie in $K$ or $L$. 
We show then that all $\tau\left(\left\{a_l^{ij}\right\};\left\{h_k^{ij}\right\}\right)$, associated to a fixed $\tilde{\sigma}$, belong to the same $G$-orbit, and that also the simplices associated to
either $g\tilde{\sigma}$ with $g\in\pi_1M$ or to $\widetilde{g\sigma}$ with $g\in G$ belong to the same $G$-orbit. We define then $r\left(G\sigma\right)=G\tau$.\\
 
Assumption (iv) implies that the universal cover $\widetilde{K\cup L}$ is a submulticomplex
of the universal cover $\tilde{M}$. Assumption (ii) together with assumption (iv) gives that the 0-skeleton of $\widetilde{K\cup L}$ is the whole 0-skeleton of $\widetilde{M}$.\\ 
Let $\pi:\widetilde{K\cup L}\rightarrow K\cup L$ be the projection. We will need a specific section $s$ of $\pi$ on the 1-skeleton
$$s:\left(K\cup L\right)_1\rightarrow\left(\widetilde{K\cup L}\right)_1$$
$$\sigma\rightarrow s\left(\sigma\right)=\tilde{\sigma}$$
defined as follows:

Fix a vertex $p\in A_0$ and some lift $\tilde{p}\in \tilde{A}_0\subset\tilde{M}_0$.
For any vertex $v$ of $K\cup L$, there is some edge $e\in K_1\cup L_1$ connecting 
$p$ to $v$, because $K$ and $L$ are complete. There are unique lifts $\tilde{e}$ and $\tilde{v}$ such that
$\tilde{e}$ has boundary points $\tilde{p}$ and $\tilde{v}$. This defines $\sim$ on the 0-skeleton, and also on some 1-simplices. Now, for all other 1-simplices $e\in K_1\cup L_1$ with boundary points $v$ and $w$, possibly $v=p$, we fix the unique lift $\tilde{e}$ in $\widetilde{K\cup L}$  with 0-th vertex $\tilde{v}$. (Note that $\underline{in\ K\cup L}$ there are no edges with one vertex in $K_0-A_0$, the other vertex in $L_0-A_0$.) \\
It should be noted: if $e$ has vertices $v_0$ and $v_1$, then $\tilde{e}$ has vertices $\tilde{v_0}$ and $h_1\tilde{v_1}$ with, a priori, $h_1\in\pi_1\left(K\cup L\right)$. Assume that $e$ is an edge in $K$. We have a unique edge connecting $\tilde{v_1}$ to $h_1\tilde{v_1}$. This edge projects to an edge $f$
with both vertices $v_1$. Since $v_1$, as a vertex of $e$, belongs to $K$,
we conclude that $f\in K_1$. This implies $h_1\in\pi_1K\subset \pi_1\left(K\cup L\right)$. In a similar way, if $e$ is an edge in $L$, we conclude that $h_1\in\pi_1L$.\\
As a consequence, we get the following observation.

(A): {\em if $g\tilde{e}$ is an edge with boundary points $h_0\tilde{v_0}$ and $h_1\tilde{v_1}$, then
$g=h_0$ and 

$h_1h_0^{-1}$ is either $1$, or an element of $\pi_1K$, or an element of $\pi_1L$.}\\
Indeed, we have just seen this for $g=1$. The general case follows after applying $g^{-1}$ to $g\tilde{e}$.\\
%r construction this is true for $g=1$: consider the 2-simplex $\tau$
%with vertices $\tilde{p}$,$\tilde{v_0}$ and $h_1\tilde{v_1}$, which exists because $\tilde{p}$ can be connected to any point by an edge from $K\cup L$, and because in $\tilde{K\cup L}$ any 1-skeleton determines a simplex. $\tilde{tau}$ projects to a simplex either in $K$ or in $L$, say in $K$.
%
% and $v_0=p$. For $g=1$ and $v_0,v_1$ arbitrary, we note that $e\in K_1\cup K_2$ has boundary points $v_0,v_1$, hence $v_0$ and $v_1$ belong both to $K$ or both to $L$, say $K$.
%By construction, $\tilde{e}$ connects $\tilde{v_0}$ to some $k\tilde{v_1}$, as it is the lift of an
%edge in $K$. (A) follows by applying $g^{-1}$ to $g\tilde{e}$.\\
Moreover, if $g_1\tilde{e_1}$ is a 1-simplex with boundary points $h_{01}\tilde{v}_{01}$ and $h_{11}\tilde{v}_{11}$,
and $g_2\tilde{e_2}$ is a 1-simplex with boundary points $h_{02}\tilde{v}_{02}$ and $h_{12}\tilde{v}_{12}$, then:

(B): {\em if $g_1\tilde{e_1}$ and $g_2\tilde{e_2}$ have a common boundary point $h\tilde{v}=h_{11}\tilde{v}_{11}=h_{02}\tilde{v}_{02}$, 

then one of the following two possibilities holds:

- $v\in A$ or

- $h_{01}^{-1}h_{11}$ and $h_{02}^{-1}h_{12}$ either belong both to $\pi_1K$ or belong both to $\pi_1L$.}\\
Indeed, if $v\not\in A$, then $v$ is not adjacent to both, edges of $K$ and edges of $L$. \\

{\bf Minimizing pathes:}
 
Let $v_0,v_1$ be vertices of $\widetilde{K\cup L}$. By a {\bf path} from $v_0$ to $v_1$ we mean a sequence of
1-simplices $e_1,\ldots,e_r$ such that $v_0$ is a vertex of $e_1$, $e_i$ and $e_{i+1}$ have a vertex
in common for $1\le i\le r-1$ and, $v_1$ is a vertex of $e_r$. 
 
Given two vertices $v_1,v_2\in \tilde{M}_0$, they belong to $\left(\widetilde{K\cup L}\right)_0$ because of (iii) and (v), and we may represent them as $v_1=g_1\tilde{w_1}, v_2=g_2\tilde{w_2}$ with $g_i\in\pi_1\left(K\cup L\right)$ and $w_i\in \left(K\cup L\right)_0$ for $i=1,2$.

$\pi_1\left(K\cup L\right)=\pi_1K*_{\pi_1A}\pi_1L$ is an
amalgamated product, hence $g_1g_2^{-1}$
either belongs to $\pi_1A$ or it can be decomposed as
$g_1g_2^{-1}=h_1\ldots h_m$, where $h_i$ are elements of $\pi_1K-\pi_1A$ or
$\pi_1L-\pi_1A$ and, $h_i\in\pi_1K$ iff $h_{i+1}\in\pi_1L$. Such an expression is called a {\bf normal form} of $g_1g_2^{-1}$. If $h_1\ldots h_m$ and $h_1^\prime\ldots h_l^\prime$ are two normal forms of $g_1g_2^{-1}$, then necessarily $l=m$ and for $i=1,\ldots,m$ belong $h_i$ and $h_i^\prime$ to the same equivalence class modulo $\pi_1A$. 
 
We call a path $e_0,\ldots,e_{m-1}$ from $g_1\tilde{w}_1$ to $g_2\tilde{w}_2$ {\bf minimizing} if it satisfies the following:

there is a {\em normal form} $g_1g_2^{-1}=h_1\ldots h_m$ and a {\em set of vertices} $a_0,\ldots,a_m\in A_0$, with $a_i\not=a_{i+1}$ for $i=0,\ldots,m-1$,
such that:\\ - $e_0$ has vertices $g_2\tilde{w}_2$ and $g_2\tilde{a}_m$\\ 
- $e_i$ has vertices $h_{m-i+1}\ldots h_{m}g_2\tilde{a}_{m-i}$ and $h_{m-i+2}\ldots h_mg_2\tilde{a}_{m-i+1}$ for $i=1,\ldots,m$\\
- $e_{m+1}$ has vertices $g_1\tilde{w}_1$ and $g_1\tilde{a}_0$.\\
One should note that all these edges exist in $\widetilde{K\cup L}$ because all neighboring points project to distinct points in $K$ resp.\ $L$ and can therefore 
be joined by an edge in $K$ resp.\ $L$, by completeness. Moreover, the construction should be understood such that we skip $e_0$ resp.\ $e_{m+1}$ if $w_2\in A$ resp.\ $w_1\in A$.

It follows from (A) and (B), that these pathes are length-minimizing in the sense of being exactly the pathes with a minimum number of edges between $v_1$ and $v_2$. Since this latter characterisation depends only on $v_1$ and $v_2$, we conclude: for different sections $s_1$ and $s_2$, there is a bijection between the corresponding sets of minimizing pathes from $v_1$ to $v_2$.

Since $\tilde{K}$ and $\tilde{L}$ are universal covers of
{\em minimally complete} multicomplexes,
there is at most one edge
between two vertices, and therefore a path of length $m$ 
becomes uniquely determined after fixing its $m+1$ vertices.
Hence, after fixing $a_0,\ldots,a_m\in A_0$ and a normal form $g_1g_2^{-1}=h_1\ldots h_m$, we get a unique path, to be denoted $p\left(a_0,\ldots,a_m;h_1,\ldots,h_m\right)$.

We note for later reference the following obvious observations: \\
(C1) {\em Subpathes of minimizing pathes are minimizing.}\\
(C2) {\em If $e_1,\ldots,e_k$ is a minimizing path, $e_k$ projects to an edge in $K$, $e_{k+1}$ projects to an edge in $L$, $e_k$ and $e_{k+1}$ have a common vertex, then $e_1,\ldots,e_k,e_{k+1}$ is a minimizing path.}\\
%(C) {\em If $v_1$ and $v_2$ are two vertices of $K\cup L$, then any path from $v_1$ to $v_2$ has to 
%intersect some minimizing path from $a_1v_1$ to $a_2v_2$ for some $a_1,a_2\in G$.}\\
%This follows immediately from uniqueness modulo $\pi_1A$ of the the above mentioned representation 
%$g_1g_2^{-1}=h_1\ldots h_m$.\\

{\bf Intersection with simplices:}
 
Given a set of vertices $\left\{v_i\right\}_{1\le i\le n}$ we consider the set 
$$\bigcup_{0\le i,j\le n}P\left(i,j\right)=\left\{r_{ij}^k: 1\le i<j\le n, r_{ij}^k\in P\left(i,j\right)\right\}, $$
where $P\left(i,j\right)$ is the set of 
minimizing paths
from $v_i$ to $v_j$.
 
Let $\tilde{\tau}$ be any simplex in $\widetilde{K\cup L}$. We claim
that $\left\{\tilde{\tau}\cap r_{ij}^k: r_{ij}^k\in P\left(i,j\right)\right\}$
is the full 1-skeleton of a subsimplex of $\tilde{\tau}$. We have to check the following claim:\\
\hspace*{1in}{\em if $\left[x,y\right], \left[z,w\right]\in\tilde{\tau}_1\bigcap \cup_{ij}P\left(i,j\right)$, then also $\left[x,z\right], \left[x,w\right], $\\
\hspace*{1in}$\left[y,z\right]$ and $\left[y,w\right]$
belong to $\tilde{\tau}_1\bigcap \cup_{ij}P\left(i,j\right)$.}\\
Indeed, assume that there is a minimizing path $\left\{e_0,\ldots,e_{m+1}\right\}$
with $e_i$ having vertices $h_{m-i+1}\ldots h_{m}g_2\tilde{a}_{m-i}$ and $h_{m-i+2}\ldots h_mg_2\tilde{a}_{m-i+1}$ for $i=1,\ldots,m$ such that $\left[x,y\right]=e_l$, that is, 
$x=h_{m-l+1}\ldots h_mg_2\tilde{a}_{m-l}$ and $y=h_{m-l+2}\ldots h_mg_2\tilde{a}_{m-l+1}$.
Assume also
that there is 
a minimizing path
$\left\{e_0^\prime,\ldots,e_{m^\prime}^\prime\right\}$
with analogous notations such that $\left[z,w\right]=e_l^\prime$, that is
$z=h_{m^\prime-l^\prime+1}^\prime\ldots h_{m^\prime}^\prime g_2^\prime\tilde{a}_{m^\prime-l^\prime}^\prime$ and 
$w=h_{m^\prime-l^\prime+2}^\prime\ldots h_{m^\prime}^\prime g_2^\prime\tilde{a}_{m^\prime-l^\prime+1}^\prime$.

Note that all simplices in $\widetilde{K\cup L}$ project to simplices in $K$ or in $L$. Assume that $\tilde{\tau}$ projects to $K$. By the discussion preceding observation (A),
this means that 
$h_{m-l+1}$ and $h_{m^\prime-l^\prime+1}^\prime$ belong both to $\pi_1K-\pi_1A$.
It follows that $h_{m-l},h_{m-l+2},h_{m^\prime-l^\prime}^\prime,h_{m^\prime-l^\prime+2}^\prime$ belong to $\pi_1L-\pi_1A$. But this implies, for example, that  
$\left[x,z\right]$ is part of some minimizing path, 
namely the path $$\left\{e_0,\ldots,e_{l-1},\left[x,z\right],e_{l^\prime-1}^\prime,\ldots,e_0^\prime\right\}.$$

In a similar way, we see that
[x,w], [y,z] and [y,w] belong to the intersection of $\tilde{\tau}$ with minimizing pathes between suitable pairs of $v_i$'s.\\
The same argument works if $\tau$ projects to $L$.\\

{\bf 'Central' simplices:}
 
We are given vertices $v_0=g_0\tilde{w_0},\ldots,v_n=g_n\tilde{w_n}\in\tilde{M}_0$, $n\ge 2$.\\
{\em We claim:} if we fix, for each index pair $\left\{i,j\right\}$, a normal form
$g_ig_j^{-1}=h_1^{ij}\ldots h_{m_{ij}}^{ij}$,\\
\hspace*{0.7in}vertices $a_0^{ij},\ldots,a_{m_{ij}}^{ij}\in A_0$, and the minimizing path $p_{ij}\in P\left(i,j\right)$,\\
\hspace*{0.7in}then there is {\em at most one} 
{\bf n-dimensional} simplex $\tilde{\tau}\in \widetilde{K\cup L}$ \\
\hspace*{0.7in}such that the intersection of $\tilde{\tau}$
with $\cup_{0\le i,j\le n}p_{ij}$ is the 1-skeleton\\
\hspace*{0.7in}of an {\bf n-dimensional} simplex,
i.e., is the full 1-skeleton of $\tilde{\tau}$.\\
(In fact, for such an n-simplex to exist, the $h_{ij}$'s as well as the $a_{ij}$'s have to satisfy obvious compatibilty conditions. We will not make use of these explicit conditions in our proof.)

We prove the claim.
Assume there are                               
two such simplices $\tilde{\tau}_1\not=\tilde{\tau}_2$ with $dim\left(\tilde{\tau}_1\right)=dim\left(\tilde{\tau}_2\right)=n$. 
By assumption (vi), it suffices to show that $\tilde{\tau}_1$ and $\tilde{\tau}_2$ have the same 1-skeleta. 
We have to distinguish the cases that $\tilde{\tau}_1$ and $\tilde{\tau}_2$ have no common vertex or that they have a common subsimplex.

Assume first that $\tilde{\tau}_1$ and $\tilde{\tau}_2$ have no vertex in common. 
In the following, 'minimizing path' will mean the unique minimizing path with respect to our fixed choice of normal forms and of vertices in $A$. 
We will frequently use the following fact: each edge of $\tilde{\tau}_1$ (resp.\ $\tilde{\tau}_2$) is contained in the minimizing path from $v_i$ to $v_j$ for a unique pair 
$\left\{i,j\right\}$ of indices. This is true by a counting argument: there are 
$\frac{n\left(n+1\right)}{2}$ minimizing pathes and $\frac{n\left(n+1\right)}{2}$ edges of $\tilde{\tau}_1$, each edge belongs to some minimizing path by assumption, and no minimizing path can have two consecutive edges projecting both to $K$ or both to $L$, by the 
definition of normal forms.\\
For each $k$ and $l$, the minimizing path from $v_k$ to $v_l$ passes through $\tilde{\tau}_1$ as well as
through $\tilde{\tau}_2$. 
Let $\left[w_k^1,w_l^1\right]$ resp.\ $\left[w_k^2,w_l^2\right]$ be the 
intersections of this minimizing path with $\tilde{\tau}_1$ resp.\ $\tilde{\tau}_2$.\\
We claim: all minimizing pathes from $v_k$ to some $v_i$ with $i\not=k$ pass through $w_k^1$ {\em and} $w_k^2$.
To prove the claim, note that, by (C1), the subpath from $v_k$ to $w_k^1$ is minimizing,
that is, the corresponding sequence $h_1,\ldots,h_m$ is a normal form (for $\Pi_{i=1}^mh_i$) with 
$h_m\in\pi_1L$ if $\tau_1$ projects to $K$ or vice versa. It follows from the definition of normal forms that also $\Pi_{i=1}^{m+1}h_i$ is a normal form if $h_{m+1}\in\pi_1K$ is the element corresponding to an edge of $K$ having $w_k^1$ as a vertex. Since normal forms are unique up to multiplication of the $h_i$'s with elements of $\pi_1A$, we conclude that there is no minimizing path
from $v_k$ to some vertex of $\tau_1$ which does not pass through $w_k^1$.
By the same argument,
all minimizing pathes from $v_k$ to some vertex of $\tau_2$ have to pass through $w_k^2$. In particular, they have to contain the unique minimizing path from $w_k^1$ to $w_k^2$ as a 
subpath, by (C1). This, in turn, implies that all minimizing pathes from $v_k$ to any $v_i, i\not=k$, contain the minimizing path from $w_k^1$ to $w_k^2$ and, in particular, contain the same edge
of $\tilde{\tau}_1$. But, by the above counting argument, it may not happen that several minimizing pathes
pass through the same edge of $\tilde{\tau}_1$. 
This gives the contradiction.
$$\pspicture(0,0)(8,2)
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\uput[0](1.7,1.3){$w_k^1$}
\uput[0](2.7,1.3){$w_l^1$}
\uput[0](1.9,0.6){$\tilde{\tau}_1$}
\uput[0](5.3,0.8){$\tilde{\tau}_2$}
\uput[0](4.4,1.3){$w_k^2$}
\uput[0](-0.8,2){$v_k$}
\uput[0](8,1){$v_l$}
\uput[0](7,0){$v_i$}
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It remains to discuss the case that $\tilde{\tau}_1$ and $\tilde{\tau}_2$ have a proper boundary face in common. 
Let $w_k$ be a vertex of $\tau_1$ which is not a vertex of $\tau_2$ and $w_l$
a vertex of both, $\tau_1$ and $\tau_2$.
After reindexing, there are, by the above counting argument, $v_k$ and $v_l$ 
such that the minimizing path from $v_k$ to $v_l$ contains the edge $\left[w_k,w_l\right]\subset \tau_1$.
By the same argument as above, all minimizing pathes from $v_k$ to any 
$v_i$ pass through $w_k$ as well as through $w_l$, that is, they all
contain the edge $\left[w_k,w_l\right]$ giving a contradiction.
%Let again $0\le m\le n$ such that $w_m^1$ and $w_m^2$ do not belong to this common face. Removing this common face from $P$, gives a disconnected multicomplex with components $B$ and $C$.
%An argument
%completely analogous to the first case allows to conclude that $v_m\in B$ and $v_m\in C$, yielding a contradiction.\\
This finishes the proof of the claim. Since it will be used again in the proof that $r$ is a chain map, we write down the following observation, which we have just proved.\\
(D): {\em If $v_0,\ldots,v_n\in\widetilde{M}_0$ are vertices of $M$, then their central simplex $\tilde{\tau}$ to 

some choice of $\left\{h_k^{ij}\right\},\left\{a_l^{ij}\right\}$, if it exists, has vertices $w_0,\ldots,w_n$ such that 

for 
any $i$ and $j$ the minimizing path from $v_i$ to $v_j$ passes through $w_i$
and $w_j$.}\\
\\
We denote by $\tau\left(\left\{a_l^{ij}\right\};\left\{h_k^{ij}\right\}\right)$ the projection of $\tilde{\tau}$ to $K\cup L$.\\
Next we claim: if we still are given $v_0=g_0\tilde{w_0},\ldots,v_n=g_n\tilde{w_n}\in\tilde{M}_0$, but vary $a_0,\ldots,a_m\in A_0$ and the normal forms $g_ig_j^{-1}=h_1^{ij}\ldots h_{m_{ij}}^{ij}$,
then all $\tau\left(\left\{a_l^{ij}\right\};\left\{h_k^{ij}\right\}\right)$ belong to the same $G$-orbit.

Let us consider $\tau\left(\left\{a_l\right\};\left\{h_k^{ij}\right\}\right)$
and $\tau\left(\left\{a_l^\prime\right\};\left\{h_k^{ij}\right\}\right)$. 
The same argument which showed uniqueness of $\tilde{\tau}\left(\left\{a_l\right\};\left\{h_k^{ij}\right\}\right)$ lets us conclude that,
representing the vertices of $\tilde{\tau}\left(\left\{a_l\right\};\left\{h_k^{ij}\right\}\right)$ as $\gamma_0\tilde{a}_0,\ldots,\gamma_n\tilde{a}_n$ and the vertices of $\tilde{\tau}\left(\left\{a_l^\prime\right\};\left\{h_k^{ij}\right\}\right)$ as $\gamma_0^\prime\tilde{a_0}^\prime,
\ldots,\gamma_n^\prime\tilde{a_n}^\prime$, we must have $\gamma_0=\gamma_0^\prime,\ldots,\gamma_n=\gamma_n^\prime$. Hence, 
corresponding edges of $\tau$ are $A$-related 
in the sense of definition 4
and belong, by assumption (viii), to the same $G$-orbit.

Now we consider $\tau\left(\left\{a_l\right\};\left\{h_k^{ij}\right\}\right)$
and $\tau\left(\left\{a_l\right\};\left\{h_k^{ij\prime}
\right\}\right)$, where $g_ig_j^{-1}=h_1^{ij}\ldots h_m^{ij}=h_1^{ij\prime}\ldots h_m^{ij\prime}$ are different normal forms. Since we may argue succesively, it suffices to consider the case that different normal forms occur for only one index pair $i,j$. 
%We will henceforth omit the index $\left\{i,j\right\}$.
For the same reason, it suffices to consider the case that there is $1\le s\le m-1$ such that $h_s^{ij\prime}=h_s^{ij}a^{-1}, h_{s+1}^{ij\prime}=ah_{s+1}^{ij}$ and $h_l^{ij\prime}=h_l^{ij}$ otherwise. Now, from the above construction,
it follows that one of the following two possibilities takes place:\\
- if 
$h_l^{ij}\ldots h_m^{ij}g_2^{ij}\tilde{a}_{l-1}^{ij}$ and $h_{l+1}^{ij}\ldots h_m^{ij}g_2^{ij}\tilde{a}_l^{ij}$
are not vertices of $\tilde{\tau}\left(\left\{a_l^{ij\prime}\right\};\left\{h_k^{ij}\right\}\right)$, then $\tilde{\tau}\left(\left\{a_l^{ij\prime}\right\};\left\{h_k^{ij\prime}\right\}\right)=\tilde{\tau}\left(\left\{a_l^{ij\prime}\right\};\left\{h_k^{ij}\right\}\right)$.\\
- if one resp.\ both of $h_l^{ij}\ldots h_m^{ij}g_2^{ij}\tilde{a}_{l-1}^{ij}$ and $h_{l+1}^{ij}\ldots h_m^{ij}g_2^{ij}\tilde{a}_l^{ij}$ are vertices of 
$\tilde{\tau}\left(\left\{a_l^{ij}\right\};\left\{h_k^{ij}\right\}
\right)$, then  $\tilde{\tau}\left(\left\{a_l^{ij}\right\};\left\{h_k^{ij\prime}\right\}\right)$ has $n$ resp.\ $n-1$ vertices in common with 
$\tilde{\tau}\left(\left\{a_l^{ij}\right\};\left\{h_k^{ij}\right\}
\right)$, and has moreover as remaining vertices one or both 
of $h_l^{ij}\ldots h_m^{ij}g_2^{ij}\tilde{a}_{l-1}^{ij}$ and $h_{l+1}^{ij}\ldots h_m^{ij}g_2^{ij}\tilde{a}_l^{ij}$. \\
It follows that the 1-skeleta of 
$\tilde{\tau}\left(\left\{a_l^{ij}\right\};\left\{h_k^{ij}\right\}
\right)$ and $\tilde{\tau}\left(\left\{a_l^{ij}\right\};\left\{h_k^{ij\prime}\right\}
\right)$, considered as tuples of 1-simplices, are $A$-related in the sense of definition 4. Indeed, that the concatenations are trivial in $\pi_1\left(K\cup L\right)$ follows from the assumption that $K$ and $L$ are aspherical in the sense of definition 3, which forces the concatenations to bound a 2-simplex resp.\ a union of two 2-simplices. By assumption (viii), we get that $\tau\left(\left\{a_l\right\};\left\{h_k^{ij}\right\}\right)$
and $\tau\left(\left\{a_l\right\};\left\{h_k^{ij\prime}
\right\}\right)$ belong to the same $G$-orbit.\\

{\bf Retraction in treelike complex:}

We are going to define $r$. Given an n-simplex $\sigma\in M$, $n\ge2$, we consider a lift 
$\tilde{\sigma}\in\tilde{M}$ which projects to $\sigma$. Its vertices
$v_0,\ldots v_n$ belong to $\widetilde{M}_0=
\left(\widetilde{K\cup L}\right)_0$.\\
Now we run the above constructions with $v_0,\ldots,v_n$ and look for simplices $\tau$ with $dim\left(\tau\right)=n$.
 
If there is no simplex $\tau$ with $dim\left(\tau\right)=n$, we define $r\left(\sigma\right)=0$.
Otherwise, there is a unique $G$-orbit
of simplices $\left\{g\tau:g\in G\right\}$ with
$dim\left(g\tau\right)=n$ for all $g\in G$, and we define $r\left(G\sigma\right):=G\tau$.
 
We have to check that the definition of $r$ does not depend on the choice of $\tilde{\sigma}$ neither on the choice of $\sigma$ in its $G$-orbit. 

Observe the following: if $\tilde{f}:\tilde{M}\rightarrow \tilde{M}$ is a simplicial self-map of the universal cover, such that $\pi\tilde{f}$ maps $K$ to $K$ and
$L$ to $L$, then $\tilde{f}$ 
maps minimizing pathes from $v_i$ to $v_j$ to minimizing pathes from 
$\tilde{f}\left(v_i\right)$ to $\tilde{f}\left(v_j\right)$. Hence, simplices $\tilde{\tau}$ intersecting the
family of minimizing pathes associated to $v_0,\ldots,v_n$ in the full 1-skeleton of an $n$-simplex are mapped by $\tilde{f}$ to simplices $\tilde{f}\left(\tilde{\tau}\right)$ intersecting the
family of minimizing pathes associated to $\tilde{f}\left(v_0\right),\ldots,\tilde{f}\left(v_n\right)$ in the full 1-skeleton of an $n$-simplex.
Thus, if $\tilde{\tau}$ belongs to the family of 'central' simplices associated to some simplex
$\tilde{\sigma}$, then $\tilde{f}\left(\tilde{\tau}\right)$  belongs to the family of 'central' simplices associated to
$\tilde{f}\left(\tilde{\sigma}\right)$. \\
We conclude: $r\left(G\sigma\right)$ does not depend on the choice of $\sigma$ in its $G$-orbit,
 neither on the choice of $\tilde{\sigma}$, for fixed $\sigma$, in the orbit of the deck group.

Finally, the two desired conditions are clearly satisfied: \\
we have constructed $\tilde{\tau}$ with the help of the condition that it is a 
simplex in $\widetilde{K\cup L}$, hence $r\left(\sigma\right)=\tau$ is a simplex either in $K$ or in $L$. If $\sigma$ is a simplex in $K$ or $L$, then clearly 
$r\left(\sigma\right)=\sigma$, hence $r$ is leftinverse to $i_{K*}$ and
$i_{L*}$.\\

{\bf Compatibility with $\partial$-operator:}
 
It remains to show that $r$ is a chain map, i.e., that $\partial r\left(G\sigma\right)=r\left(G
\partial\sigma\right)$ holds for all simplices $\sigma$ in $M$.
 
First we consider the case $r\left(G\sigma\right)\not=0$.\\
Let $v_0,\ldots,v_n$ be the vertices of a lift $\tilde{\sigma}$, and $\left\{h_k^{ij}\right\}$, $\left\{a_l^{ij}\right\}$ such that a central simplex $\tilde{\tau}$ exists.
It is obvious, if we consider the set of vertices without $v_k$ and the corresponding restricted sets of normal forms and vertices, that we get the $k$-th face of $\tilde{\tau}$ as a central simplex.
That implies $r\left(G\partial_k\sigma\right)=\partial_kr\left(G\sigma\right)$ for all $k$, and hence $r\left(G\partial\sigma\right)=\partial r\left(G\sigma\right)$.
%$w_0,\ldots,w_n$ the vertices of some $\tau$ with $s\left(\tau\right)=n$, labelled such that
%$w_i$ is the vertex of shortest distance to $v_i$, and denote
%$\left(w_i,w_j\right)^0$ the unique 1-simplex with boundary points $w_i$ and $w_j$ such that $\left(w_i,w_j\right)^0$
%is a subsimplex of $\tau$. It follows that the 1-simplex
%$\left(w_i,w_j\right)^0$ is part of some $r_{ij}^0$, a minimizing path from $v_i$ to $v_j$. Let $\partial_k\tau$ be the unique (n-1)-simplex whose edges are $\left\{\left(w_i,w_j\right)^0: i\not=k, j\not= k\right\}$.
%The set of pathes $\left\{r_{ij}^0\right\}_{i,j\not=k}$ intersects $\partial_k\tau$ in the full 1-skeleton of the
%(n-1)-simplex                                                      
%$\partial_k\tau$, hence $\partial_k\tau\left(v_0,\ldots,v_n\right)$ is actually one of
%the central (n-1)-simplices corresponding to $v_0,\ldots,v_{k-1},v_{k+1},\ldots,v_n$.\\
%Since there are $3^n$ central simplices to $v_0,\ldots,v_n$, but $3^{n-1}$ central
%simplices to each (n-1)-tupel $v_0,\ldots,v_{k-1},v_{k+1},\ldots,v_n$, the
%factors match to get 
%$r\left(\partial\sigma\right)=\partial r\left(\sigma\right)$.
 
We consider now the case $r\left(G\sigma\right)=0$.\\
If $r\left(G\partial_k\sigma\right)=0$ for all faces $\partial_k\sigma$ of $\sigma$, we conclude $r\left(G\partial\sigma\right)=0$.\\
So assume that for some face $\partial_k\sigma$ of $\sigma$ we have
$r\left(G\partial_k\sigma\right)=
G\tau$ for some (n-1)-simplex $\tau$.
That means that, for the vertices $v_0,\ldots,v_{k-1},v_{k+1},
\ldots,v_n$ and some choice of $\left\{h_m^{ij}\right\},\left\{a_l^{ij}\right\}$ we have the central simplex $\tilde{\tau}$. Here, the $h_m^{ij}$ and $a_l^{ij}$ are only chosen for $i\not=k,j\not=k$. By observation (D),
$\tilde{\tau}$ has vertices $w_0,\ldots,w_{k-1},w_k,\ldots,w_n$
such that the minimizing path from $v_i$ to $w_j$ passes through $w_i$ for all 
$i\not=k\not=j$. \\
For $i\not=k$ consider the set $P_{ik}^\prime$ of the minimizing pathes 
from $v_k$ to $w_i$. There exists some vertex $v$,  with the property that 
for each $i$ exists a path in $P_{ik}^\prime$ containing $\left[v,w_i\right]$ as last edge.
If $v$ were not one of $w_0,\ldots,w_{k-1},w_{k+1},\ldots,w_n$, then this would give us a central simplex to the vertices 
$v_0,\ldots,v_{k-1},v_k,v_{k+1},\ldots,v_n$, contradicting the assumption.
Hence $v=w_j$ for some $j$. But this implies that
%Considering the union $\mathcal{P}$
%of the union of minimizing pathes from $v_i$ to $v_j$ ($i\not=k\not=j$) with the union of the minimizing pathes from $v_k$ to $w_i$, we get that
% $v_0,\ldots,v_{k-1},v_{k+1},
%\ldots,v_n$ belong to pairwise distinct connected components of the complement of
%$\tilde{\tau}$ in $\mathcal{P}$.
%However, $v_k$ must be in the same connected component as some $v_i$, because otherwise 
%there would exist a central n-simplex to $v_0,\ldots,v_n$.  Hence,
the central simplices to $v_0,\ldots,v_{k-1},v_{k+1},
\ldots,v_n$ are exactly the central simplices to
$v_0,\ldots,v_{j-1},v_{j+1},
\ldots,v_n$, and the corresponding simplices have the same orientation if and only if
$j-k$ is {\em odd}. Therefore, $r\left(\partial_k\sigma\right)$ cancels against $r\left(\partial_j\sigma\right)$. As we find such a $j$ for any $k$ with
$r\left(\partial_k\sigma\right)\not=0$, we get
$r\left(\partial \sigma\right)=0$.\\
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\uput[0](1.8,1.7){$\tilde{\tau}\left(v_0,v_1,v_2\right)$}
\uput[0](1.8,1.3){$=\tilde{\tau}\left(v_0,v_1,v_3\right)$}
\uput[0](6,0.8){$\tilde{\tau}\left(v_0,v_2,v_3\right)$}
\uput[0](6,0.4){$=\tilde{\tau}\left(v_1,v_2,v_3\right)$}
\uput[0](-0.5,2){$v_0$}
\uput[0](0.5,0){$v_1$}
\uput[0](8,1){$v_2$}
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\endpspicture$$

{\bf Relative construction:}
 
Finally, we consider the general case $M^\prime\not=\emptyset$. For the
universal
cover $\pi:\tilde{M}\rightarrow M$ denote the subcomplex
$\tilde{M}^\prime:=\pi^{-1}\left(M^\prime\right)$.
Since $A\subset M^\prime$, the definition of 'minimizing pathes' implies that, for $x,y\in \tilde{M}^\prime_0$, all minimizing pathes from $x$ to $y$ are subsets of 
$\tilde{M}^\prime$. As a consequence, for $v_0,\ldots,v_n\in \tilde{M}^\prime_0$, any central simplex, if it exists, must belong to $\tilde{M}^\prime$. Hence,
$\tau\subset M^\prime\cap K=K^\prime$ or $\tau\subset M^\prime\cap L=L^\prime$.
This proves that $r$ induces a chain map $r:C_*\left(M,M^\prime\right)
\rightarrow C_*\left(K,K^\prime\right)\oplus C_*\left(L,L^\prime\right)$, finishing the proof of lemma 5. \end{pf}   
\begin{cor}: Let $\left(M,M^\prime\right),\left(K,K^\prime\right),\left(L,L^\prime\right)$ 
satisfy all assumptions of lemma 5. Let
$\gamma_1\in H_b^p\left(K,K^\prime\right), \gamma_2\in
H_b^p\left(L,L^\prime\right)$ be
bounded cohomology classes, $p\ge 3$, such that the action of $G$ fixes $\gamma_1$ and $\gamma_2$. Then exists a class
$\gamma\in H^p_b\left(M,M^\prime
\right)$ satisfying
$\parallel \gamma\parallel\le max\left\{\parallel\gamma_1\parallel,
\parallel\gamma_2\parallel\right\}$, such that the restrictions to
$K$ and $L$ give back $\gamma_1$ and $\gamma_2$ and that $\gamma$ is fixed by $G$.\end{cor}
\begin{pf} Let $c_1$ resp.\ $c_2$ be bounded cocycles representing $\gamma_1$ resp.\ $\gamma_2$.
To define a bounded cocycle $c$, it suffices to define its value on simplices.
So let $\sigma$ be a simplex in $M$. Define:\\
- $c\left(\sigma\right):=c_1\left(r\left(\sigma\right)\right)$ if $r\left(G\sigma\right)\in G\backslash C_*\left(K\right)$,\\
- $c\left(\sigma\right):=c_2\left(r\left(\sigma\right)\right)$ if $r\left(G\sigma\right)\in G\backslash C_*\left(L\right)$,\\
- $c\left(\sigma\right):=0$ else.\\
If $p\ge3$, we get an induced map in bounded cohomology. Indeed, let
$c_1-\tilde{c_1}=\delta b_1$,\\
$c_2-\tilde{c_2}=\delta b_2$ for (bounded) relative (p-1)-cochains
$b_1,b_2$ and define again $b\left(\tau\right)$\\
as $b_1\left(r\left(\tau\right)\right), b_2\left(r\left(\tau\right)\right)$ or 
0 according to whether $r\left(\tau\right)$ is in $G\backslash C_*\left(K\right)$,
in $G\backslash C_*\left(L\right)$ or 0. (This definition would not work for $p=2$.) An
obvious calculation yields $c-\tilde{c}=\delta b$.\\
All claims of the corollary are obvious except possibly that $c$ is indeed a relative cocycle, i.e.,
that $\delta c$ vanishes on (p+1)-simplices in $M^\prime$. Assume
w.l.o.g. that $r\left(\sigma\right)\in C_*\left(K^\prime\right)$. Then $c_1\left(\partial r\left(\sigma\right)\right)=0$ because $c_1$ is a relative cocycle and, therefore
$\delta c\left(\sigma\right)=c\left(\partial\sigma\right)=c_1\left(r\left(\partial\sigma\right)\right)
=c_1\left(\partial r\left(\sigma\right)\right)=0$.\end{pf}                               

\subsection{The 'HNN'-case}


\begin{lem}: Assume that $ \left(K,K^\prime\right)$
is a pair of path-connected, minimally complete multicomplexes, and that $A_1, A_2$ are disjoint minimally complete submulticomplexes of $K^\prime$ such that there exists a simplicial isomorphism $F:A_1\rightarrow A_2$.
Let $\left(L,L^\prime\right)$ be the pair of multicomplexes obtained from $\left(K,K^\prime\right)$ by identifying $\sigma$ and $F\left(\sigma\right)$ for all simplices $\sigma$ in $A_1$ and let $P:\left(K,K^\prime\right)\rightarrow\left(L,L^\prime\right)$ be the canonical projection.\\
Moreover let $\left(M,M^\prime\right)$ be a pair of multicomplexes. Assume that\\
(i) $L$ is a submulticomplex of $M$ with inclusion $i_L:L\rightarrow M$,\\
(ii) $M_0=L_0$,\\ 
(iii) $\pi_1A_1\rightarrow \pi_1K$ and $\pi_1A_2\rightarrow\pi_1K$ are injective,\\
(iv) the inclusion $L\rightarrow M$ induces an isomorphism $\pi_1 L\rightarrow\pi_1M$,\\
(v) $K$ and $L$ are aspherical in the sense of definition 3,\\
(vi)$L^\prime=M^\prime\cap L$.\\
Assume moreover that a group $G$ acts simplicially on $\left(M,M^\prime\right)$ as well as on $\left(K,K^\prime\right)$ such that \\
(vii) the action of $G$ commutes with $i_LP$, as well as with $F$,\\
(viii) $G$ acts transitively on $j$-simplices of $A_1$, for any $j\ge 0$.\\
Then there is a relative chain map $r:G\backslash C_*\left(M,M^\prime\right)\rightarrow
G\backslash C_*\left(K,K^\prime\right)$ in degrees $*\ge2$ such that\\
- if $G\sigma$ is the $G$-orbit of a simplex in $M$, then $r\left(G\sigma\right)$ is the $G$-orbit of a simplex in $K$,\\
- $r{i_L}_*P_*=id_{G\backslash C_*\left(K,K^\prime\right)}$.\end{lem}

\begin{pf}
The proof of lemma 6 parallels in several aspects the proof of lemma 5. We 
will then be somewhat briefer in the explanations.\\
By (iv), the universal covering $\tilde{L}$ is a submulticomplex of $\tilde{M}$. By (ii) and (iv), $\tilde{L}_0=\tilde{M}_0$.\\
Denote $A$ the image of $A_1$ (or $A_2$) in $L\subset M$. Fix a vertex $p\in A$ and some lift $\tilde{p}$. A vertex $v\in L_0$ is the image of some vertex of $K$, to be denoted $v$ by abuse of notation.
There is some edge in $K$ with boundary points $p$ and $v$, because $K$ is complete. 
Its image under the projection $P$ is an edge in $L$ with boundary points $p$ and $v$.
There are unique lifts $\tilde{e}\in \tilde{L}_1$ and $\tilde{v}\in\tilde{L}_0$ such that $\tilde{e}$ has boundary points $\tilde{p}$ and $\tilde{v}$. If $v$ happens to be in $A$, we construct $\tilde{e}$ and $\tilde{v}$ by choosing an edge $e$ which remains in $A$. This is possible because $A$ is complete.  \\
Recall that $\pi_1L$ is an HNN-extension of
$\pi_1K$. We consider $\pi_1K$ as subgroup of $\pi_1L$ and denote by $t$ the extending element of $$\pi_1L=<\pi_1K,t\mid t^{-1}at=F_*a\ \ \forall a\in\pi_1A>.$$
If $e\in L_1$ is an edge with boundary points $v$ and $w$, we fix 
the unique lift $\tilde{e}\in\tilde{L}_1$ with 0-th vertex $\tilde{v}$. It will be 
crucial that the 1-th vertex of $\tilde{e}$ is then necessarily of the form $g\tilde{w}$ for some $w\in L_0$ with either $g\in \pi_1K$ or $g=t$. This is true because: the edge $f$ with vertices 
$\tilde{w}$ and $g\tilde{w}$ projects to a closed edge 
in $L$, which is either the image of a closed edge in $K$ or the 
image of an edge in $K$ with endpoints $b\in A_1,c\in A_2$ 
such that $F\left(b\right)=c$. In the first case, $g\in\pi_1K$, in 
the second case $g=t$.\\

We have defined a map $\sim:L^1\rightarrow \tilde{L}^1$, satisfying

(A): {\em if $g\tilde{e}$ is an edge with boundary points $h_0\tilde{v_0}$ and $h_1\tilde{v_1}$, then $g=h_0$ and $h_1h_0^{-1}$ is either 1 or is 
an element of $\pi_1K$ or equals $t$.}\\
Moreover, if $g_1\tilde{e_1}$ is a 1-simplex with boundary points $h_{01}\tilde{v}_{01}$ and $h_{11}\tilde{v}_{11}$,
and $g_2\tilde{e_2}$ is a 1-simplex with boundary points $h_{02}\tilde{v}_{02}$ and $h_{12}\tilde{v}_{12}$, then:
 
(B): {\em if $g_1\tilde{e_1}$ and $g_2\tilde{e_2}$ have a common boundary point $h\tilde{v}=h_{11}\tilde{v}_{11}=h_{02}\tilde{v}_{02}$,
 
then one of the following two possibilities holds:
 
- $v\in A$ or
 
- $h_{01}^{-1}h_{11}$ and $h_{02}^{-1}h_{12}$ belong both to $\pi_1K$ or equal both $t$.}\\

{\bf Minimizing pathes:}

Given two vertices $v_0,v_1\in\tilde{M}_0=\tilde{L}_0$, we may represent them
as $v_1=g_1\tilde{w_1}, v_2=g_2\tilde{w}_2$ with $g_i\in\pi_1 L$ and $w_i\in M_0$ for $i=1,2$. $\pi_1L$ is an HNN-extension of $\pi_1K$, hence $g_1g_2^{-1}$
has an expression $g_1g_2^{-1}=h_1\ldots h_m$
with $h_i\in\pi_1K$ or $h_i\in t$ such that $h_i\in\pi_1K$ implies $h_{i+1}\not\in\pi_1K$. (But we allow $h_i=h_{i+1}=t$.)
This expression, which we will call a {\bf normal form},
is unique up to compatible changes of
the $h_i\in\pi_1K$ in their equivalence class 
modulo $\pi_1 A$.

We call a path $e_1,\ldots,e_r$ {\bf minimizing} if 
there are $a_0,\ldots,a_m\in A$ and a normal form $g_1g_2^{-1}=h_1\ldots h_m$ such that\\
- $e_0$ has vertices $g_2\tilde{w_2}$ and $g_2\tilde{a}$,\\
- $e_i$ has vertices  $h_{m-i+1}\ldots h_{m}g_2\tilde{a}$ and $h_{m-i+2}\ldots h_mg_2\tilde{a}$ for $i=1,\ldots,m$,\\
- $e_{m+1}$ has vertices $g_1\tilde{w_1}$ and $g_1\tilde{a}$.\\
One should note that the above edges exist in $\widetilde{L}$, since $K$ is complete.
The construction should be understood that we skip $e_0$ resp.\ $e_{m+1}$ if $w_2\in A$ resp.\ $w_1\in A$.\\
It follows from (A) and (B), that these pathes are length-minimizing in the sense of being exactly the pathes between $v_1$ and $v_2$ with a minimum number of edges.
Since this latter characterisation depends only on $v_1$ and $v_2$, we conclude: for different sections, there is a bijection between the corresponding sets of minimizing pathes from $v_1$ to $v_2$. 

Since 
there is at most one edge
between two vertices, a minimizing path
becomes uniquely determined after fixing its vertices.
So the only freedom in the choice of the minimizing path
consists \\
- in the choice of $a_0,\ldots,a_m$, and\\
- in the choice of the $h_i$ in their equivalence class modulo $\pi_1A$.\\
The unique path corresponding to such a choice will be denoted
$p\left(a_0,\ldots,a_m;h_1,\ldots,h_m\right)$.\\

{\bf Intersection with simplices:}

Let $v_0,\ldots,v_n$ be vertices of $\tilde{L}$.
Defining $P\left(i,j\right)$ like in the proof of lemma 5, 
we want to check that, for any simplex $\tau$ in $\tilde{ L}$,
$\left\{\tau\cap r_{ij}^k: r_{ij}^k\in P\left(i,j\right)\right\}$ is the full 1-skeleton of some subsimplex of $\tau$.\\
Assume that $\left[x,y\right]$ and $\left[z,w\right]$ are edges of $\tau$,
with 
$x=h_{m-l+1}\ldots h_mg_2\tilde{a}$, $y=h_{m-l+2}\ldots h_mg_2\tilde{a}$, 
$z=h_{m^\prime-l^\prime+1}^\prime\ldots h_{m^\prime}^\prime g_2^\prime\tilde{a}^\prime$ and
$w=h_{m^\prime-l^\prime+2}^\prime\ldots h_{m^\prime}^\prime g_2^\prime\tilde{a}^\prime$.  
By the discussion preceding observation (A), we get that $h_{m-l+2}$ and $h_{m^\prime-l^\prime+2}^\prime$ either belong both to $\pi_1K-\pi_1A$ or are both equal to $t$, and that the other of these two posibilities must hold true for $h_{m-l+1},h_{m-l+3},h_{m^\prime-l^\prime+1}^\prime,h_{m^\prime-l^\prime+3}^\prime$. This implies that $\left[x,z\right]$ is part of the minimizing path $e_0,\ldots,e_l,\left[x,z\right],e_{l^\prime}^\prime,\ldots,e_0^\prime$, similarly for the other edges.\\

{\bf 'Central' simplices:}

We are given vertices $v_0=g_0\tilde{w_0},\ldots,v_n=g_n\tilde{w_n}\in\tilde{M}_0$, $n\ge 2$.\\
We fix $a_0,\ldots,a_m\in A_0$ and normal forms $g_ig_j^{-1}=h_1^{ij}\ldots h_{m_{ij}}^{ij}$. Hence, we have unique minimizing pathes $p_{ij}$ from $v_i$ to $v_j$.
Then there is at most one {\bf n-dimensional} simplex $\tilde{\tau}\in \widetilde{ L}$ such that the intersection of $\tilde{\tau}$ with $\cup_{0\le i,j\le n}p_{ij}$ is
the 1-skeleton of an {\bf n-dimensional} simplex, i.e., is the full 1-skeleton of $\tilde{\tau}$.
This is proved by literally the same argument as in the corresponding part of the proof of lemma 5, to which we refer.\\
We point out that the edges of $\tilde{\tau}$ are of the form $\left[h_0\tilde{w}_0,h_1\tilde{w}_1\right]$ with $h_1h_0^{-1}\in\pi_1K$. Indeed, the definition of minimizing pathes implies that either $h_1h_0^{-1}\in\pi_1K$ or $h_1h_0^{-1}=t$. But the latter case would contradict the assumption $n\ge 2$, because then $\left[h_0\tilde{w}_0,th_0\tilde{w}_0\right]$ would be the only edge in $\tilde{\tau}$ having $h_0\tilde{w}_0$ as a 
vertex.\\ 
We consider the projection $\tau^\prime$ of $\tilde{\tau}$ to $L$. By construction, the edges of
$\tau^\prime$ are projections of 1-simplices of $K$. Assumption (v) implies then that $\tau^\prime$ is the projection of some simplex $\tau\in K$.
\\
The resulting simplex in $K$ will be denoted as
$\tau\left(\left\{a_l^{ij}\right\};\left\{h_k^{ij}\right\}\right)$. Literally the same argument as in the proof of lemma 5 shows:
if we fix 
$v_0,\ldots,v_n$, but  vary $a_0,\ldots,a_m\in A_0$ and the normal forms $g_ig_j^{-1}=h_1^{ij}\ldots h_{m_{ij}}^{ij}$,then all $\tau\left(\left\{a_l^{ij}\right\};\left\{h_k^{ij}\right\}\right)$ belong to the same $G$-orbit.\\

{\bf Retraction in treelike complex:}

Given an $n$-simplex $\sigma\in M$, we consider a lift $\tilde{\sigma}\in\tilde{M}$ with vertices $v_0,\ldots,v_n$, and we let $r\left(G\sigma\right)=G\tau$
if the above construction gives the $G$-orbit of a simplex $\tau$ with $dim\left(\tau\right)=n$,
otherwise we define $r\left(G\sigma\right)=0$.
The definition of $r$ does neither depend on the choice of the lift $\tilde{\sigma}$ nor on the choice of $\sigma$ in its $G$-orbit, and it satisfies the two desired conditions.\\
The same argument as in the proof of lemma 5 shows that $r\left(G\partial\sigma\right)=G\partial r\left(G\sigma\right)$.\\
From the assumptions follows $A\subset M^\prime$. It is then clear from the definition that minimizing pathes between points of $M^\prime$ remain in $M^\prime$. Thus $r$ maps $C_*\left(M^\prime\right)$ to $C_*\left(K\cap P^{-1}M^\prime\right)=C_*\left(K^\prime\right)$.
\end{pf}\\
\\
In an analogous manner to corollary 2, we conclude
\begin{cor} 
 Let $\left(M,M^\prime\right),\left(K,K^\prime\right),\left(L,L^\prime\right)$
satisfy all assumptions of lemma 6. Denote $P$ the composition of the projection $K\rightarrow L$ with the inclusion $L\rightarrow M$.\\
Let
$\gamma_1\in H_b^p\left(K,K^\prime\right)$ be
a bounded cohomology class, $p\ge 3$, such that the action 
of $G$ fixes $\gamma_1$. Then exists a class
$\gamma\in H^p_b\left(M,M^\prime
\right)$, satisfying
$\parallel \gamma\parallel\le \parallel\gamma_1\parallel$,
such that $P^*\gamma=\gamma_1$
and that $\gamma$ is fixed by $G$.\end{cor}           

\section{Glueing along amenable boundaries}
\subsection{Dualizing the problem}
\begin{lem}: (i): Let $M_1,M_2$ be two compact n-manifolds with boundary, $A_1,A_2$ connected
(n-1)-dimensional submanifolds of $\partial M_1$ resp.\ $\partial M_2$,
$f:A_1\rightarrow A_2$ a homeomorphism,
$M=\glu$ the glued
manifold, $A\subset M$ the image of the $A_i$, and $j_1:\left(M_1,\partial M_1\right)\rightarrow
\left(M,\partial M\cup A\right), j_2:\left(M_2,\partial M_2\right)\rightarrow\left(M,\partial M\cup A\right)$ the inclusions. \\
Assume that the
following holds: For all $\gamma_1\in H^n_b\left(M_1,
\partial M_1\right), \gamma_2\in H^n_b\left(M_2,
\partial M_2\right)$ one can find $\gamma\in H^n_b\left(
M,\partial M\cup A\right)$ such that $j_1^*\gamma=\gamma_1,
j_2^*\gamma=\gamma_2$ and $\parallel\gamma\parallel
\le max\left\{\parallel\gamma_1\parallel,\parallel\gamma_2\parallel\right\}$.\\
Then $\parallel M,\partial M\parallel\ge\parallel M_1, \partial M_1\parallel
+\parallel M_2,\partial M_2\parallel$.\\
(ii): Let $M_1$ be a compact n-manifold with boundary, $A_1,A_2$ disjoint connected (n-1)-dimensional
submanifolds of $\partial M_1$, $f:A_1\rightarrow A_2$ a homeomorphism,
$M=M_1/f$ the glued
manifold, $A\subset M$ the image of the $A_i$, and $P:\left(M_1,\partial M_1\right)
\rightarrow \left(M,\partial M\cup A\right)$ the canonical projection. \\
Assume that the
following holds: For all $\gamma_1\in H^n_b\left(M_1,
\partial M_1\right)$ one can find $\gamma\in H^n_b\left(
M,\partial M\cup A\right)$ such that $P^*\gamma=\gamma_1$
and $\parallel\gamma\parallel \le \parallel\gamma_1\parallel$.\\
Then $\parallel M,\partial M\parallel\ge\parallel M_1, \partial M_1\parallel
$.
\end{lem}
\begin{pf}
(i): First consider the case that $M_1$ and $M_2$ have nontrivial simplicial volume.
Then, by 1.2, the relative fundamental cocycles
have preimages $\beta_1\in H_b^n\left(M_1,\partial
M_1\right)$ and $\beta_2\in H_b^n\left(M_2,\partial
M_2\right)$. Consider for $i=1,2$
$$\gamma_i:=\parallel M_i,\partial M_i\parallel\beta_i$$
By 1.2, we have $\parallel\gamma_i\parallel=1$.
 
By assumption, we get
$\gamma
\in H_b^n\left(M,\partial M\cup A\right)$ satisfying
$\parallel\gamma\parallel\le
max\left\{\parallel\gamma_1\parallel,\parallel\gamma_2\parallel\right\}=1$
and $j_1^*\gamma=\gamma_1, j_2^*\gamma=
\gamma_2$.
 
Let $i:\left(M,\partial M\right)\rightarrow\left(M,\partial M\cup A\right)$ be the inclusion.
In $H_n\left(M,\partial M\cup A\right)$ we have $i_*\left[M,\partial M\right]={j_1}_*
\left[M_1,\partial M_1\right]+{j_2}_*\left[M_2,\partial M_2\right]$.
 
Hence, $$i^*\gamma\left(\left[M,\partial M\right]\right)=
\gamma_1\left(\left[M_1,\partial M_1\right]\right)+
\gamma_2\left(\left[M_2,\partial M_2\right]\right)=
\sima+\simb$$
Thus, $\beta=\frac{1}{\sima+\simb}i^*\gamma$
is the relative fundamental cocycle of $\left(M,\partial M\right)$ and, by
$\parallel\beta\parallel\le\frac{1}{\sima +\simb}$ and duality follows
$\parallel M,\partial M\parallel\ge\parallel M_1,\partial M_1\parallel
+\parallel M_2,\partial M_2\parallel$.
 
Now consider the case $\sima\not=0$ and $\simb =0$. Consider $\gamma_2=0$ and
again $\gamma_1=\sima\beta_1\in H^n_b\left(M_1,\partial M_1\right)$.
Then we find $\gamma\in H^n_b\left(M,\partial M\cup A\right)$ with
$j_1^*\gamma=\gamma_1, j_2^*\gamma=0$ and
$\parallel \gamma\parallel\le\parallel\gamma_1\parallel$. The same
way as in the first case we get
that $\frac{1}{\sima}i^*\gamma$ represents $\beta$ and thus
$\parallel M,\partial M\parallel\ge\sima$.\\
Finally the case $\sima=\simb=0$ is trivial anyway.\\
(ii): We suppose $\parallel M_1,\partial M_1\parallel\not=0$,
since otherwise the claim is
trivially true. Then the relative fundamental cocycle
has preimage $\beta_1\in H_b^n\left(M_1,\partial
M_1\right)$. Consider
$\gamma_1:=\parallel M_1,\partial M_1\parallel\beta_1$.
By 1.2, $\parallel\gamma_1\parallel =1$.
 
We find $\gamma
\in H_b^n\left(M,\partial M\cup A\right)$ satisfying
$\parallel\gamma\parallel\le
\parallel\gamma_1\parallel=1$
and $P^*\gamma=\gamma_1$.
 
Let $i:\left(M,\partial M\right)\rightarrow\left(M,\partial M\cup A\right)$ be the
inclusion. In $H_n\left(M,\partial M\cup A\right)$ we have $i_*\left[M,\partial M\right]=P_*\left[M_1,\partial M_1\right]$.
Hence,
$i^*\gamma\left[M,\partial M\right]=\gamma P_*
\left[M_1,\partial M_1\right]=P^*\gamma\left[M_1,\partial M_1\right]
=\gamma_1\left[M_1,\partial M_1\right]=\parallel M_1,\partial M_1\parallel$.\\
Thus, $\beta=\frac{1}{\sima}i^*\gamma$
is the relative fundamental class of $\left(M,\partial M\right)$ and by
$\parallel\beta\parallel\le\frac{1}{\sima }$ and duality follows
$\parallel M,\partial M\parallel\ge\parallel M_1,\partial M_1\parallel$.\end{pf}




\subsection{Multicomplexes associated to glueings}

{\bf The 'amalgamated' case.}\\
We are going to consider the following situation: $X_1, X_2$ are 
topological spaces, $A_1\subset X_1, A_2\subset X_2$ path-connected subspaces,
$f:A_1\rightarrow A_2$ is a homeomorphism such that $f_*:\pi_1A_1\rightarrow\pi_1A_2$ restricts to an isomorphism from $ker\left(\pi_1A_1\rightarrow\pi_1X_1\right)$ to
$ker\left(\pi_1A_2\rightarrow\pi_1X_2\right)$. Let $X=X_1\cup_fX_2$.\\

The assumption on $f_*$ implies that $\pi_1X_1,\pi_1X_2$ inject into $\pi_1X$.
By proposition 3, it follows that \\
- (i) $K\left(X_1\right)$ and $K\left(X_2\right)$
are submulticomplexes of $K\left(X\right)$. \\
Concerning the 0-skeleta, we have \\
- (ii) $K\left(X\right)_0=X=X_1\cup X_2=K\left(X_1\right)_0\cup K\left(X_2\right)_0$.

Let $A=X_1\cap X_2$ be the intersection of $X_1$ and $X_2$ as subspaces of 
$X$. There is an obvious homomorphism $\pi_1A\rightarrow\pi_1X$. Hence, 
there is a map from $K\left(A \right)$ to $K\left(X_1\right)\cap
K\left(X_2\right)$, the intersection of $K\left(X_1\right)$ and $K\left(X_2\right)$ in $K\left(X\right)$, which identifies pathes in $A$ whose composition gives an element of $ker\left(\pi_1A\rightarrow \pi_1X\right)$. 
Since this projection kills exactly the kernel of $\pi_1K\left(A_i\right)\rightarrow \pi_1K\left(X_i\right)$ we get that \\
- (iii) $\pi_1\left(K\left(X_1\right)\cap
K\left(X_2\right)\right)$ injects into $\pi_1K\left(X_1\right)$ resp.\ $\pi_1K\left(X_2\right)$.\\
In particular, $\pi_1\left(K\left(X_1\right)\cup K\left(X_2\right)\right)$ is the amalgamated product of 
$\pi_1K\left(X_1\right)\simeq\pi_1X_1$ and $\pi_1K\left(X_2\right)\simeq\pi_1X_2$, amalgamated over $\pi_1
\left(K\left(X_1\right)\cap K\left(X_2\right)\right)\simeq im\left(\pi_1A_i\rightarrow\pi_1X_i\right)$. But this amalgamated product is isomorphic to $\pi_1X\simeq\pi_1K\left(X\right)$ and we conclude\\
- (iv) the inclusion $K\left(X_1\right)\cup K\left(X_2\right)\rightarrow K\left(X\right)$
induces an isomorphism of fundamental groups.

Moreover, it follows from proposition 2 that \\
- (v) $K\left(X_1\right)$ and $K\left(X_2\right)$ are aspherical,\\
and we clearly have\\
- (vi) $K_i^\prime=K\left(X_i\right)\cap K^\prime\subset K\left(X\right)$ 
for $i=1,2$,\\
where $K_i^\prime$ is the image of $K\left(A_i\right)$ in $K\left(X_i\right)$ and $K^\prime$ is the image of $K\left(A\right)$ in $K\left(X\right)$.

Finally, from injectivity of $\pi_1X_i\rightarrow\pi_1X$, one gets that the canonical map $\Pi_{X_i}\left(A_i\right)\rightarrow\Pi_X\left(A\right)$ is an 
isomorphism for $i=1,2$. Consider the action of $G=\Pi_X\left(A\right)$
on $K\left(X\right)$. \\
- (vii) $G$ maps $K\left(X_i\right)$ to $K\left(X_i\right)$  and $K_i^\prime$ to $K_i^\prime$ for $i=1,2$.\\
As a consequence, the action of $\Pi_X\left(A\right)$ on $K\left(X\right)$ preserves 
$K\left(X_1\right)\cap K\left(X_2\right)\subset K\left(X\right)$.
Even though $\pi_1\left(X_1\cap X_2\right)$ may not inject into $\pi_1X$, analogously to lemma 3, we get:\\
- (viii) $G$ acts transitively on $\left(K\left(X_1\right)\cap K\left(X_2\right)\right)$-related
tuples of 1-simplices in $K\left(X\right)$.\\
\\                                                                               
\\
{\bf The 'HNN'-case.}\\
Let $X_1$ be a topological space, $A_1,A_2$ path-connected
subspaces of $X_1$, $f:A_1\rightarrow A_2$ a homeomorphism 
such that $f_*$ restricts to an isomorphism from $ker\left(\pi_1A_1\rightarrow \pi_1X_1\right)$ to $ker\left(\pi_1A_2\rightarrow\pi_1X_1\right)$.
Let $X=X_1/_\sim$, where $x_1\sim x_2$ if $x_1\in A_1, x_2\in A_2$ and $x_2=f\left(x_1\right)$. \\

We have canonical, not necessarily injective, maps from $K\left(A_1\right)$ and $K\left(A_2\right)$ to $K\left(X_1\right)$. For brevity, let us denote $K_i^\prime\subset K\left(X_1\right)$ the image
of the map from $K\left(A_i\right)$, for $i=1,2$. \\
$f$ induces a simplicial isomorphism $F:K_1^\prime\rightarrow K_2^\prime$. Let $L$ be the multicomplex obtained from $K\left(X_1\right)$ by identifying $\sigma$ and $F\left(\sigma\right)$ for any
simplex $\sigma$ in $K_1^\prime$. \\
We have then\\
(i) a canonical embedding $i_L:L\rightarrow K\left(X\right)$.\\
Indeed, the map $K\left(X_1\right)\rightarrow K\left(X\right)$, induced by the projection, factors over $L$.\\
One checks easily:\\
(ii) $K\left(X\right)_0=L_0$,\\
(iii) $\pi_1K_1^\prime\rightarrow\pi_1K\left(X_1\right)$ and $\pi_1K_2^\prime\rightarrow\pi_1K\left(X_1\right)$ are injective,\\
(iv) the inclusion $L\rightarrow K\left(X\right)$ induces an isomorphism of $\pi_1$'s,\\
(v) $K\left(X\right)$ and $L$ are aspherical in the sense of definition 3.\\
Let $A$ be the image of $A_1$ in $X$ and $K^\prime$ the image of $K\left(A\right)$
in $K\left(X\right)$. Then\\
(vi) the projection from $K\left(X_1\right)$ to $L$ maps $K_i^\prime$ to $K^\prime$.

Finally, $\pi_1X$ is an HNN-extension of $\pi_1X_1$, i.e.\ $\pi_1X_1\rightarrow\pi_1X$ is injective, and
one gets that the canonical map $\Pi_{X_i}\left(A_1\right)\rightarrow\Pi_X\left(A\right)$ is an isomorphism for $i=1,2$. Consider the action of $G=\Pi_X\left(A\right)$ on $K\left(X\right)$. We use the isomorphisms $\Pi_{X_1}\left(A_1\right)\simeq\Pi_X\left(A\right)\simeq\Pi_{X_1}\left(A_2\right)$ to get identifications with $G$ and observe that
after these identifications the action of $G$ commutes with $F$. Moreover,\\
(vii) the action of $G$ commutes with $i_LP$, where $P:K\left(X_1\right)\rightarrow L$ is the canonical projection.\\
Similarly to lemma 3, we get \\
(viii) $G$ acts transitively on $K^\prime$-related
tuples of 1-simplices in $K\left(X\right)$.\\ 




                                                                                 
 
 
 
 
 
\subsection{Proof of Theorem 1} 

In this section, we prove superadditivity for simplicial volume of manifolds 
with boundary with respect to glueing along amenable subsets of the boundary.
A similar result for open manifolds is the Cutting-off-theorem in \cite{gro}.
One should note that, at least for manifolds with boundary, the opposite inequality need not hold. As a counterexample one may glue solid tori along disks to get a handlebody.   
\begin{lem}: (i): Let $M_1,M_2$ be two compact, connected n-manifolds,
$A_1,A_2$ (n-1)-dimensional submanifolds of $\partial M_1$ resp.$\partial M_2$,
$f:A_1\rightarrow A_2$ a homeomorphism
and $M=\glu$ the glued manifold.\\
If $f_*$ maps
$ker\left(\pi_1A_1\rightarrow\pi_1M_1\right)$ isomorphically to
$ker\left(\pi_1A_2\rightarrow\pi_1M_2\right)$, and if\\
$im\left(\pi_1A_1\rightarrow\pi_1M_1\right)$ is amenable,
then $\parallel M,\partial M\parallel\ge\sima+\simb$.\\
(ii):  Let $M_1$ be a compact, connected n-manifold,
no component of which is a 1-dimensional closed intervall,
$A_1, A_2$ disjoint (n-1)-dimensional submanifolds of $\partial M_1$,
$f:A_1\rightarrow A_2$ a homeomorphism
and $M=M_1/f$ the glued manifold. \\
If
$im\left(\pi_1A_1\rightarrow\pi_1M_1\right)$ is
amenable, and $f_*:ker\left(\pi_1A_1\rightarrow\pi_1M_1\right)\rightarrow
ker\left(\pi_1A_2\rightarrow\pi_1M_1\right)$ is an isomorphism, then $\parallel M,\partial M\parallel\ge
\parallel M_1,\partial M_1\parallel$.\end{lem}
\begin{pf} (i): For manifolds of dimensions $\le 2$ one
checks easily that there is no counterexample. So we are going
to assume that $n\ge3$.\\
We want to check the assumption of lemma 7.
We can restrict to the case that $A_1$ and $A_2$ are path-connected, since we may argue succesively
for their path-connected components.

First, we make the {\bf restrictive assumption} that {\bf $\pi_1\partial M_i$ and $\pi_1
A_i$ should inject into
$\pi_1M_i$} for $i=1,2$. We will show afterwards how to handle the general case.
The advantage of this assumption is that, by proposition 3, we may
assume $K\left(\partial M_i\right)$ to be a submulticomplex of $K\left(M_i\right)$ and
$K\left(\partial M\cup A\right)$ to be a submulticomplex of $K\left(M\right)$.
Denoting by $j_1,j_2,k_1,k_2$ the embeddings and by $I,I_1,I_2$ the isometric isomorphisms from Prop.2(ii),
we claim that the following diagram commutes.
$$\begin{diagram}
\bigoplus_{i=1}^2 H_b^n\left(M_i,\partial M_i\right) &\lTo^{j_1^*\oplus j_2^*}&H_b^n\left(M,\partial M\cup A\right)\\
\uTo^{I_1\oplus I_2}& &\uTo^{I}\\
\bigoplus_{i=1}^2 H_b^n\left(K\left(M_i\right),K\left(\partial M_i\right)\right)&\lTo^{k_1^*\oplus k_2^*}&H_b^n\left(K\left(M\right),K\left(\partial M\cup A\right)\right)\\
\end{diagram}$$
 
To see that the diagram commutes, recall that $I$ was constructed (in section 1.3) as a 
composition 
$I=\left(S^*\right)^{-1}\left(h^*\right)^{-1}p^*$. $S^*$ was induced by 
the projection $S:\widetilde{K}\left(X\right)\rightarrow X$, hence
$\left(S_i^*\right)^{-1}j_i^*=j^*\left(S^*\right)^{-1}$ follows from the 
obvious relation $Sj_i=jS_i$. Also,  $h_i^*j_i^*=j^*h^*$ follows from the fact that $h^*$ is induced by an inclusion of chain complexes. Finally, $p^*$ is induced by the projection $p:\tilde{K}\left(X\right)\rightarrow K\left(X\right)$ described in 1.3. which clearly commutes with the inclusions coming from proposition 3.

Consider the following commutative diagram:
$$\begin{diagram}
\bigoplus_{i=1}^2 H_b^n\left(K\left(M_i\right),K\left(\partial M_i\right)\right)&\lTo^{k_1^*\oplus k_2^*}&H_b^n\left(K\left(M\right),K\left(\partial M\cup A\right)\right)\\
\uTo^{p_1\oplus p_2}& &\uTo^{p}\\
\bigoplus_{i=1}^2 H_b^n\left(K\left(M_i\right)^{\Pi_{M_i}\left(A_i\right)},K\left(\partial M_i\right)^{\Pi_{M_i}\left(A_i\right)}\right)&\lTo^{k_1^*\oplus k_2^*}&H_b^n\left(K\left(M\right)^{\Pi_M\left(A\right)},K\left(\partial M\cup A\right)^{\Pi_M\left(A\right)}\right)
\end{diagram}$$
By lemma 3, 
$\Pi_{M_1}\left(A_1\right),
\Pi_{M_2}\left(A_2\right), \Pi_M\left(A\right)$ are amenable. Hence, by lemma 2(i), $p_1,p_2,p$
have left inverses $Av_1,Av_2,Av$ of norm $=1$.
For i=1,2 define $$\gamma^\prime_i:=Av_i\left(I_i\right)^{-1}\gamma_i
\in H_b^n\left(
K\left(M_i\right)^{\Pi_{M_i}\left(A_i\right)},
K\left(\partial M_i\right)^{\Pi_{M_i}\left(A_i\right)}\right).$$ They satisfy
$\parallel\gamma^\prime_i\parallel=\parallel \gamma_i\parallel$.\\
It follows from the discussion in 3.2.\ that we can apply lemma 5 and corollary 2. Hence, we get $$\gamma^\prime\in H_b^n\left(K\left(M\right)^{\Pi_M\left(A\right)},
K_A\left(\partial M\right)^{\Pi_M\left(A\right)}\right),$$
satisfying $\parallel\gamma^\prime\parallel\le max\left\{
\parallel \gamma_1^\prime\parallel, \parallel \gamma_2^\prime\parallel
\right\}= max\left\{\parallel \gamma_1\parallel,\parallel \gamma_2\parallel
\right\}$ and \\
$k_i^*\gamma^\prime=\gamma_i^\prime$ for i=1,2.\\
Let
$$\gamma:=Ip^*\gamma^\prime      \in H_b^*\left(M,\partial M\right)$$
It satisfies
$\parallel\gamma\parallel=\parallel \gamma^\prime\parallel\le
max\left\{\parallel\gamma_1\parallel,\parallel\gamma_2\parallel\right\}=1$
and 

$j_1^*\gamma=
j_1^*Ip^*\gamma^\prime=I_1k_1^*p^*\gamma^\prime=I_1p_1^*k_1^*\gamma^\prime=
I_1p_1^* Av_1 \left(h_1^*\right)^{-1}\gamma_1$.
 
It is easy to see that $\Pi_{M_i}\left(A_i\right)$ are connected and that the actions
of $\Pi_{M_i}\left(A_i\right)$ on $K\left(M_i\right)$ are continuous. 
Hence, all elements of $\Pi_{M_i}\left(A_i\right)$ are, as mappings from
$K\left(M_i\right)$ to itself, homotopic to the identity. From part (ii) of lemma 2, we get that $p_i^*\circ Av_i=id$.
Hence, we obtain $j_1^*\gamma=\gamma_1$. The same way, $j_2^*\gamma=\gamma_2$.
 
Thus, we have checked the assumptions of lemma 7.\\

We are now going to consider the {\bf general case}, i.e., we do not assume any longer
injectivity of fundamental groups. 

Let $Y\subset X$ with $ker\left(\pi_1Y\rightarrow \pi_1X\right)\not=0$. In 
this case, $\tilde{K}\left(Y\right)$ doesn't embed into $\tilde{K}\left(X\right)$. However,
$\tilde{K}\left(.\right)$ is clearly functorial in the sense that continuous mappings $Y\rightarrow X$
induce simplicial maps $\tilde{K}\left(Y\right)\rightarrow
\tilde{K}\left(X\right)$. In particular, the embedding induces simplicial maps
$\tilde{f}:\tilde{K}\left(Y\right)\rightarrow \tilde{K}\left(X\right)$ and
$f:K\left(Y\right)\rightarrow K\left(X\right)$. We consider its image 
$f\left(K\left(Y\right)\right)$ as a submulticomplex of $K\left(X\right)$.

We want to show that there are maps
$$ H:H_b^n\left(K\left(X\right),f\left(K\left(Y\right)\right)\right)\rightarrow H_b^n\left(X,Y\right)$$
$$\overline{H}:H_b^n\left(X,Y\right)\rightarrow H_b^n\left(K\left(X\right),f\left(K\left(Y\right)\right)\right)$$
{\bf of norm $\le 1$}. (They are not isomorphisms, even though we
will have $H\overline{H}=id$.)

To this behalf, we 
explain more in detail the construction of
$I^{-1}=Av\circ h^*\circ S^*$
in proposition 2. \\
$S^*$ is induced by a weak homotopy equivalence $S:\tilde{K}_X\rightarrow X$.
From the definition of $S$ on
\cite{gro}, p.42, it is clear that $S$ maps $\tilde{f}\left(\tilde{K}_Y\right)$
to $Y$. Hence $S$ induces a map $S^*$ of norm $\le 1$
from $H_b^n\left(X,Y\right)$ to $H_b^n\left(\mid\tilde{K}_X\mid,\mid \tilde{f}\left(\tilde{K}_Y\right)\mid\right)$.\\
The isomorphism $Av:H_b^n\left(\tilde{K}_X\right)\rightarrow H_b^n\left(K\left(X\right)\right)$ is induced by averaging over the
amenable group $\Gamma_1/\Gamma_n$, where $\Gamma_i$ is the group of
simplicial automorphisms which are the identity on the i-skeleton, cf. \cite{gro}, p.46. 
We get a map
$Av: H_b^n\left(\tilde{K}_X,\tilde{f}\left(\tilde{K}_Y\right)\right)\rightarrow
H_b^n\left(K_X,f\left(K_Y\right)\right)$ which has norm $\le 1$ by definition of the averaging.\\
Finally, by proposition 1 we get an isometry $h^*:
H_b^n\left(\mid\tilde{K}_X\mid,\mid \tilde{f}\left(\tilde{K}_Y\right)\mid\right)
\rightarrow H_b^n\left(\tilde{K}_X,\tilde{f}\left(\tilde{K}_Y\right)\right)$. Hence, we may define $\overline{H}=Av\circ h^*\circ S^*$ and $H=j^*\circ \left(h^*\right)^{-1}\circ p^*$.

We will call $H_1,H_2,H$ and $\overline{H_1},\overline{H_2},\overline{H}$ the maps
corresponding to $\left(X,Y\right)=\left(M_1,\partial M_1\right),
\left(M_2,\partial M_2\right)$ resp.\ $\left(M,\partial M\cup A\right)$.\\
\\
The action of $\Pi_{A_i}\left(M_i\right)$
on $K\left(M_i\right)$ preserves $f\left(K\left(\partial M_i\right)\right)$.
We get the commutative diagram
$$\begin{diagram}
\bigoplus_{i=1}^2 H_b^n\left(M_i,\partial M_i\right) &\lTo^{j_1^*\oplus j_2^*}&H_b^n\left(M,\partial M\cup A\right)\\
\dTo^{\overline{H_1}\oplus \overline{H_2}}& &\uTo^{H}\\
\bigoplus_{i=1}^2 H_b^n\left(K\left(M_i\right),f\left(K\left(\partial M_i\right)\right)\right)&\lTo^{k_1^*\oplus k_2^*}&H_b^n\left(K\left(M\right),f\left(K\left(\partial M\cup A\right)\right)\right)\\
\uTo^{p_1^*\oplus p_2^*}& &\uTo^{p^*}\\
\bigoplus_{i=1}^2 H_b^n\left(K\left(M_i\right)^{\Pi_{M_i}\left(A_i\right)},K\left(\partial M_i\right)^{\Pi_{M_i}\left(A_i\right)}\right)&\lTo^{k_1^*\oplus k_2^*}&H_b^n\left(K\left(M\right)^{\Pi_M\left(A\right)},K\left(\partial M\cup A\right)^{\Pi_M\left(A\right)}\right).\\
\end{diagram}$$
One checks easily that all arguments in the first part of the proof go through, finishing the proof of part (i).\\

(ii):
In dimensions $\le 2$ we check
that the closed interval is the only connected counterexample. Assume then $n\ge3$.
Again we may suppose $A_1, A_2$ connected.\\
We will again assume that $\pi_1\left(\partial M_1\right)\rightarrow \pi_1\left(M_1\right)$ and
$\pi_1\left(\partial M\cup A\right)\rightarrow \pi_1\left(M\right)$ are injective. The generalisation to the case of compressible boundary follows then
by arguments completely analogous to those in the proof
of part (i).\\
Like in part (i), we get a commutative diagram, where $P,Q,R$ are the obvious projections.
$$\begin{diagram}
H_b^n\left(M_1,\partial M_1\right) &\lTo^{P^*}&H_b^n\left(M,\partial M\cup A\right)\\
\uTo^{I_1}& &\uTo^{I}\\
H_b^n\left(K\left(M_1\right),K\left(\partial M_1\right)\right)&\lTo^{R^*}&H_b^n\left(K\left(M\right),K\left(\partial M\cup A\right)\right)\\
\uTo^{p_1}& &\uTo^{p}\\
H^n_b\left(K\left(M_1\right)^{\Pi_{M_1}\left(A_1\cup A_2\right)},K\left(\partial M_1\right)^{\Pi_{M_1}\left(A_1\cup A_2\right)}
\right)&\lTo^{Q^*}&H_b^n\left(K\left(M\right)^{\Pi_M\left(A\right)},K\left(\partial M\right)^{\Pi_M\left(A\right)}\right)\\
\end{diagram}$$

Given $\gamma_1
\in H^*_b\left(M_1,\partial M_1\right)$,
define $$\gamma^\prime_1:=Av_1I_1^{-1}\gamma_1
\in H_b^*\left(
(K\left(M_1\right)^{\Pi_{M_1}\left(A_1\cup A_2\right)}
,K\left(\partial M_1\right)^{\Pi_{M_1}\left(A_1\cup A_2\right)}
\right).$$
We check that the assumptions of corollary 3 are satisfied and get $$\gamma^\prime\in     H_b^n\left(K\left(M\right)^{\Pi_M\left(A\right)},K\left(\partial M\right)^{\Pi_M\left(A\right)}\right).$$
Then define 
$\gamma:=Ip\gamma^\prime\in H_b^n\left(M,\partial M\cup A\right)$. Analogously to the proof of part (i) we get that $P^*\gamma=\gamma_1$ and $\parallel \gamma\parallel\le\parallel \gamma_1\parallel$ holds.

Thus, we can apply lemma 7 to finish the proof.\end{pf} 

\begin{lem}: \\
i) Let $M_1,M_2$ be compact manifolds, $A_1$ resp.\ $A_2$ connected components
of $\partial M_1$ resp.\ $\partial M_2$ and assume that there exist connected sets
$A_i^\prime\subset M_i$ with $A_i^\prime\supset A_i$ and $\pi_1A_i^\prime$ amenable.
Let $f:A_1\rightarrow A_2$ be a homeomorphism and
$M=M_1\cup M_2/f$ the glued manifold.
Then $\parallel M,\partial M\parallel\le\parallel M_1,\partial M_1\parallel + \parallel M_2,\partial M_2\parallel$.\\
ii) Let $M^\prime$ be a compact manifold, $A_1,A_2$ connected components of
$\partial M^\prime$ with $\pi_1A_i$ amenable. Let $f:A_1\rightarrow A_2$ be a homeomorphism and
$M=M^\prime/f$ the glued manifold. Then $\parallel M,\partial M\parallel\le
\parallel M^\prime,\partial M^\prime\parallel$.\end{lem}
\begin{pf} ii) is reduced to i) via the homeomorphism $M=M^\prime\cup_{\left(id,0\right)+\left(f,1\right)}\left(A_1\times I\right)$. (Note that $\parallel A_1\times I,A_1\times\left\{0,1\right\}\parallel =0$, since $\pi_1A_1$ is amenable.)
 
To prove i), we need the following reformulation of a theorem of Matsumoto-Morita.
For a space $X$  and $q\in N$ let $C_q\left(X\right)$ be the group of singular chains
and $B_q\left(X\right)$ the subgroup of boundaries. By theorem 2.8.\ of \cite{mat} the following
two statements are equivalent:\\
a) there exists a number $K>0$ such that for any boundary $z\in B_q\left(X\right)$ there
is a
chain $c\in C_{q+1}\left(X\right)$ satisfying $\partial c=z$ and $\parallel c\parallel <
K\parallel z\parallel$,\\
b) the homomorphism $H_b^{q+1}\left(X\right)\rightarrow H^{q+1}\left(X\right)$ is injective.
 
Now let $\sum_{i=1}^ma_i\sigma_i$ and $\sum_{j=1}^nb_j\tau_j$
be representatives of $\left[M_1,\partial M_1\right]$ and
$\left[M_2,\partial M_2\right]$ with $$\sum_{i=1}^m\mid
a_i\mid\le\sima+\epsilon$$ and $$\sum_{j=1}^n\mid
b_j\mid\le\simb+\epsilon.$$
By proposition 4 we may suppose that
$\partial\left( \sum_{i=1}^ma_i\sigma_i\right)\mid_{A_1}\in C_*\left(\partial
M_1\right)$ and
$\partial\left( \sum_{j=1}^nb_j\tau_j\right)\mid_{A_2}\in C_*\left(\partial
M_2\right)$ have norm smaller than
$\frac{\epsilon}{2K}$. (Note that $\pi_1 A_i
\rightarrow\pi_1M_i$ factors over
$\pi_1A_i^\prime$, hence has amenable image.)
 
Let $A^\prime$ be the image of $A^\prime_1$ in $M$.
As $\pi_1A^\prime$ is amenable, $H_b^{q+1}\left(A^\prime\right)=0$ for $q\ge 0$ (\cite{gro},\cite{iva}),
hence, $H_b^{q+1}\left(A^\prime\right)\rightarrow H^{q+1}\left(A_i^\prime\right)$ is
clearly injective and we get a constant $K$ with the property in a).\\
Therefore,
we find
$c\in C_*\left(A^\prime\right)\subset C_*\left(M\right)$ with $\parallel
c\parallel\le\epsilon$ and $$\partial c=\partial\left(\sum_{i=1}^ma_i\sigma_i+
\sum_{j=1}^nb_j\tau_j\right).$$
 
Then
$z=\sum_{i=1}^ma_i\sigma_i + \sum_{j=1}^nb_j\tau_j-c\in
C_*\left(\glu\right)$ is a fundamental
cycle of norm smaller than $\parallel M_1,\partial M_1\parallel+\parallel M_2,\partial M_2\parallel+3\epsilon$.\end{pf}\\
{\bf Remark}: The assumption of lemma 9 is in particular satisfied if
$im\left(\pi_1\partial M_1\rightarrow\pi_1M_1\right)$ and
$im\left(\pi_1\partial M_2\rightarrow\pi_1M_1\right)$ are amenable and
the (singular) compression disks can be chosen to be disjoint. For
3-manifolds $M_i$, by a theorem of Jaco, cf.\ [4], there is $A_i^\prime\subset M_i$ with
$\pi_1A_i^\prime=im\left(\pi_1A_i\rightarrow\pi_1M_i\right)$                      
if $im\left(\pi_1A_i\rightarrow\pi_1M_i\right)$ is finitely presented.\\
\begin{thm}: \\
(i):Let $M_1,M_2$ be two compact n-manifolds,
$A_1$ resp.$A_2$ connected components of $\partial M_1$ resp.$\partial M_2$,
$f:A_1\rightarrow A_2$ a homeomorphism,
$M=\glu$ the glued manifold. \\
If $\pi_1A_1$ and $\pi_1A_2$ are amenable \\
and
$f_*: ker\left(\pi_1A_1\rightarrow\pi_1M_1\right)\rightarrow
ker\left(\pi_1A_2\rightarrow\pi_1M_2\right)$ is an isomorphism, \\
then $\parallel M,\partial M\parallel=\sima+\simb$.\\
(ii): Let $M_1$ be a compact n-manifold, no
connected component of $M_1$ a 1-dimensional closed intervall,
$A_1,A_2$ connected components of $\partial M_1$,
$f:A_1\rightarrow A_2$ a homeomorphism,
$M=M_1/f$ the glued manifold. \\
If
$\pi_1A_1$ is
amenable and $f_*:ker\left(\pi_1A_1\rightarrow\pi_1M_1\right)\rightarrow
ker\left(\pi_1A_2\rightarrow\pi_1M_1\right)$ is an isomorphism, then $\parallel M,\partial M\parallel=\parallel
M_1,\partial M_1\parallel$.\end{thm}
\begin{pf} Theorem 1 follows from lemma 8 and 9.\end{pf}
\begin{cor}: (i) Let A be a properly embedded
annulus in a compact 3-manifold M. If 
$im\left(\pi_1A_1\rightarrow\pi_1M_A\right)\approx
im\left(\pi_1A_2\rightarrow\pi_1M_A\right)$ for the two 
images $A_1, A_2$ of $A$
in $M_A$, then
$\parallel M_A,\partial M_A\parallel\le\parallel
M,\partial M\parallel$.\\
(ii) Let T be an embedded torus in a compact 3-manifold M.
If \\
$im\left(\pi_1T_1\rightarrow\pi_1M_T\right)\approx
im\left(\pi_1T_2\rightarrow\pi_1M_T\right)$ for the two
images $T_1, T_2$ of $T$ in $M_T$, then 
$\parallel M_T,\partial M_T\parallel
=\parallel M,\partial M\parallel$.\end{cor}
 
Remark: If $\partial M$ consists of tori and A is an incompressible
annulus, then even $\parallel M_A,\partial M_A\parallel=\parallel
M\parallel$ holds by a theorem of \cite{som1}.\\
\\
In \cite{som1}, a version of corollary 4 has been proved for the special case that $\partial M$
consists of tori. The principal ingredient in the proof is the following statement:\\
{\bf Proposition 5} (\cite{som1}, Lemma 1): {\em Let $M$ be a compact 3-manifold whose boundary $\partial M$ consists of
tori and $H$ be a 3-dimensional compact submanifold of $int\left(M\right)$. Suppose
$int\left(H\right)$ is a hyperbolic 3-manifold and $\partial H$ is incompressible in $M$.Then we have $\parallel M,\partial M\parallel\ge\parallel H,\partial H\parallel$.}\\
In \cite{thul}, this proposition is stated for closed $M$ as theorem 6.5.5. without writing a proof. In
\cite{som1}, it is then derived for
$M$ with toral boundary, using the doubling argument, see the proof of lemma 1
in \cite{som1}. \\
Hence, our proof seems to be the first written proof of corollary 4 and proposition 5. (It
is easy to see that corollary 4 implies proposition 5.)\\
We want to mention that, according to Agol, an alternative proof of proposition 5 (hence, corollary 4)
should be possible using the methods of \cite{agol}.\\
We refer to \cite{som1} to see that corollary 4 actually allows to compute the simplicial
volumes of all Haken 3-manifolds with (possibly empty) toral boundary.
\subsection{Counterexamples}
The first example shows that the condition                            
$ker\left(\pi_1A_1\rightarrow\pi_1M_1\right)
\approx ker\left(\pi_1A_2\rightarrow\pi_1M_2\right)$ can not be weakened.
 
{\bf Example 1:} {\em Dehn fillings}\\
Let $K$ be a knot in $S^3$ such that $S^3 - K$ admits a hyperbolic metric of finite
volume. Let $V\subset S^3$ be a regular neighborhood of $K$. $\partial V$ is
a torus, hence, $\pi_1\partial V$ is amenable. But
$$\parallel S^3\parallel=0<
\parallel S^3 - V,\partial V\parallel+\parallel V,\partial V\parallel.$$
More generally, it is known(\cite{agol}) that $$\parallel M,\partial M\parallel < \parallel M - V,\partial V\cup\partial M\parallel=\parallel M - V,\partial V\cup\partial M\parallel+\parallel V,\partial V\parallel$$ holds if both $int\left(M\right)$ and $int\left(M-V\right)$ admit a hyperbolic metric of finite volume, i.e., if $M$ is obtained by performing hyperbolic Dehn filling at $M-V$.\\
Lemma 8 does not apply because the meridian of $\partial V$
maps to zero in $\pi_1V$, but it doesn't so in
$\pi_1\left(S^3 - V\right)$ resp.\ $\pi_1\left(M - V\right)$. \\
\\
The second example shows that the assumption '$A_i$ connected'
in lemma 12
can not be weakened to assume only $A_i\subset A_i^\prime$. (In particular, one
can not just glue along an
amenable subset $A_i\subset\partial M_i$.)

{\bf Example 2:} {\em Heegard splittings}\\
Any 3-manifold can be decomposed into two handlebodies $H$ and $H^\prime$, to be identified
along their boundaries. Let $g$ be the genus of $H$ and $H^\prime$ and consider a
set of properly embedded disks $D_1,\ldots,D_g\subset H^\prime$ such that
$H^\prime - \cup_{i=1}^g V_i$ is a 3-ball $B$, where $V_i$ are disjoint open regular neighborhoods
of $D_i$. Denote $A_i=V_i\cap\partial H^\prime$.\\
$M$ is then obtained as follows: $V_1,\ldots,V_g$ are glued to $H$ along the annuli $A_i$, afterwards
$B$ is glued along its whole boundary.\\
Of course, $\parallel V_i,\partial V_i\parallel =0$ and $\parallel B,\partial B\parallel=0$.
Thus, if
lemma 9 were applicable to the
annuli $A_i$, we would get that
$\parallel M,\partial M\parallel<\parallel H_g,\partial
H_g\parallel$. \\
But there are 3-manifolds of arbitrarily large
simplicial volume which admit Heegard splittings of a given genus.
To give an explicit example,
let $f$ be a pseudo-Anosov diffeomorphism on a surface of genus g,
and let $M_n$ be the mapping tori of the iterates $f^n$.
By Thurstons hyperbolization theorem, $M_1$ is hyperbolic. Hence,
$\parallel M_1\parallel>0$ and $\parallel M_n\parallel=n\parallel M_1\parallel$
becomes arbitrarily large. On the other hand, all $M_n$ admit a Heegardsplitting of
genus 2g+1.                                                                    












\section{Doubling 3-manifolds reduces simplicial volume}

\noindent
We recollect the theory of 3-manifolds necessary for
the proof of theorem 2.

It is well-known, cf., \cite{hem} that any compact 3-manifold is a connected sum of finitely many manifolds which are irreducible or $S^2\times S^1$, where a 3-manifold is termed irreducible if each embedded 2-sphere bounds an
embedded 3-ball in $M$, and that  
any 3-manifold can be cut along finitely many disks to get pieces which have 
incompressible (i.e., $\pi_1$-injective) boundary. 

Now let $M$ be a compact irreducible 3-manifold with incompressible 
boundary. We use the Jaco-Shalen-Johannson theorem to decompose $M$, 
following \cite{neus}.\\
In section 2 of \cite{neus}, there is defined a so-called W-decomposition,
which is a family of disjoint properly embedded incompressible tori and
annuli. Denoting $M_i$ the connected manifolds obtained after cutting
$M$ along the tori and annuli from the W-decomposition, proposition
3.2.\ of \cite{neus} asserts that the $M_i$ 
are either Seifert fibered (i.e., finitely covered by $S^1$-bundles), 
I-bundles or simple, where simple in the terminology of \cite{neus}
means that any properly {\em immersed} incompressible torus or annulus 
can be isotoped into $\partial M_i$.

By Thurston's hyperbolization theorem, the simple pieces admit a 
hyperbolic metric. Moreover, we have\\
{\bf Proposition 6} (\cite{thua}, Theorem 3): {\em Let $M$ be a hyperbolic manifold such that $int\left(M\right)$ admits a complete hyperbolic metric.
There is a totally geodesic surface in $int\left(M\right)$
for each non-torus component of $\partial M$ if and only if any incompressible, properly
embedded annulus is boundary-parallel and $\partial M$ is incompressible.}\\
(Note that the condition "$\partial M$ incompressible", missing in \cite{thua},
is necessary to exclude handlebodies. It is needed in the proof to
guarantee that $DM$ is irreducible.)

Therefore the simple pieces admit an (incomplete) hyperbolic metric
such that the toral boundary components correspond to cusps and the boundary components of higher
genus are totally geodesic.\\
\\
\\ 
For an oriented manifold $M$, let $DM$ denote the double of $M$, defined by glueing two
differently oriented copies of $M$ via the identity of $\partial M$. It is trivial that
$\parallel DM\parallel\le 2\parallel M,\partial M\parallel$. Theorem 1 implies,
of course:
if $M$ is a compact 3-manifold with $\parallel \partial M\parallel=0$,
then $\parallel DM\parallel=2\parallel M,\partial M\parallel$. We will show that
this is actually an if-and-only-if condition.
\begin{lem}: Let $N$ be a compact manifold of dimension $\ge3$ whose
interior admits a hyperbolic metric of finite volume. Let $F\subset N$
be a closed codimension-1 submanifold which is totally geodesic for the
hyperbolic metric on $N$. Then $\parallel N,\partial N\parallel^F >
\parallel N,\partial N\parallel$.\end{lem}
\begin{pf} If $\partial N=\emptyset$, this follows from \cite{jun}.
If $\partial N\not=\emptyset$, it follows from the main result of our paper \cite{k}.\end{pf}
\begin{cor}: 
Let $n\ge3$ and let $M$ be a compact n-manifold with
boundary $\partial M=\partial_0M\cup\partial_1M$, such that
$M-\partial_0 M$ admit incomplete hyperbolic metrics of finite volume
with totally geodesic boundary $\partial_1M$. \\
If $\partial_1 M\not=\emptyset$, then $\parallel DM\parallel < 2\parallel M,\partial M\parallel$.\end{cor}
\begin{pf}: Let $N$ be the manifold obtained from two copies of $M$
by glueing via the identity only of $\partial_1M$. $N$ is obtained from $DM$
by cutting along $\partial_0M$, hence, $\parallel N,\partial N\parallel=
\parallel DM\parallel$ by theorem 1. (Note that $\pi_1\partial_0M$ is amenable
because $\partial_0M$ is covered by an (n-1)-torus, see p.156 in \cite{bp},
for example.)\\
$N$ carries a hyperbolic metric of finite volume, in which $\partial_1 M$
is totally geodesic. Hence, we get $\parallel N,\partial N\parallel < 
\parallel N,\partial N\parallel^{\partial_1M}=2\parallel M,\partial M\parallel$.\end{pf}


 
\begin{thm}: If M is a compact 3-manifold with
$\parallel\partial M\parallel >0$, then \\
$\parallel DM\parallel < 2\parallel M,\partial M \parallel$.\end{thm}
\begin{pf}
First, we want
to reduce the claim to compact irreducible manifolds. For this purpose, note
that $\parallel M_1\sharp M_2,\partial\left(M_1\sharp M_2\right)\parallel
=\parallel M_1,\partial M_1\parallel + \parallel M_2,\partial M_2\parallel$
holds
also for manifolds with boundary (of dimension $\ge3$). Indeed, defining a
fundamental class of the wedge $M_1\vee M_2$ as $\left[M_1\vee M_2,\partial
\left(M_1\vee M_2\right)\right]:={i_1}_*\left[M_1,\partial M_1\right]
+{i_2}_*\left[M_2,\partial M_2\right]$, for the inclusions $i_1:M_1\rightarrow
M_1\vee M_2$ and $i_2:M_2\rightarrow M_1\vee M_2$, we can define
the simplicial volume $\parallel M_1\vee M_2,\partial\left(M_1\vee M_2
\right)\parallel$ as the infimum over the $l^1$-norms of relative cycles
representing $\left[M_1\vee M_2,\partial\left(M_1\vee M_2\right)\right]$
and it is implicit in the proof of theorem 1 that with this definition
$\parallel M_1\vee M_2,\partial\left(M_1\vee M_2\right)\parallel=\\
\sima +\simb$ holds.
Consider the projection from $M_1\sharp M_2$ to $M_1\vee M_2$ which 
pinches the connecting sphere to a point. It induces an isomorphism of
fundamental groups (this is the point, where dimension $\ge 3$ is needed)
and has degree 1. By the same argument as in the proof of lemma 1,
we get $\parallel M_1\sharp M_2,\partial\left(M_1\sharp M_2\right)\parallel
=\parallel M_1\vee M_2,\partial\left(M_1\vee M_2\right)\parallel$.

In the same way we get generally that identification of pairs of points in 
manifolds doesn't change the simplicial volume. In particular one has\\
$\parallel D\left(M_1\sharp M_2\right)\parallel = \parallel DM_1\parallel
+ \parallel DM_2\parallel$.

Any compact 3-manifold is a connected sum of 3-manifolds which are either irreducible
or $S^1\times S^2$. Since $\parallel S^1\times S^2\parallel=0$, we can reduce the claim to irreducible 3-manifolds because of
$\parallel D\left(M_1\sharp M_2\right)\parallel=\parallel DM_1\parallel + \parallel DM_2\parallel$
and \\
$\parallel M_1\sharp M_2,\partial\left(M_1\sharp M_2\right)\parallel =\parallel
M_1,\partial M_1\parallel + \parallel M_2,\partial M_2\parallel$. 

 
By the discussion at the beginning of
this section, we can cut $M$ along properly embedded disks and incompressible 
properly embedded annuli and tori such that the obtained pieces are either 
Seifert fibered, I-bundles or simple, 
where the simple pieces admit an (incomplete) hyperbolic metric
such that the toral boundary components correspond to cusps and the boundary components of higher
genus are totally geodesic.

We argue that these pieces $M_i$ satisfy the claim of theorem 2. For a Seifert
fibration, the boundary consists of tori, hence, there is nothing to prove.
If $M_i$ is an $I$-bundle, then $DM$ is an $S^1$-bundle and $\parallel DM_i\parallel=0$ holds by corollary 6.5.3 of \cite{thul}. (But, if $\parallel \partial M_i\parallel >0$, then
$\parallel M_i\parallel \ge\frac{1}{3}\parallel \partial M_i\parallel >0$ by the argument as in the remark on handlebodies below.) Finally, if 
$M_i$ is "simple" and $\parallel\partial M_i\parallel >0$, then
the totally geodesic boundary of the hyperbolic structure is non-empty (not
all boundary components can correspond to cusps), and $\parallel DM_i\parallel < 2\parallel M_i,\partial M_i\parallel$ holds by lemma 10.

If $M$ is a compact irreducible 3-manifold with $\parallel \partial M\parallel>0$, then clearly $\parallel \partial M_i\parallel > 0$ holds for at least one of
the pieces in its Jaco-Shalen-Johannson decomposition. To finish the
proof
of theorem 2, we still need the following lemma 11, where $M_F$ is defined as in the introduction.
\begin{lem}: Let $M$ be a compact 3-manifold 
and $F$ an incompressible, properly embedded annulus, torus or disk. \\
If $\parallel D\left(M_F\right)\parallel < 2\parallel M_F,\partial M_F\parallel$,
then $\parallel DM\parallel < 2\parallel M,\partial M\parallel$.\end{lem}
\begin{pf} The claim follows from corollary 3 if $F$ is a torus. \\
From now on, consider $F=A$ an annulus. \\
The claim will follow from the
somewhat paradoxical observation that \\
$\parallel M_A,\partial M_A\parallel
\le\parallel M,\partial M\parallel$, but $\parallel D\left(M_A\right)\parallel
\ge\parallel DM\parallel$.
 
$DM_A$ is obtained from $DM$ by cutting off
an incompressible torus
$T=DA$ and identifying afterwards in a different way the
pairs of incompressible annuli which are "halves" of
the same copy of T. In other words $\left(D\left(M_A\right)\right)_{2A}=\left(DM\right)_T$.
Look at the following picture, where dimension has been lowered by one.\\
\psset{unit=0.06\hsize}
$$\pspicture(-2,-2)(15,2)
\psarc(0,0){2}{15}{345}
\psarc(-0.5,0){0.4}{180}{360}
\psarc(-0.5,-0.1){0.3}{0}{180}
\psarc(2,0){0.5}{90}{270}
\psarc(1.5,0){0.7}{310}{50}
\psbezier[linewidth=2pt,linearc=.25](-0.2,-0.1)(1,0.7)(1.7,0.3)
\psbezier[linewidth=2pt,linearc=.25,linestyle=dashed](-0.6,-0.5)(1,-0.7)(2.1,-0.3)
\uput[0](0.7,0.7){A}
\uput[0](0.7,-2.2){M}
\psline{->}(3,0)(3.8,0.5)(4.6,-0.5)(5.4,0.5)(6.2,0)
\psbezier(-2,0)(-1.7,-0.2)(-1.4,-0.2)(-0.8,-0.2)
\psbezier[linestyle=dashed](-2,0)(-1.7,0.2)(-1.4,0.2)(-0.9,0)
\psarc(10,0){2}{90}{270}
\psarc(10,0){2}{315}{45}
\psbezier(10,2)(12.5,2.5)(13,2.5)(13.7,2.9)
\psbezier(11.5,1.5)(12.5,1.8)(13,1.8)(14,1.5)
\psbezier(10,-2)(12.5,-2.5)(13,-2.5)(13.8,-2.9)
\psbezier(11.5,-1.5)(12.5,-1.8)(13,-1.8)(14,-1.5)
\pscircle(14,2.2){0.7}
\pscircle(14,-2.2){0.7}
\psarc[linewidth=2pt,linearc=.25](14,2.2){0.7}{180}{360}
\psarc[linewidth=2pt,linearc=.25](14,-2.2){0.7}{0}{180}
\uput[0](14,0){2A}
\uput[0](9,-2.3){$M_A$}
\psbezier(8,0)(9,0.3)(11,0.3)(12,0)
\psbezier[linestyle=dashed](8,0)(9,-0.3)(11,-0.3)(12,0)
\psline{->}(0.3,-2.5)(0.8,-2.9)(0.3,-3.3)(0.8,-3.7)
\psline{->}(9.5,-2.8)(9.8,-3.1)(9.5,-3.4)(9.8,-3.7)
\endpspicture$$\\
\\
\\
\psset{unit=0.05\hsize}
$$\pspicture(-2,-2)(17,2)
\psarc(0,0){2}{15}{345}
\psarc(-0.5,0){0.4}{180}{360}
\psarc(-0.5,-0.1){0.3}{0}{180}
\psarc(2,0){0.5}{80}{100}
\psarc(2,0){0.5}{260}{280}
\psarc(4,0){2}{195}{165}
\psarc(4.5,0){0.4}{180}{360}
\psarc(4.5,-0.1){0.3}{0}{180}
\psbezier[linewidth=2pt,linearc=.25,linestyle=dashed](-0.2,-0.1)(1,0.3)(1.7,0.3)
(1.9,0.2)(2.7,0.2)(3.5,0.1)(4.2,-0.1)                         
\psbezier[linewidth=2pt,linearc=.25](-0.3,-0.3)(1,0.1)(2.3,0.1)(4.3,-0.3)
\uput[0](0.5,0.6){T}
\uput[0](1,-2.4){DM}
\psbezier(-2,0)(-1.7,-0.2)(-1.4,-0.2)(-0.8,-0.2)
\psbezier[linestyle=dashed](-2,0)(-1.7,0.2)(-1.4,0.2)(-0.9,0)
\psbezier(6,0)(5.7,-0.2)(5.4,-0.2)(4.8,-0.2)
\psbezier[linestyle=dashed](6,0)(5.7,0.2)(5.4,0.2)(4.9,0)
\psarc(10,0){2}{90}{270}
\psarc(10,0){2}{315}{45}
\psbezier(10,2)(12.5,2.5)(13,2.5)(13.7,2.8)
\psbezier(11.5,1.5)(12.5,1.8)(13,1.8)(14,1.5)
\psbezier(10,-2)(12.5,-2.5)(13,-2.5)(14,-3)
\psbezier(11.5,-1.5)(12.5,-1.8)(13,-1.8)(14,-1.5)
\psarc(14,2.2){0.7}{80}{180}
%\psarc[linestyle=dashed](14,2.2){0.7}{0}{90}
\psarc(14,-2.2){0.7}{180}{280}
%\psarc[linestyle=dashed](14,-2.2){0.7}{270}{360}
\psbezier(8,0)(9,0.3)(11,0.3)(12,0)
\psbezier[linestyle=dashed](8,0)(9,-0.3)(11,-0.3)(12,0)
\psarc(16,0){2}{270}{90}
\psarc(16,0){1.5}{110}{250}
\psbezier(16,2)(15.5,2.5)(14.5,2.5)(14.2,3)
\psbezier(15.5,1.5)(15,1.8)(14.5,1.8)(14,1.5)
\psbezier(16,-2)(15.5,-2.5)(14.5,-2.5)(14.2,-2.9)
\psbezier(15.5,-1.5)(15,-1.8)(14.5,-1.8)(14,-1.5)
\psbezier(14.5,0)(15,0.3)(17,0.3)(18,0)
\psbezier[linestyle=dashed](14.5,0)(15,-0.3)(17,-0.3)(18,0)
\psarc[linewidth=2pt,linearc=.25](14,2.2){0.7}{180}{270}
\psarc[linewidth=2pt,linearc=.25,linestyle=dashed](14,2.2){0.7}{270}{360}
\psarc[linewidth=2pt,linearc=.25,linestyle=dashed](14,-2.2){0.7}{0}{90}
\psarc[linewidth=2pt,linearc=.25](14,-2.2){0.7}{90}{180}
\uput[0](13.4,0){2A}
\uput[0](9,-2.5){$D\left(M_A\right)$}
\endpspicture$$
\\
\\
\\
Hence, we get $\parallel D\left(M_A\right)\parallel \ge \parallel\left(DM\right)_T\parallel
=\parallel DM\parallel$, what implies the claim of lemma 11 because of $\parallel M_A,\partial M_A\parallel\le\parallel M,\partial M\parallel$.\\
If $F$ is a disk, the argument is the same\end{pf}\\
This finishes the proof of theorem 2.\end{pf}\\
\\
\\
{\bf Handlebodies}. \\
Note that for any $n$-dimensional compact manifold holds:\\
$\parallel M,\partial M\parallel\ge\frac{1}{n}\parallel \partial M\parallel$. Namely, the 
boundary operator $\partial:H_n\left(M,\partial M\right)\rightarrow H_{n-1}\left(
\partial M\right)$ maps the relative fundamental class $\left[M,\partial M\right]$
to the fundamental class $\left[\partial M\right]$. It is obvious that $\parallel\partial\parallel\le n+1$. For a representative $\sum_{i=1}^ra_i\sigma_i$ of $\left[M,\partial M\right]$                                          
one even gets $\parallel \partial\sum_{i=1}^ra_i\sigma_i\parallel\le n\parallel\sum_{i=1}^r
a_i\sigma_i\parallel$, because each $\sigma_i$ has to have at least one face not in $\partial M$,
cancelling against some other face.\\
Let $H_g$ denote the 3-dimensional handlebody of genus $g$. $H_g$ is a (g-1)-fold
covering of $H_2$, hence, $\parallel H_g,\partial H_g\parallel =C\left(g-1\right)$.
From the above argument follows $C\ge\frac{4}{3}$. (In fact, we showed in \cite{kue}, by constructing
triangulations of handlebodies, that $\frac{4}{3}\le C\le 3$.)
 
The double of $H_g$ is the g-fold connected sum $S^2\times S^1\sharp\ldots\sharp S^2\times S^1$,
whose simplicial volume vanishes.
This shows that there may be manifolds
$M$ of arbitrarily large simplicial volume with $\parallel DM\parallel =0$.\\
\\
In fact, we can give a precise condition when $\parallel DM\parallel=0$ holds 
for a compact 3-manifold $M$. Recall 
that any compact 3-manifold can be cut along disks into finitely many pieces 
$M_j$ which have incompressible boundary. We claim that $\parallel DM\parallel=0$
if and only if all these $M_j$ have a
Jaco-Shalen-Johannson-decomposition without simple pieces in the sense
of \cite{neus}.

Namely, if $M_j$ has a JSJ-decomposition without simple pieces, one easily gets
$\parallel DM_j\parallel =0$, thus, $DM_j$ is a graph manifold by \cite{som1}.
$DM$ is obtained from $\cup DM_j$ by cutting off some 3-balls and identifying
their boundaries in pairs. By the same argument as in the proof of lemma 4
in \cite{som1}, $DM$ is then a graph manifold and $\parallel DM\parallel =0$.

To prove the other implication, assume that $M_j$ had a simple piece $H$, on which we put a hyperbolic metric with totally geodesic boundary. Inside $DM$
this gives us a submanifold $H^\prime$, obtained from two
copies of $H$ by glueing via the identity on {\em a submanifold of}
$\partial H$. Clearly, $H^\prime$ admits a hyperbolic metric with totally
geodesic boundary. From proposition 5, one can conclude $\parallel DM\parallel >0$.\\
\\
\\
\\


  




\begin{thebibliography}{29}
\bibitem{agol}I.\ AGOL, 'Topology of hyperbolic 3-manifolds`, PhD thesis, UCSD, 1998.
\bibitem{bp}R.\ BENEDETTI and C.\ PETRONIO, {\em Lectures on
Hyperbolic Geometry} (Universitext, Springer-Verlag, Berlin, 1992). 
\bibitem{gro}M.\ GROMOV, 'Volume and Bounded
Cohomology', {\em Public.\ Math.\ IHES} 56 (1982) 5-100.
\bibitem{hem}J.\ HEMPEL, {\em 3-Manifolds} (Princeton University Press, 1976).
\bibitem{iva}N.\ IVANOV, 'Foundations of the theory of
bounded cohomology', {\em J.\ Sov.\ Math.}\ 37 (1987) 1090-1114.
\bibitem{jun}D.\ JUNGREIS, 'Chains that realize the Gromov
invariant of hyperbolic manifolds',  {\em Ergodic Theory and Dynamical Systems} 17 (1997) 643-648.
\bibitem{kue}T.\ KUESSNER, 'Gromov volume of compact manifolds', diploma thesis, FU Berlin, 1996; http://www.mathematik.uni-tuebingen.de/~thilo/dipn.ps
\bibitem{k}T.\ KUESSNER, 'Efficient fundamental cycles of
cusped hyperbolic manifolds', preprint; http://front.math.ucdavis.edu/math.GT/0007003
\bibitem{mat}S.\ MATSUMOTO and S.\ MORITA, 'Bounded Cohomology of
Certain Groups of Homeomorphisms', {\em Proc.\ Amer.\ Math.\ Soc.}\ 94 (1985) 539-544.
\bibitem{neus}W.\ NEUMANN and G.\ SWARUP, 'Canonical decompositions of
3-manifolds', {\em Geometry and Topology} 1 (1997) 21-40.
\bibitem{som1}T.\ SOMA, 'The Gromov invariant of links', 
{\em Invent.\ Math.}\ 64 (1981) 445-454.
\bibitem{thul}W.\ THURSTON, 'The Geometry and Topology of
3-Manifolds', lecture notes; http://msri.org/publications/books/gt3m 
\bibitem{thua}W.\ THURSTON, 'Hyperbolic geometry and 3-manifolds',
{\em Low-dimensional topology}, Proc.\ Conf., Bangor/Engl.\ 1979, Vol.\ 1,
London Math.\ Soc.\ Lecture Note Series 48 (1982), pp.9-25.
\end{thebibliography}                                














\end{document}


math0109057.extracted_bib
\begin{thebibliography}{29}
\bibitem{agol}I.\ AGOL, 'Topology of hyperbolic 3-manifolds`, PhD thesis, UCSD, 1998.
\bibitem{bp}R.\ BENEDETTI and C.\ PETRONIO, {\em Lectures on
Hyperbolic Geometry} (Universitext, Springer-Verlag, Berlin, 1992). 
\bibitem{gro}M.\ GROMOV, 'Volume and Bounded
Cohomology', {\em Public.\ Math.\ IHES} 56 (1982) 5-100.
\bibitem{hem}J.\ HEMPEL, {\em 3-Manifolds} (Princeton University Press, 1976).
\bibitem{iva}N.\ IVANOV, 'Foundations of the theory of
bounded cohomology', {\em J.\ Sov.\ Math.}\ 37 (1987) 1090-1114.
\bibitem{jun}D.\ JUNGREIS, 'Chains that realize the Gromov
invariant of hyperbolic manifolds',  {\em Ergodic Theory and Dynamical Systems} 17 (1997) 643-648.
\bibitem{kue}T.\ KUESSNER, 'Gromov volume of compact manifolds', diploma thesis, FU Berlin, 1996; http://www.mathematik.uni-tuebingen.de/~thilo/dipn.ps
\bibitem{k}T.\ KUESSNER, 'Efficient fundamental cycles of
cusped hyperbolic manifolds', preprint; http://front.math.ucdavis.edu/math.GT/0007003
\bibitem{mat}S.\ MATSUMOTO and S.\ MORITA, 'Bounded Cohomology of
Certain Groups of Homeomorphisms', {\em Proc.\ Amer.\ Math.\ Soc.}\ 94 (1985) 539-544.
\bibitem{neus}W.\ NEUMANN and G.\ SWARUP, 'Canonical decompositions of
3-manifolds', {\em Geometry and Topology} 1 (1997) 21-40.
\bibitem{som1}T.\ SOMA, 'The Gromov invariant of links', 
{\em Invent.\ Math.}\ 64 (1981) 445-454.
\bibitem{thul}W.\ THURSTON, 'The Geometry and Topology of
3-Manifolds', lecture notes; http://msri.org/publications/books/gt3m 
\bibitem{thua}W.\ THURSTON, 'Hyperbolic geometry and 3-manifolds',
{\em Low-dimensional topology}, Proc.\ Conf., Bangor/Engl.\ 1979, Vol.\ 1,
London Math.\ Soc.\ Lecture Note Series 48 (1982), pp.9-25.
\end{thebibliography}