A Combination Theorem for PD(n)-Pairs

TL;DR

This paper establishes a new combination theorem for PD(n)-pairs, advancing the understanding of their structural properties in geometric group theory.

Contribution

It introduces a novel combination theorem specifically for PD(n)-pairs, expanding the toolkit for analyzing their algebraic and topological features.

Findings

Proves a new combination theorem for PD(n)-pairs

Provides tools for analyzing PD(n)-pair structures

Enhances understanding of their geometric properties

Abstract

We prove a combination theorem for PD(n)-pairs.

Equations

$$H^k(G;A)\stackrel{e\cap }{\to }{H}_{n-k}(G,\mathbf{S};C\otimes A)$$

$$H^k(G,\mathbf{S};A)\stackrel{e\cap }{\to }{H}_{n-k}(G;C\otimes A)$$

$$(M,\partial M)\stackrel{i}{\to }(M,\partial M\cup Y)\stackrel{j}{\leftarrow }(X,\partial X)$$

$$\to H_k(T)\to H_k(M,\partial M)\stackrel{({\varphi }_X,-{\varphi }_Y)}{\to }{H}_k(X,\partial X)\oplus H_k(Y,\partial Y)\to H_{k-1}(T)\to$$

$$\to H_k(U,\partial U)\to H_k(G,\partial G)\to H_k(G_1,\partial G_1)\oplus H_k(G_2,\partial G_2)\to H_{k-1}(U,\partial U)\to$$

$$\begin{array}{ccccccc}{H}_k(\partial U)& \stackrel{0}{\to }& H_{k-1}(\partial U)& \stackrel{(+1,-1)}{\to }& H_{k-1}(\partial U)\oplus H_{k-1}(\partial U)& \to & H_{k-1}(\partial U)\\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ H_k(U)& \to & H_k(G,\overline{\mathbf{S}})& \to & H_k(G_1,\overline{{\mathbf{S}}_1}\cup U)\oplus H_k(G_2,\overline{{\mathbf{S}}_2}\cup U)& \to & H_{k-1}(U)\\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ H_k(U,\partial U)& \to & H_k(G,\partial G)& \to & H_k(G_1,\partial G_1)\oplus H_k(G_2,\partial G_2)& \to & H_{k-1}(U,\partial U)\\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ H_{k-1}(\partial U)& \stackrel{0}{\to }& H_{k-2}(\partial U)& \stackrel{(+1,-1)}{\to }& H_{k-2}(\partial U)\oplus H_{k-2}(\partial U)& \to & H_{k-2}(\partial U)\end{array}$$

$$H_k(\partial G,\overline{\mathbf{S}})\to H_k(G,\overline{\mathbf{S}})\to H_k(G,\partial G)\to H_{k-1}(\partial G,\overline{\mathbf{S}})$$

$$H_k({\mathbf{S}}_1\cup {\mathbf{S}}_2\cup (W_1\underset{\partial U}{\ast }{W}_2),{\mathbf{S}}_1\cup {\mathbf{S}}_2\cup W_1\cup W_2)$$

$$H_k(\partial G_1,\overline{{\mathbf{S}}_1}\cup U)\to H_k(G_1,\overline{{\mathbf{S}}_1}\cup U)\to H_k(G_1,\partial G_1)\to H_{k-1}(\partial G_1,\overline{{\mathbf{S}}_1}\cup U)$$

$$H_k({\mathbf{S}}_1\cup (W_1\underset{\partial U}{\ast }U),{\mathbf{S}}_1\cup W_1\cup U)$$

$$0\to H_n(U,\partial U)\to H_n(G,\partial G)\to H_n(G_1,\partial G_1)\oplus H_n(G_2,\partial G_2)\to H_{n-1}(U,\partial U)\to$$

Full text

Keywords: PD(n)-pair, graph of groups, $n$-manifold, commutative diagram.

1. Introduction

A group $G$ is an orientable $PD(n)$-group if for any coefficient $ZG$-module $M$ it has Poincaré duality, i.e. there exists a fundamental class $[G]\in H_{n}(G,\mathbf{Z})$ such that the cap product with $[G]$ induces an isomorphism $H^{i}(G,M)\overset{[G]\cap}{\rightarrow}H_{n-i}(G,M)$, cf. [2] p.136.

A group $G$ and a family $\mathbf{S}=\{S_{i},i\in I\}$ of not necessary distinct subgroups of $G$, denoted $(G,\mathbf{S})$, is called a group pair. The definition of an orientable $PD(n)$-group has been generalized to group pairs as follows.

Definition 1**.**

cf. [1], Definition 4.1.

A group pair $(G,\mathbf{S})$ is called a duality pair of dimension $n$ if there exists a $G$-module $C$, called the dualizing module of $(G,\mathbf{S})$, and an element $e\in H_{n}(G,\mathbf{S};C)$ such that the cap-products

[TABLE]

and

[TABLE]

are isomorphisms for all $G$-modules $A$ and all $k\in\mathbf{Z}$.

The dualizing module $C$ is isomorphic as a $G$-module to $H^{n}(G,\mathbf{S};\mathbf{Z}G))\cong H_{0}(G;C\otimes\mathbf{Z}G)$, cf. [1], p.299.

The paper [4] has an excellent description of $PD(3)$-pairs.

Definition 2**.**

cf. [1], p. 302.

A duality pair $(G,\mathbf{S})$ of dimension $n$ is called a Poincaré duality pair of dimension $n$, in short a $PD(n)$-pair, if its dualizing module $C$ is isomorphic to $\mathbf{Z}$ as an Abelian group. In this case $C$ will be denoted $\tilde{\mathbf{Z}}$ and for any $G$-module $A$ the module $\tilde{\mathbf{Z}}\otimes A$ (with diagonal action) will be denoted $\tilde{A}$. If the $G$-action on $\tilde{\mathbf{Z}}$ is trivial, the $PD(n)$-pair is orientable, otherwise it is non-orientable.

The main result of this paper is motivated by the topology of manifolds.

Let $M_{1}$ and $M_{2}$ be compact $n$-manifolds with non-empty boundaries. Let $\{N^{i}_{1},k\geq i\geq 1\}$ and $\{N^{j}_{2},l\geq j\geq 1\}$ be the boundary components of $M_{1}$ and $M_{2}$ respectively. Let $L_{1}$ and $L_{2}$ be compact submanifolds of $N^{1}_{1}$ and $N^{1}_{2}$ such that there exists a homeomorphism $f:(L_{1},\partial L_{1})\rightarrow(L_{2},\partial L_{2})$. Then the space $M_{1}\underset{f(L_{1})=L_{2}}{\cup}M_{2}$ is an $n$-manifold with the boundary components $(N^{1}_{1}-L_{1})\underset{f(\partial L_{1})=\partial L_{2}}{\cup}(N^{1}_{2}-L_{2})$, $\{N^{i}_{1},k\geq i\geq 2\}$, and $\{N^{j}_{2},l\geq j>2\}$.

In order to formulate the corresponding statement for groups we need the following construction.

Definition 3**.**

To simplify the exposition we consider a graph to be an oriented one-dimensional $CW$-complex. Let $G_{1}$ and $G_{2}$ be the fundamental groups of graphs of groups $\Gamma_{1}$ and $\Gamma_{2}$ respectively. For any edge in those graphs which is a loop, insert an additional trivial vertex group in its middle. Choose vertices $v_{i}$ in $\Gamma_{i},i=1,2$ which have the same degree $n$ and let $\{e_{i}^{j},i=1,2,n\geq j\geq 1\}$ be all the edges in $\Gamma_{i}$ beginning at $v_{i}$. Let $U_{1}$ and $U_{2}$ be the vertex groups of $v_{1}$ and $v_{2}$ respectively, and let $\{P^{i}_{j},i=1,2,n\geq j\geq 1\}$ be the edge groups of $\{e_{i}^{j},i=1,2,n\geq j\geq 1\}$. Assume that there exists an isomorphism $\phi:(U_{1},\overset{n}{\underset{j=1}{\cup}}P^{j}_{1})\rightarrow(U_{2},\overset{n}{\underset{j=1}{\cup}}P^{j}_{2})$.

Define a new graph of groups, called the amalgamation of $\Gamma_{1}$ and $\Gamma_{2}$ along $\phi$, denoted by $\Gamma_{1}\underset{\phi}{*}\Gamma_{2}$, as follows.

1. (1)

Form the disjoint union of $\Gamma_{1}$ and $\Gamma_{2}$. 2. (2)

Remove the vertices $v_{i}$ from $\Gamma_{i}$. 3. (3)

Identify each pair of the edges $e_{1}^{j}$ and $e_{2}^{j},n\geq j\geq 1$ by an orientation-reversing homeomorphism.

Note that $\Gamma_{1}\underset{\phi}{*}\Gamma_{2}$ need not be connected.

The paper [3] has an excellent discussion of graphs of groups.

Theorem 1**.**

Let $(G_{1},\mathbf{S_{1}}\cup(U_{1}\underset{\partial U_{1}}{*}W_{1}))$ and $(G_{2},\mathbf{S_{2}}\cup(U_{2}\underset{\partial U_{2}}{*}W_{2}))$ be $PD(n)$-pairs such that $U_{1}\underset{\partial U_{1}}{*}W_{1}$ and $U_{2}\underset{\partial U_{2}}{*}W_{2}$ are fundamental groups of graphs of groups $\Gamma_{1}$ and $\Gamma_{2}$ respectively. Let $v_{i},i=1,2$ be vertices in $\Gamma_{i},i=1,2$ which have the same degree $n$ and let $\{e_{i}^{j},i=1,2,n\geq j\geq 1\}$ be all the edges in $\Gamma_{i}$ beginning at $v_{i}$. Let $U_{1}$ and $U_{2}$ be the vertex groups of $v_{1}$ and $v_{2}$ respectively, and let $\{P^{i}_{j},i=1,2,n\geq j\geq 1\}$ be the edge groups of $\{e_{i}^{j},i=1,2,n\geq j\geq 1\}$. Assume that there exists an isomorphism $\phi:(U_{1},\overset{n}{\underset{j=1}{\cup}}P^{j}_{1})\rightarrow(U_{2},\overset{n}{\underset{j=1}{\cup}}P^{j}_{2})$. Let $G$ be the amalgamated free product $G_{1}\underset{\phi(U_{1})=U_{2}}{\ast}G_{2}$. Note that the group $W_{1}\underset{\phi(\partial U_{1})=\partial U_{2}}{*}W_{2}$ is the set of the fundamental groups of the components of $\Gamma_{1}\underset{\phi}{*}\Gamma_{2}$. If all the subgroups $\{P^{i}_{j},i=1,2,n\geq j\geq 1\}$ are $PD(n-2)$-groups then $(G,(W_{1}\underset{\phi(\partial U_{1})=\partial U_{2}}{*}W_{2})\cup\mathbf{S_{1}}\cup\mathbf{S_{2}})$ is a $PD(n)$-pair.

Remark 1**.**

Bieri and Eckmann proved a special case of Theorem 1 for $(U_{i}\underset{\partial U_{i}}{*}W_{i})=U_{i},i=1,2$, cf. [1], Theorem 8.1. The main technical tool of their proof is Theorem 3.5 of [1] which has the following topological interpretation. Let $X$ and $Y$ be compact $n$-manifolds such that the boundary of $X$ is $\partial X=\partial_{1}X\cup T_{X}$ and the boundary of $Y$ is $\partial Y=\partial_{1}Y\cup T_{Y}$ where $T_{X}$ and $T_{Y}$ are homeomorphic compact $(n-1)$-manifolds. Let $M=X\underset{T_{X}=T_{Y}}{\cup}Y$ and let $T$ be the image of $T_{X}$ (which is also the image of $T_{Y}$) in $M$. Note that the inclusions

[TABLE]

induce homomorphisms on the homology $H_{k}(M,\partial M)\overset{i_{*}}{\rightarrow}H_{k}(M,\partial M\cup Y)\overset{j_{*}}{\leftarrow}H_{k}(X,\partial X)$, where the map $j_{*}$ is the excision isomorphism. Hence there exists a map $\phi_{X}:H_{k}(M,\partial M)\rightarrow H_{k}(X,\partial X)$. Similarly there exists a map $\phi_{Y}:H_{k}(M,\partial M)\rightarrow H_{k}(Y,\partial Y)$. The topological analog of Theorem 3.5 of [1] is that the following sequence

[TABLE]

is exact. When $k=n$ the first group in that sequence is trivial, the second and the fourth groups in that sequence are isomorphic to $\mathbf{Z}$, and the third group is isomorphic to $\mathbf{Z\oplus Z}$.

In order to prove Theorem 1 we will need the following technical result.

Theorem 2**.**

Let $(G_{i},\mathbf{S_{i}}),i=1,2$ be group pairs of dimension $n$. Let $U_{i}\underset{\partial U_{i}}{*}W_{i}$ be a subgroup of $G_{i},i=1,2$ such that $(U_{1},\partial U_{1})$ is isomorphic to $(U_{2},\partial U_{i})$. Let $\overline{\mathbf{S_{i}}}=\mathbf{S_{i}}\cup W_{i},i=1,2$ and let $G=G_{1}\underset{U_{1}=U_{2}}{*}G_{2}$. Denote the image of $(U_{1},\partial U_{1})$ in $G$ (which is equal to the image of $(U_{2},\partial U_{2})$ in $G$) by $(U,\partial U)$. Denote $\overline{\mathbf{S}}=\overline{\mathbf{S_{1}}}\cup\overline{\mathbf{S_{2}}}\cup W_{1}\cup W_{2}$. Denote $\partial G=\mathbf{S_{1}}\cup\mathbf{S_{2}}\cup(W_{1}\underset{\partial U_{1}=\partial U_{2}}{*}W_{2})$ and for $i=1,2$ denote $\partial G_{i}=\mathbf{S_{i}}\cup(U_{i}\underset{\partial U_{i}}{*}W_{i})$. Then the following long sequence

[TABLE]

is exact.

2. Proof of Theorem 2

Consider the following commutative diagram

[TABLE]

Theorem 3.5 of [1] implies that the second row of this diagram is exact. The first and the last columns of this diagram are exact because they are long exact sequences of pairs. The first and the last rows of the diagram split into short exact sequences.

Compare $H_{k}(G,\overline{\mathbf{S}})$ and $H_{k}(G,\partial G)$ using the exact sequence of the triple $(G,\partial G,\overline{\mathbf{S}})$.

[TABLE]

where the first term is, by definition,

[TABLE]

which is isomorphic by excision to $H_{k}(\partial U\times I,\partial U\times\partial I)\cong H_{k-1}(\partial U)$. It follows that the second column of the diagram is exact.

Compare $H_{k}(G_{1},\overline{\mathbf{S_{1}}}\cup U)$ and $H_{k}(G_{1},\partial G_{1})$ using the exact sequence of the triple $(G_{1},\partial G_{1},\overline{\mathbf{S_{1}}}\cup U)$.

[TABLE]

where the first term is, by definition,

[TABLE]

which is isomorphic by excision to $H_{k}(\partial U\times I,\partial U\times\partial I)\cong H_{k-1}(\partial U)$. It follows that the third column of the diagram is exact.

Using diagram chasing, it is easy to show that the third row of the diagram is exact, completing the proof of Theorem 2.

Note that if $G$ is a $PD(k)$-group then $(G,\{G,G\})$ is a $PD(k+1)$-pair.

3. Proof of Theorem 1

We use the notation of Theorem 2.

For $i=1,2$ let $\Gamma^{\prime}_{i}$ be the the graph $\Gamma_{i}-v_{i}-(\overset{n}{\underset{j=1}{\cup}}e_{i}^{j})$, and let $W_{i}$ be the fundamental group of $\Gamma^{\prime}_{i}$. Note that $\Gamma^{\prime}_{i}$ need not be connected. Hence for $i=1,2$ the group $U_{i}\underset{\partial U_{i}}{*}W_{i}$ can be considered as the fundamental group of the graph of groups with two vertex groups, namely $U_{i}$ and $W_{i}$, and $n$ edge groups, namely $\overset{n}{\underset{j=1}{\cup}}P^{j}_{i}$. By assumption, $U_{i}\underset{\partial U_{i}}{*}W_{i}$ is a $PD(n-1)$-group. Also by assumption, the subgroups $\{P^{i}_{j},i=1,2,n\geq j\geq 1\}$ are $PD(n-2)$-groups, hence part (ii) of Theorem 8.4 of [1] implies that $(U_{1},\overset{n}{\underset{j=1}{\cup}}P^{j}_{1})$ and $(U_{2},\overset{n}{\underset{j=1}{\cup}}P^{j}_{2})$ are $PD(n-1)$-pairs.

For $i=1,2$ Theorem 8.4 of [1] also implies that the dualizing modules of $U_{i}\underset{\partial U_{i}}{*}W_{i}$ and $(U_{i},\overset{n}{\underset{j=1}{\cup}}P^{j}_{i})$ are isomorphic to $H^{n}((U_{i}\underset{\partial U_{i}}{*}W_{i}),\mathbf{Z}(U_{i}\underset{\partial U_{i}}{*}W_{i}))$, considered as $U_{i}$-modules by restriction. Denote this dualizing module $C$. It has a $G$-module structure.

Theorem 2 states that the sequence

[TABLE]

is exact.

Choose fundamental classes $e_{i}\in H_{n}(G_{i},\partial G_{1}),i=1,2$ and $e_{0}\in H_{n-1}(U,\partial U)$ such that $\partial e_{1}=\partial e_{2}=e_{0}$, and consider the element $e\in H_{n}(G,\partial G)$ which maps onto $(e_{1},-e_{2})$. The diagram in the proof of Theorem 2 shows that $(G,\partial G)$ is a $PD(n)$-pair with dualizing module $C$ and fundamental class $e$, completing the proof of Theorem 1.

4. Acknowledgment

We would like to thank Peter Scott for helpful discussions.