On the decomposition of 3-dimensional Poincaré duality complexes

John Crisp

18 April 1996

School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia

ABSTRACT

We show that if the fundamental group of an orientable P D_{3}-complex has infinitely many ends then it is either a proper free product or virtually free of finite rank. It follows that every P D_{3}-complex is finitely covered by one which is homotopy equivalent to a connected sum of aspherical P D_{3}-complexes and copies of S^{1} × S^{2}. Furthermore, any torsion element of the fundamental group of an orientable P D_{3}-complex has finite centraliser.

AMS Subject Classification (1991): Primary 57P10 Key words and phrases: Poincaré duality complex, graph of groups, tree.

§0. INTRODUCTION

Let P be a P D_{3}-complex, P̃ its universal covering space, and π = π_{1}(P). It was known by Wall [6] that either π is finite and P̃ ≃ S^{3}, or π has one end and P̃ is contractible (P aspherical), or π has two ends and P is homotopy equivalent to one of P_{3}#P_{3}, S^{1} × P_{2}, S^{1} × S^{2} or S^{1} ˜×S^{2}, or π has infinitely many ends. Turaev since showed that if π is a proper free product, then P is a nontrivial connected sum of P D_{3}-complexes [5], but it is still not known whether every orientable P D_{3}-complex that has a fundamental group with infinitely many ends is such a nontrivial connected sum, as is the case for orientable 3-manifolds. If this question, of [6], were to be answered in the affirmative, then it would mean that every orientable P D_{3}-complex P decomposes as the connected sum of P D_{3}-complexes P_{i} such that P̃_{i} is either a homotopy 3-sphere, a homotopy 2-sphere, or contractible. Theorem 3.4 implies that this is at least true of some finite cover of P. Theorem 3.7 gives still further constraints on the occurrence of torsion in π. However, this is not enough to completely settle Wall’s question. In particular, the question raised in [3] as to whether S_{3} ∗_{C_{2}} S_{3} may be the fundamental group of an orientable P D_{3}-complex remains unanswered. Here S_{3} denotes the symmetric group on three elements.

The approach here is motivated by ideas in Hillman’s paper [3]. There the groups H_{∗}(C, H̄^{1}(π, Zπ)), for C a cyclic subgroup of π, are known by duality and a spectral sequence argument. In §1 we outline an interpretation for such homologies in the more general setting of groups acting on trees. The calculations in that context are independent of any duality properties. In §2 we make precise the application of this to P D_{3}-complexes, via the accessibility of π, and the Mayer-Vietoris sequence of [1] for graphs of groups. These observations lead naturally to the results in §3 where we also recover the main result of [3] as Corollary 3.8.

§1. TREES WITH ∞-VERTICES

In this section we introduce the notion of a module Π being presented by a tree X with ∞-vertices, and proceed to calculate the homology groups H_{∗}(C, Π) of a prime order cyclic group C acting by automorphisms on X. These turn out to be precisely determined by properties of the subtree of fixed points of X under the action of C.

We follow [2] in defining a graph X to be the disjoint union of a pair of sets EX and V X, called the edge and vertex sets respectively, together with a pair of functions o, t : EX → V X which specify for each edge ϵ an original vertex o(ϵ), and a terminal vertex t(ϵ). A nonempty graph X is called a tree if there exists a unique reduced path (called a geodesic) between any two distinct vertices, or equivalently if X is connected and contains no closed loops. A graph automorphism g : X → X is a bijection such that g(EX) = EX, g(V X) = V X, and, for ϵ ∈ EX, o(g(ϵ)) = g(o(ϵ)) and t(g(ϵ)) = g(t(ϵ)). These compose (on the left) in the obvious way, and if a group G acts on the tree (or graph) X by automorphisms then we shall call X a G-tree (or G-graph respectively).

1.1 DEFINITION : A G-tree with ∞-vertices is a tree X with a distinguished G-invariant subset V_{f}X ⊂ V X consisting of finite valence vertices (i.e: having only finitely many adjacent edges). Vertices which do not lie in V_{f}X are said to be ∞-vertices. (Note that an ∞-vertex need not have infinite valence).

Henceforth we shall assume every tree X to have this extra structure, and any graph automorphism of X to preserve the set V_{f}X. Note that an ∞-vertex structure is also inherited by any subtree of X simply by restriction of the set V_{f}X.

To any tree X with ∞-vertices we may associate a Z-module Π[X], which is said to be presented by the tree X, as follows. Let Z[V_{f}X] and Z[EX] denote the free Z-modules with bases V_{f}X and EX respectively. Then Π[X] is defined to be the cokernel of the map Δ : Z[V_{f}X] ↪ Z[EX] defined for each ν ∈ V_{f}X by the formula

Δ(ν) = ∑_{{ϵ|t(ϵ)=ν}} ϵ − ∑_{{ϵ|o(ϵ)=ν}} ϵ .  (1)

Furthermore, if X is a G-tree then Π[X] naturally inherits a left ZG-module structure.

1.2 EXAMPLE : In §2 to follow we find that if G is an accessible group then the ZG-module H^{1}(G, ZG) is presented by any terminal G-tree. This includes the case that G is finitely presentable ([2], Theorem VI.6.3) and, in particular, the case that G is the fundamental group of a P D_{3}-complex.

1.3 DEFINITION : For a tree X define the set of ends EX to be the set of classes of rays (i.e: infinite paths from a vertex in X) which differ as sets only within a bounded set. (Explicitly , rays γ_{1} and γ_{2} are equivalent if γ_{1} ∪ γ_{2} \ γ_{1} ∩ γ_{2} is a finite subset of X). Consequently we may think of there being a unique geodesic from any vertex in X to any end in EX, or for that matter from any end to any other end. Let e(X) = |EX| denote the number of ends of X, and ∞(X) = |V X \ V_{f}X| the number of ∞-vertices, each of which may be infinite. Finally write

ξ(X) = e(X) + ∞(X) − 1 .

For convenience we refer to an element of the union EX ∐ V X\V_{f}X as an end/∞-vertex. Thus ξ(X) + 1 measures the number of end/∞-vertices of X.

1.4 LEMMA : Let X be a tree with ∞-vertices. Then Π[X] is a free Z-module of (possibly infinite) rank R = max{ξ(X), 0}.

Proof : Suppose that x = ∑_{ϵ∈EX} m_{ϵ}ϵ represents a nontrivial element [x] ∈ Π[X]

such that n[x] = 0 for some n ≠ 0. That is n.x = ∑_{ν∈V_{f} X} n_{ν}Δ(ν), for some

n_{ν} which are zero for all but finitely many vertices. We shall formally extend such an expression to a sum over all of V X by setting n_{ν} = 0 whenever ν is an ∞-vertex. By a choice of the representative x, we may assume that 0 ≤ n_{ν} < n for each ν ∈ V X. But then, since n.m_{ϵ} = n_{t(ϵ)} −n_{o(ϵ)}, our choice forces m_{ϵ} = 0 for each ϵ ∈ EX. That is x = 0 contradicting the nontriviality of [x]. Thus Π[X] has no torsion elements and so is free as a Z-module.

Write rk[X] for the rank of Π[X] as a (free) Z-module. Given any edge ϵ_{0} in X define the subtrees X_{+} and X_{−} to be the components of X \ {ϵ_{0}} containing the vertices t(ϵ_{0}) and o(ϵ_{0}) respectively, with V_{f}X_{±} defined by restriction of V_{f}X. Any edge ϵ_{0} of a given tree X represents an element of Π[X] which we shall write as [ϵ_{0}]_{X}, suppressing the subscript whenever there is no ambiguity. We have [ϵ_{0}] = 0 exactly when we may write ϵ_{0} = ∑_{ν∈V X} n_{ν}Δ(ν) where n_{ν} is

zero except for finitely many vertices ν ∈ V_{f}X. By connectedness of trees this is equivalent to

n_{ν} = \{ \begin{matrix}n_{o(ϵ_{0})} & if ν ∈ X_{−} \\ n_{o(ϵ_{0})} + 1 & if ν ∈ X_{+} , \end{matrix}

since, for ϵ ∈ EX, n_{t(ϵ)} − n_{o(ϵ)} = 1 if ϵ = ϵ_{0} and is 0 otherwise. Thus

[ϵ_{0}] = 0 ⟺ one of X_{+} or X_{−} is finite with no ∞-vertices

⟺ one of X_{+} or X_{−} has no ends and no ∞-vertices ⟺ either ξ(X_{+}) = −1 or ξ(X_{−}) = −1 .

Note that finite vertices have finite valence, and a finite valence graph with no ends must be finite.

If rk[X] > 0 we may choose an edge ϵ_{0} such that [ϵ_{0}] ≠ 0. Then each of X_{+} and X_{−} has either an ∞-vertex or an end, and so ξ(X) > 0. Conversely, if ξ(X) > 0 there are at least two distinct end/∞-vertices, and choosing ϵ_{0} to lie on the geodesic joining them we arrange that X_{+} contains one end/∞-vertex and X_{−} the other. But then [ϵ_{0}] ≠ 0 and rk[X] > 0. Thus we have shown that rk[X] = 0 ⇔ ξ(X) ≤ 0.

Let X_{+}^{∘} and X_{+}^{•} denote the trees X_{+} ∪{ϵ_{0}, o(ϵ_{0})} with V_{f}X_{+}^{∘} = V_{f}X_{+} and V_{f}X_{+}^{•} = V_{f}X_{+} ∪ {o(ϵ_{0})}. Similarly define X_{−}^{∘} and X_{−}^{•}. Observe that, for any choice of edge ϵ_{0} in X, the free Z-module Π[X] may be expressed thus as the quotient of a sum of free Z-modules by a single relation :

Π[X] ≅ \frac{Π[X_{+}^{∘}] ⊕ Π[X_{−}^{∘}]}{([ϵ_{0}]_{X_{+}^{∘}}, 0) ∼ (0, [ϵ_{0}]_{X_{−}^{∘}})} , or ⟨Π[X_{+}^{∘}], Π[X_{−}^{∘}] | [ϵ_{0}]_{X_{+}^{∘}} = [ϵ_{0}]_{X_{−}^{∘}}⟩ .

If we may assume that either [ϵ_{0}]_{X_{+}^{∘}} ≠ 0 or [ϵ_{0}]_{X_{−}^{∘}} ≠ 0 then, the relation above being non-vacuous, it follows that

rk[X] = rk[X_{+}^{∘}] + rk[X_{+}^{∘}] − 1 .

Suppose that rk[X] > 0, and choose ϵ_{0} such that [ϵ_{0}]_{X} ≠ 0. Then the assumption above is valid and since, in fact, both [ϵ_{0}]_{X_{+}^{∘}} ≠ 0 AND [ϵ_{0}]_{X_{−}^{∘}} ≠ 0 we may equally well argue that

rk[X_{±}^{•}] = rk[X_{±}^{∘}] + rk[∘−^{ϵ_{0}}•] − 1
= rk[X_{±}^{∘}] − 1 .

Thus we have rk[X] = rk[X_{+}^{•}] + rk[X_{−}^{•}] + 1 . On the other hand, it is easily checked that ξ(X) = ξ(X_{+}^{•}) + ξ(X_{−}^{•}) + 1 where ξ(X_{±}^{•}) = ξ(X_{±}) ≥ 0, as [ϵ_{0}]_{X} is nontrivial. We may now argue by induction on m = min{rk[X], ξ(X)} to show that rk[X] = ξ(X) whenever m is finite and non-negative. But this leaves only the case rk[X] = ξ(X) = ∞ and we are done. •

1.5 COROLLARY : If G is an infinite group and X a terminal G-tree (see §2) then

ends(G) = e(X) + ∞(X) .

Proof : This follows immediately from Lemma 2.1 and Lemma 1.4 above. •

Suppose now that X is a C-tree, where C = ⟨g⟩ denotes a finite cyclic group of prime order p, and write Π = Π[X] for the ZC-module presented by X. Note that the set X^{C} of fixed points of X under the action of C is also a tree with ∞-vertices, in fact a subtree of X, where we set V_{f}X^{C} = V_{f}X ∩ X^{C}. To compute the homology H_{q}(C, Π), for q > 0, we need now to define the norm map N̄ : Π_{C} → Π^{C} with respect to C, which is induced by the map N : Π → Π^{C} given by N(x) = x + g(x) + ... + g^{p−1}(x) for x ∈ Π. The fixed subtree X^{C} alone shall determine precisely the kernel and cokernel of N̄.

For ν ∈ V X^{C}, let O_{ν} (resp. T_{ν}) denote the set of edges of X which originate (resp. terminate) at ν, and write E_{ν} = O_{ν} ∪ T_{ν}. For ϵ ∈ E_{ν}, write X_{ϵ} for the tree (with ∞-vertices) obtained from the connected component of X \ {ν} containing ϵ by replacing ν with an infinite vertex. Then it follows that

Π[X] ≅ ( ⨁_{ϵ∈E_{ν}} Π[X^{ϵ}])/Z⟨r⟩ as a Z-module,

where the relator r is just ∑_{ϵ∈T_{ν}}[ϵ]− ∑_{ϵ∈O_{ν}}[ϵ] when ν ∈ V_{f}X and trivial otherwise.

Since g fixes ν it clearly acts on the sets O_{ν} and T_{ν}, and permutes the connected components of X \ {ν} hence inducing Z-isomorphisms Π[X_{ϵ}] →^{≅} Π[X_{g(ϵ)}] such that, in particular, g([ϵ]) = [g(ϵ)]. Write x ∈ Π[X] as x = [x_{ϵ}]_{ϵ∈E_{ν}} whenever x is represented by (x_{ϵ})_{ϵ∈E_{ν}} ∈ ⨁_{ϵ∈E_{ν}} Π[X_{ϵ}].

1.6 CLAIM : Let X be a C-tree for C = ⟨g⟩ a cyclic group of prime order p, fix ν ∈ V X^{C} and define X_{ϵ}, for ϵ ∈ E_{ν}, as above. Suppose that x = [x_{ϵ}]_{ϵ∈E_{ν}} is an element of Π[X]^{C}. Then g(x_{ϵ}) = x_{g(ϵ)} holds in Π[X_{g(ϵ)}] for each ϵ ∈ E_{ν}.

Proof : The hypothesis is that x − g(x) = 0, or equivalently that [x_{ϵ} − g(x_{g^{−1}(ϵ)})]_{ϵ∈E_{ν}} = 0 in Π[X]. The result then follows trivially in the case ν is an ∞-vertex, and otherwise we must have (x_{ϵ} − g(x_{g^{−1}(ϵ)}))_{ϵ∈E_{ν}} = (m_{ϵ}[ϵ])_{ϵ∈E_{ν}} where m_{ϵ} = m if ϵ ∈ T_{ν} and m_{ϵ} = −m if ϵ ∈ O_{ν}, for m ∈ Z fixed. Thus for fixed ϵ ∈ T_{ν} the following p equations hold in Π[X_{ϵ}],

g^{p−i}(x_{g^{i}(ϵ)} − g(x_{g^{i−1}(ϵ)})) = g^{p−i}(m[g^{i}(ϵ)]) = m[ϵ] , for i = 1, .., p .

Adding these gives p.m[ϵ] = 0 in Π[X_{ϵ}]. But since Π[X_{ϵ}] is torsion free (by Lemma 1.4) it follows that m[ϵ] = 0 for each ϵ ∈ T_{ν}. By a similar argument for ϵ ∈ O_{ν} we are done in all cases. •

1.7 REMARK : It follows from Claim 1.6 that if x ∈ Π[X]^{C} is represented as a sum of elements from only those Π[X_{ϵ}] such that g(ϵ) ≠ ϵ, then x = N(y) for some y ∈ Π[X].

1.8 LEMMA : Let X be a C-tree for C = ⟨g⟩ a cyclic group of prime order p, and let Π = Π[X] be the ZC-module presented by X. Then the norm map N̄ : Π_{C} → Π^{C} has cokernel (Z/pZ)^{R} with R = rk(Π[X^{C}]) = max{(ξ(X^{C}), 0}.

Proof : Write Π̂ = Π[X^{C}]. We may define a quotient map ϕ : Π → Π̂ with kernel K̂ = ⟨[ϵ] : ϵ ∈ EX \ X^{C}⟩. Since ϕ is a ZC-map (K̂ a ZC-submodule) it induces ϕ^{C} : Π^{C} → Π̂ = Π̂^{C} with kernel K̂^{C} = K̂ ∩ Π^{C} , and a well-defined map N̂ : Π̂ → Π̂ such that N̂ ∘ ϕ = ϕ^{C} ∘ N. We shall show that K̂^{C} ⊂ Im(N). It follows that coker(N̄) = coker(N) = coker(ϕ^{C} ∘ N) = coker(N̂) , and, since Im(N̂) = pΠ̂, the Lemma is easily proved with the help of Lemma 1.4.

Take an arbitrary x ∈ K̂^{C}, writing x = ∑_{i=1}^{k} m_{i}ϵ_{i} with each ϵ_{i} ∈ EX \ X^{C}.

Let T denote the minimal C-subtree of X such that ϵ_{i} ∈ T for each i = 1, .., k. Note that T is finite and T ∩ X^{C} is a subtree of X^{C} (nonempty since C, being finite, always has fixed points). If ν_{∞} is an ∞-vertex in T ∩ X^{C} , then the trees X_{ϵ}, for ϵ ∈ E_{ν_{∞}} , are in fact subtrees of X. For ϵ ∈ E_{ν_{∞}} , write X_{Cϵ} for the subtree which is the union of the trees X_{g^{i}(ϵ)} , for i = 1, .., p. Then Π[X] is a direct sum, over the orbits Cϵ, of the ZC-submodules Π[X_{Cϵ}] , and each component of x, corresponding to this sum, is itself an element of K̂^{C}. Treating such components individually, the proof that x ∈ Im(N) ultimately reduces to cases (∗) where T ∩ X^{C} either consists of a single ∞-vertex ν_{0}, or contains no ∞-vertices at all.

Given ϵ ∈ X^{C} such that o(ϵ) has at least two adjacent edges and neither o(ϵ) nor t(ϵ) is an ∞-vertex, define a new C-tree X′ by removing ϵ from X and identifying t(ϵ) with o(ϵ). Clearly X′ and X present isomorphic ZC-modules, since this modification corresponds to simply eliminating the generator ϵ via the relation Δ(o(ϵ)) = 0. By a finite (or zero) number of such operations on the edges of T ∩ X^{C} we arrive, in all cases (∗), at a tree X′′ with subtree T′′ corresponding to T such that T′′ ∩X^{′′C} consists of a single vertex ν_{0}. Note that via the isomorphism Π[X] ≅ Π[X′′] the element x is now expressed in terms of edges ϵ′′_{1}, .., ϵ′′_{k} ∈ T′′. Thus applying Claim 1.6 at the vertex ν_{0} in X′′, and the Remark 1.7, we have that x ∈ Im(N). •

1.9 LEMMA : Let X be a C-tree for C = ⟨g⟩ a cyclic group of prime order p, and let Π = Π[X] be the ZC-module presented by X. Then the norm map N̄ : Π_{C} → Π^{C} has kernel Z/pZ if ξ(X^{C}) = −1 , and 0 otherwise.

Proof : For convenience we shall write M = Z[EX], K = Δ(Z[V_{f}X]) and ϕ : M → Π for the surjective map with kernel K. Now N is induced under ϕ by a similar map N on M, so that the following diagram commutes

M  \overline{N} \overline{↠} N (M) ⊂ M^{C}

[figure: p ∘ ϕ ↓ ↓; ........; ϕ′; ^{........}; ϕ|_{N (M)}]

Π_{C}  \overline{N̄} \overline{}Π^{C}

where p is the obvious projection of Π onto its C-coinvariant module. Write K′ for ker(p ∘ ϕ), and note that K + (g − 1)M ⊆ K′. Conversely, if x ∈ K′ then ϕ(x) ∈ ker p = (g−1)Π and so ϕ(x) = (g−1)ϕ(y) = ϕ((g−1)y) for some y ∈ M. Thus x ∈ K +(g−1)M, proving the equality K′ = K +(g−1)M. Furthermore, M is a direct sum of copies of the augmentation module Z and a free ⟨g⟩- module, and so H_{2}(⟨g⟩, M) = 0. That is ker N = (g − 1)M. Consequently, there exists a map ϕ′ which commutes with the diagram (as shown) and such that ker ϕ′ = N (K).

Now, by the diagram, ker N̄ = ϕ′(K^{C} ∩ N (M)) which may be identified with the quotient (K^{C} ∩ N (M))/N (K) . Note that K^{C} is freely generated as a Z-module by the set V_{f}X^{C} ∪ {N (Δ(ν)) : ν ∈ V_{f}X \ V_{f}X^{C}}, while N (K) is Z-free on {N (Δ(ν)) : ν ∈ V_{f}X}. Thus K^{C}/N (K) is simply a Z/pZ-vector space with basis V_{f}X^{C}, and ker N̄ is just the subspace of those elements whose representatives lie also in N (M).

Take any element ā ∈ kerN̄ with representative a ∈ K^{C} ∩ N (M) which we may assume to have the form a = ∑_{ν∈V X^{C}} n_{ν}Δ(ν) where 0 ≤ n_{ν} < p and n_{ν}

is zero except for finitely many ν ∈ V_{f}X^{C}. Expressing a in terms of the free Zbasis EX for M, the coefficient m_{ϵ} of ϵ ∈ EX will be given as m_{ϵ} = n_{t(ϵ)}−n_{o(ϵ)}. Since a ∈ N (M) and N (M) is generated (Z-freely) by the set

{N (ϵ) : ϵ ∈ EX} = {pϵ : ϵ ∈ EX^{C}} ∪ {(1 + g + .. + g^{p−1})ϵ : ϵ ∈ EX \ EX^{C}} ,

we must have m_{ϵ} ≡ 0 (mod p) for all ϵ ∈ EX^{C}, and hence n_{o(ϵ)} = n_{t(ϵ)} (since 0 ≤ n_{ν} < p). By the connectedness of X^{C}, the coefficients n_{ν} must then take a value n_{0} constant over the whole set of fixed vertices. It now follows that ā can be chosen to be nontrivial (that is n_{0} ≠ 0), if and only if X^{C} is finite and has no ∞-vertices, that is ξ(X^{C}) = −1. Furthermore, any such choice of ā will have order p and generate the whole of ker N̄. •

We can summarise the previous two lemmas by the following:

1.10 THEOREM : Suppose that the finite cyclic group C of prime order p acts on the tree with ∞-vertices X, and let Π = Π[X] be the left ZC-module presented by X. Then

H_{i}(C, Π) = \{ \begin{matrix}(Z/pZ)^{R_{+}} & if i odd, \\ (Z/pZ)^{R_{−}} & if i > 0 even, \end{matrix}

where R_{+} = max{ξ(X^{C}), 0} and R_{−} = max{−ξ(X^{C}), 0}.

Proof : This follows immediately from Lemmas 1.8 and 1.9 given that we may calculate the homology of C from the kernel and cokernel of the norm map. •

§2. ACCESSIBLE GROUPS

A G-tree X with ∞-vertices is said to be terminal (see [2]) if each edge stabiliser is finite and each vertex stabiliser has at most one end, and V_{f}X is taken to be precisely the set of vertices with finite stabilisers. A group G is said to be accessible if there exists a terminal G-tree. For a group G we shall consider the group cohomology H^{1}(G, ZG) as a left ZG-module with action defined in terms of the natural right action by g.m = xg^{−1} for g ∈ G and x ∈ H^{1}(G, ZG).

2.1 LEMMA : Let G be an accessible group with terminal G-tree X. Then X presents H^{1}(G, ZG) as a left ZG-module.

Proof : Recall that a graph of groups (G, Y) consists of a graph Y together with a collection G of groups G_{v}, for each v ∈ V Y, and subgroups G_{e} ⊂ G_{o(e)}, for each e ∈ EY, with injective homomorphisms ϕ_{e} : G_{e} → G_{t(e)}. Fixing a choice of maximal subtree T of Y, we define the fundamental group of (G, Y) as

π_{1}(G, Y, T) = ⟨t_{e} , G_{v} | rel G_{v} , t_{e}at_{e}^{−1} = ϕ_{e}(a) for a ∈ G_{e} , t_{e} = 1 for e ∈ ET ⟩

noting that up to isomorphism this group is independent of the choice of T.

By the Bass-Serre structure theorem ([4], I.5.4 Theorem 13) G is the fundamental group of a finite graph of groups (G, Y) where Y = G\X and the edge and vertex groups of (G, Y) are isomorphic to the corresponding edge and vertex stabilisers of X respectively. Furthermore, X is isomorphic to the G-tree X̃ defined with vertex and edge sets

V X̃ = ∐_{v∈V Y} G/G_{v} , EX̃ = ∐_{e∈EY} G/G_{e} ,

such that o(gG_{e}) = gG_{o(e)} and t(gG_{e}) = gt_{e}G_{t(e)} for g ∈ G and e ∈ EY, and with G acting by left multiplication on cosets. Since X is a terminal G-tree the vertex groups of Y have at most one end, so that the cohomology Mayer-Vietoris sequence of [1] gives rise to a short exact sequence of right ZG-modules

0 → ⨁_{v∈V_{f} Y} Z[G_{v}\G] ⟶^{Δ_{r}} ⨁_{e∈EY} Z[G_{e}\G] ⟶ H^{1}(G, ZG) → 0 ,

where V_{f}Y = {v ∈ V Y | G_{v} finite} or just the set G\V_{f} X̃. If we choose to consider this as a sequence of left ZG-modules and left ZG-maps (via the antiisomorphism g ↦ g^{−1} of G), then the first two modules are naturally isomorphic to Z[V_{f} X̃] and Z[EX̃] respectively (by taking the coset G_{v}g to g^{−1}G_{v} etc..) and one can check that Δ_{r} becomes exactly the map Δ : Z[V_{f} X̃] ↪ Z[EX̃] of Definition 1.1. Thus X̃, or equivalently X, presents H^{1}(G, ZG) as a left ZG-module. •

§3. RESULTS ON P D_{3}-COMPLEXES

A connected finitely dominated CW-complex P with a homomorphism w : π_{1}(P) → {±1} is called a connected Poincaré complex of formal dimension n, or P D_{n}-complex, if it exhibits the equivariant Poincaré duality of an ndimensional manifold with orientation class w. If Π is a left (resp. right) Zπ-module, denote by Π̄ the left Zπ-module with the same underlying abelian group, and action given by g.x = w(g)gx (resp. w(g)xg^{−1}) for all g ∈ π, x ∈ Π.

3.1 LEMMA (Hillman, [3]) : Let P be a P D_{3}-complex with infinite fundamental group π. If C is a finite cyclic subgroup of π then there are isomorphisms H_{s}(C, H̄^{1}(π, Zπ)) ≅ H_{s+3}(C, Z) for all s ≥ 1.

Proof : This follows from the spectral sequence for the projection of P̃ onto P̃/C with the information that H_{q}(P̃; Z) = Z, 0, H̄^{1}(π, Zπ), 0, ... from the duality isomorphisms and the fact that P̃ is simply connected. (Note that since P is finitely dominated we may assume, without loss, that P is a finite 3-dimensional CW-complex). •

The thrust of the earlier sections §1 and §2 was to be able to calculate these homology groups independently of Lemma 3.1 (in fact without using duality) as is seen from Theorem 1.10 and the following lemma together.

3.2 LEMMA : Given a group π and homomorphism w : π → {±1}, let Π be a Zπ-module and Π̄ as above. If C = ⟨g⟩ is a finite cyclic subgroup of π then

H_{i}(C, Π̄) ≅ \{ \begin{matrix}H_{i}(C, Π) & if i > 0 and w(g) = 1, \\ H_{i+1}(C, Π) & if i > 0 and w(g) = −1. \end{matrix}

Proof : The case for w(g) = 1 is easy since then Π̄ ≅ Π as ZC-modules. In the case w(g) = −1, Π̄ and Π are distinguished as ZC-modules only by the action of g, whereby the homology, H_{∗}(C, Π̄), of the complex

[figure: … ⟶; Π̄; \overline{1 − g}; Π̄; \overline{1 + g}; Π̄; \overline{1 − g}; Π̄; 0 ,]

is just that of the complex

[figure: … ⟶; Π; \overline{1 + g}; Π; \overline{1 − g}; Π; \overline{1 + g}; Π; 0 ,]

The lemma now follows by comparing this with the complex for H_{∗}(C, Π). •

3.3 REMARK : Given a P D_{3}-complex P with infinite fundamental group π, then since π is finitely presentable, it is accessible by Theorem VI.6.3 of [2], and there exists a terminal π-tree X, as in Lemma 2.1, which presents Π = H^{1}(π, Zπ) as a left Zπ-module. It is now clear, by Lemmas 3.1, 3.2, and Theorem 1.10, that if g is an element of π of prime order then exactly one of the following two cases must hold. Either w(g) = 1 and ξ(X^{⟨g⟩}) = −1, or w(g) = −1 and ξ(X^{⟨g⟩}) = 1. We shall apply this to prove the following two theorems.

3.4 THEOREM : Let P be an orientable P D_{3}-complex with π = π_{1}(P). Then π either has one end, is a proper free product, or is virtually free of finite rank. That is to say that P is either an aspherical complex, a nontrivial connected sum (by Turaev [5]), or finitely covered by some P̂ ≃ #^{k}(S^{1} × S^{2}).

Proof : Let X be the π-tree of Remark 3.3 and (G, Y) the associated graph of group, which may be assumed to be finite since π is finitely generated. Assume that π is not a proper free product, and hence that the edge groups of (G, Y) are all nontrivial. It now suffices to show that either the vertex groups of (G, Y) are all finite, for then π must be virtually free of finite rank (see [4], II.2.6), or π has one end.

Suppose that (G, Y) has an infinite vertex group. Then either this is the only vertex group and π has one end, or it is one of the vertex groups G_{o(e)} or G_{t(e)} for some edge e ∈ EY. But in the latter case we show that both these groups are finite thus reaching a contradiction. Since G_{e} is nontrivial and finite, we may choose a nontrivial g ∈ G_{e} ⊂ G_{o(e)} of prime order. By assumption, π is infinite and Remark 3.3 applies to show that, since P is orientable, w(g) = 1 and ξ(X^{⟨g⟩}) = −1. That is X^{⟨g⟩} is finite with no ∞-vertices. But G_{o(e)} is the stabiliser of some vertex of X which, since g ∈ G_{o(e)}, must lie in X^{⟨g⟩}. Thus G_{o(e)} must be a finite group. Finally take g ∈ ϕ_{e}(G_{e}) ⊂ G_{t(e)} to show similarly that G_{t(e)} is finite. •

3.5 COROLLARY : Let P be an orientable P D_{3}-complex. Then P is homotopy equivalent to a connected sum V #P_{1}#...#P_{m} where each P_{i}, for i = 1, .., m, is an aspherical P D_{3}-complex and V is a P D_{3}-complex with virtually free fundamental group.

Proof : This follows from Theorem 3.4 by Turaev’s theorem ([5], Theorem 1) and the fact that π_{1}(P) is finitely presented, and by observing that A ∗ B is a virtually free group if both A and B are. •

3.6 COROLLARY : Let P be an arbitrary P D_{3}-complex. Then π_{1}(P) is virtually torsion free. Indeed, P is finitely covered by an orientable P D_{3}-complex which is homotopy equivalent to a (possibly empty) connected sum of aspherical P D_{3}-complexes and copies of S^{1} × S^{2}.

Proof : This follows immediately by applying Corollary 3.5 to the orientation cover of P, and noting that aspherical P D_{3}-complexes are torsion free. • 3.7 THEOREM : Let P be a P D_{3}-complex with π = π_{1}(P). If g ∈ π is a nontrivial element of prime order p such that C_{π}(g) is infinite, then p = 2, w(g) = −1 and C_{π}(g) has 2 ends.

Proof : Again let X be the π-tree with finite edge stabilisers of Remark 3.3. Note that C_{π}(g) acts on the subtree X^{⟨g⟩}, also with finite edge stabilisers. (If x ∈ C_{π}(g), ϵ ∈ X^{⟨g⟩}, then g(xϵ) = gxϵ = xgϵ = xϵ implies that xϵ ∈ X^{⟨g⟩} also). Thus C_{π}(g) being infinite implies that X^{⟨g⟩} has infinitely many edges, and so either an ∞-vertex or an end. That is ξ(X^{⟨g⟩}) ≥ 0. It follows from the Remark 3.3 that w(g) = −1, p = 2 (necessarily) and ξ(X^{⟨g⟩}) = 1. Thus X^{⟨g⟩} has exactly two end/∞-vertices and contains a unique geodesic γ joining them. Since C_{π}(g) respects the set of ∞-vertices and also acts on EX^{⟨g⟩}, it must preserve the geodesic γ. In fact, some subgroup H of index at worst 2 in C_{π}(g) must fix each end/∞-vertex. If there is some ∞-vertex involved then the infinite group H stabilises every edge in γ which is a contradiction. Thus γ is an infinite geodesic joining two ends and H acts by translations. For any edge ϵ in γ the quotient H/stab_{H}(ϵ) must be Z. But any surjection H → Z splits, and since stab_{H}(ϵ) is finite, H must be virtually Z and have two ends. Hence C_{π}(g) is also 2-ended as required. •

It follows that if P is an orientable P D_{3}-complex then any torsion element of π_{1}(P) has finite centraliser. Thus, for example π = G_{1} ∗_{A} G_{2} is not the fundamental group of an orientable P D_{3}-complex if A is finite and a proper subgroup of both N_{G_{1}}(A) and N_{G_{2}}(A), since then N_{π}(A) is infinite, and hence so is C_{π}(a) for a ∈ A. As a corollary to this theorem we may also restate the main theorem of [3].

3.8 COROLLARY : Let P be a P D_{3}-complex, with infinite fundamental group π. If π has a nontrivial finite normal subgroup N, then P ≃ S^{1} × P_{2}.

Proof : In this case N contains a nontrivial element g of prime order p where C_{π}(g) is a subgroup of finite index in π. Thus, applying Theorem 3.7, P is non-orientable, g has order 2, and π in fact has two ends. So, by Theorem 4.4 of [6], P ≃ S^{1} × P_{2}. •

Ideally we should like to improve Theorem 3.4 by reducing the case that π is virtually free. Beyond applying Theorem 3.7 as per the examples given, it appears that some other approach is needed to eliminate the cases where π is the fundamental group of a graph of groups with finite vertex groups and some nontrivial edge groups. Turaev ([5], Theorem 5) gives an explicit algebraic characterisation of the pair (π_{1}(X), w(X)) of a P D_{3}-complex X, but it is not clear how to apply this in general, let alone to decide whether the minimal example S_{3} ∗_{C_{2}} S_{3} proposed in [3] is or is not the fundamental group of an orientable P D_{3}-complex.

REFERENCES

[1] I.M. Chiswell, Exact sequences associated with a graph of groups, J. Pure Appl. Algebra 8 (1976), 63-74.

[2] W. Dicks and M.J. Dunwoody, Groups Acting on Graphs, Cambridge Studies in Advanced Mathematics 17 (Cambridge University Press 1989).

[3] J.A. Hillman, On 3-dimensional Poincaré duality complexes and 2-knot groups, Math. Proc. Camb. Phil. Soc. 114 (1993), 215-218.

[4] J.-P. Serre, Arbres, Amalgames, SL_{2}, Asterisque no.46 (Société Math. de France, Paris 1977).

[5] V.G. Turaev, Three-dimensional Poincaré complexes: homotopy classification and splitting, Math. USSR Sbornik 67 (1990), 261-282.

[6] C.T.C. Wall, Poincaré complexes I, Ann. Math. 86 (1967), 213-245.