On Poincaré duality for pairs (G,W)

Abstract Let G be a group and W a G-set. In this work we prove a result that describes geometrically, for a Poincaré duality pair (G, W ), the set of representatives for the G-orbits in W and the family of isotropy subgroups. We also prove, through a cohomological invariant, a necessary condition for a pair (G, W ) to be a Poincaré duality pair when W is infinite.


Introduction
Bieri and Eckmann [4] introduced the concept of relative cohomology H .G; SI M / for a group pair .G; S/, where G is a group, S is a family of subgroups of G and M is a Z 2 G-module. Dicks and Dunwoody [6] worked with that concept from another point of view. Instead of dealing with S a family of subgroups, they worked with a G-set W , defining the groups H .G; W I M /. In Section 2 we recall some definitions and results about relative cohomology of groups and we describe, in details, the equivalence between the theories of Dicks-Dunwoody and Bieri-Eckmann.
By using the relative cohomology theory, Bieri and Eckmann introduced the concept of Poincaré duality pair .G; S/ and gave a topological interpretation for those pairs. Dicks and Dunwoody, with their notation for relative cohomology, also gave a topological interpretation for Poincaré duality pairs .G; W /.
In Section 3 we present some concepts about Poincaré duality pairs and a result that describes topologically, for a PD n -pair .G; W /, the set E of orbit representatives and the family S of isotropy subgroups.
In Section 4, based in [1] and [2] and by using the notation from Dicks and Dunwoody, we present a characterization of the types of Poincaré duality pairs and, through of a generalized invariant "end", a cohomological criterion for a pair .G; W / to be a Poincaré duality pair.

The equivalence between the theories of Dicks-Dunwoody and Bieri-Eckmann
Definition 2.1. Let G be a group. A Z 2 G-projective resolution of a Z 2 G-module M is an exact sequence of Z 2 Gmodules: ! M is called augmentation map and we denote the projective resolution by F M . Definition 2.2. Let G be a group, M a Z 2 G-module and F Z 2 a projective resolution of Z 2 over Z 2 G. The homology groups of G with coefficients in M are, for all n 2 Z, defined by H n .GI M / D H n .F˝G M /: The cohomology groups of G with coefficients in M are, for all n 2 Z, defined by H n .GI M / D H n .Hom G .F; M //: Let G be a group and let S = fS i ; i 2 I g be a family of subgroups of G. The pair .G; S/ is called a group pair. Consider Z 2 .G=S/ the free Z 2 -module generated by the cosets gS i , which G acts by left multiplication. The map " W Z 2 .G=S/ ! Z 2 defined on the generators by ".gS i / D 1 is called usual augmentation map and we denote by 4 the kernel of ". Now we recall the concept of relative cohomology of groups due to Bieri and Eckmann.  where M is a Z 2 S-module, i.e., a Z 2 S i -module for all i . If M is a Z 2 G-module then M is a Z 2 S-module by restrictions.
We will see now the definition of relative cohomology due Dicks and Dunwoody.
Definition 2.5. Consider W a G-set and Z 2 W the free Z 2 -module generated by W . Let Á W Z 2 W ! Z be the augmentation map, 0 D ker Á, M a Z 2 G-module and P 0 a Z 2 G projective resolution of 0 . The groups of relative cohomology of the pair .G; W /, with coefficients in M are, for all k 2 Z, defined by Theorem 2.6. Let G be a group and M a Z 2 G-module. Suppose either (i) W is a G-set, E D fw i ; i 2 I g is a set of representatives for the G-orbits in W and S D fG w i j i 2 I g is the family of isotropy subgroups; or (ii) S D fS i j i 2 I g is a family of subgroups of G, W D Proof. (i) Firstly, consider a pair .G; W / and E D fw i j i 2 I g a set of orbit representatives in W . We have Gg is the orbit of the element w i 2 E. Consider, for each w i 2 I , the isotropy subgroup G w i D fg 2 G j g w i D w i g. Using the one-to-one correspondence G.w i / $ G=G w i we have Let S D fG w i ; i 2 I g be the family of isotropy subgroups. We have Z 2 W D Z 2 .G=S/. It follows that the usual augmentation map " W Z 2 .G=S/ ! Z 2 which maps a generator gG w i 2 Z 2 .G=S/ into 1 2 Z 2 from Definition 2.3 coincides with the augmentation Á W Z 2 W ! Z 2 from Definition 2.5. Moreover 4 D ker " D ker Á. Consider a Z 2 G-projective resolution of the module Z 2 : F W ::: [5]), it follows that P k is Z 2 Gprojective and the sequence P W ::: On the other hand, To calculate the cohomology groups due to Bieri-Eckmann we consider the projective resolution (1) and, by  (2) we can form the following cochain complex: (ii) Now, consider a group pair .G; S/ where S D fS i j i 2 I g is a family of subgroups of G and let W D The set E D fw i D 1S i j i 2 I g is a set of orbit representatives and is the isotropy subgroup for w i . With this notation, and by using the same idea of the first part of the proof, we have

Poincaré duality pairs
In this section, before proving the main result, we recall some definitions and results about duality groups and pairs due to Bieri and Eckmann [4] and Dicks and Dunwoody [6].
Definition 3.1. A group G is called a duality group of dimension n, or simply a D n -group, if there exist a Z 2 Gmodule C , called the dualizing module of G, and natural isomorphisms for all integers k and all Z 2 G-modules M . In the special case where C D Z 2 , we say that G is a Poincaré duality group of dimension n, or simply a PD n -group.  (1) and (2) in Definition 3.2 are equivalent ( [3]). Hence, to show that a group pair .G; S/ is a PD n -pair it is enough to prove only one of the isomorphisms given in Definition 3.2.
In view of Remark 3.3 we can define Poincaré duality pairs in a simpler way.
for all Z 2 G-module M and all k 2 Z.
Definition 3.5. A group pair .G; S/, with S D fS i ; i 2 I g, is realised by a pair of CW-complexes .X; Y /, if X is a K.G; 1/-complex and Y is a subcomplex of X whose components Y i , i 2 I , are K.S i ; 1/ complexes, so that the maps i # W 1 .Y i / ! 1 .X / induced by the inclusions i W Y i ,! X are injective and map 1 .Y i / on S i G, after a convenient choice of paths connecting base points. The pair .X; Y / is called an Eilenberg-MacLane pair and is denoted by K.G; S; 1/.
The next result provides a topological interpretation for PD n -pairs (see [4]).
Theorem 3.6. If .G; S/ admits an Eilenberg-MacLane pair .X; Y /, where X is a compact manifold of dimension n and Y D @X , then .G; S/ is a PD n -pair.
In the following we present the definition of PD n -pair and its topological interpretation given by Dicks and Dunwoody in [6].  Hence .G; S/ is a PD n -pair according to Bieri and Eckmann. By using [4, Theorem 4.2], we have, for a PD n -pair .G; W /, the following results: (i) W falls into finitely many G-orbits.
(ii) For each w 2 W , the isotropy subgroup G w is a PD n 1 -group.
Theorem 3.9. Let X be a compact n-manifold, e X its universal covering space, and suppose that e X and the components of the boundary @ e X are all contratible; let G D 1 .X / and let W be the G-set of components of @ Q X . Then .G; W / is a Poincaré duality pair of dimension n.
The next theorem provides a relation between the topological interpretations for PD n -pairs given by the Theorem 3.6 due to Bieri Eckmann and Theorem 3.9 due to Dicks-Dunwoody, describing the set of orbit representatives in W and the family of isotropy subgroups. Theorem 3.10. Let X be a compact n-manifold, which is also a CW-complex, with boundary @X D [ i2I X i , where X i ; i 2 I; are the components of @X . Consider e X the universal covering of X and suppose that e X and the components of the boundary @ e X are all contractible. Let G D 1 .X / and W the G-set of components of @ e X . Then, e X i j i 2 I g is the family of isotropy subgroups given by E then .X; @X / is an Eilenberg-MacLane pair realising the group pair .G; S/. Hence, .G; W / is a PD n -pair according to Dicks-Dunwoody and .G; S/ is a PD n -pair according to Bieri-Eckmann.
Proof. (i) Consider the covering map p W e X ! X . Since p is a local homeomorphism we have @ e X D p 1 .@X / and so W is the set of path components of p 1 .@X /. On the other hand, N X i D p 1 .X i / consists of copies of the universal covering of X i . Thus, for each i 2 I we have a set J i and a family of path connected sets f By using the fact that G is the deck transformation group of e X , we can prove that the G-action maps a copy into a universal covering of X i on a copy of universal covering of the X i , ie, if j 2 J i and g 2 G, then g: f X ij D f X i k for any k 2 J i . Also, since G acts transitively in p 1 .x/, we have that, for all j; k 2 J i , there is g 2 G such that g: f X ij D f X i k . Thus, each component X i of @X provides only one G-orbit in W . Consider, for each i 2 I , a unique copy f X i 2 f f X ij j j 2 J i g of the universal covering of X i . It follows from above considerations that E D f f X i ; i 2 I g is a set of orbit representatives in W . Namely, (ii) Consider S D fS i D G e X i j i 2 I g, the family of the isotropy subgroups. We will prove that S i D 1 .X i /. For each i 2 I , let p i W f X i ! X i be the restriction of p to X i . Since f X i is a universal covering of X i , it follows that Since g: f X i D f X i and g 2 A. e X ; p/, we have the following commutative diagram iii) It follows from the hypotheses that X is a K.G; 1/ and X i D K.S i ; 1/-subcomplex of X for all i 2 I . Therefore, .X; @X / is an Eilenberg-MacLane-pair realising .G; S/. Finally, by the hypotheses of the theorem it follows that .G; W / is a PD n -pair according to Dicks-Dunwoody and from .i i i / it follows that .G; S/ is a PD n -pair according to Bieri-Eckmann.

A cohomological criterion for Poincaré duality pairs
Let .G; S/ a group pair with S D fS i j i 2 I g a family of subgroups with infinite index in G and consider the  This cohomological criterion can be revised in the notation of Dicks-Dunwoody. Before proving the result we have to make some remarks.
Let G be a group, S a subgroup of G, M a Z 2 G-module and i W S ! G the inclusion map. Consider the restriction map res G S W H 1 .GI M / ! H 1 .SI M /: Let .G; W / be a pair where G is a group and W is a G-set. Consider w; u 2 W representatives for the same G-orbit in W . If G u and G w are the correspondent isotropy subgroups then ker res G G u D ker res G G w .
Proof. Consider w; u 2 W representatives for the same G-orbit in W . Then there exists 2 G such that u D w and it is easy to see that G u D G Then res G G w D ' ı res G G u and so, ker res G G u D ker res G G w .  By using this Lemma, we can adapt the definition of the invariant E.G; S/ to the notation from Dicks and Dunwoody.
Definition 4.4. Let .G; W / be a pair where G is a group and W is a G-set such that OEG W G w D 1 for all w 2 E, where E is a set of orbit representatives in W . Consider the restriction map res G Remark 4.5. E.G; W / is an invariant in the category C whose objects are pairs .G; W / where G is a group, W is a G-set such that OEG W G w D 1 for all w 2 E, where E is a set of representatives for the G-orbits of W , and whose morphisms are maps W .G; W / ! .
(where E and E 0 are sets of orbit representatives for the G-orbits of W and the G 0 -orbits of W 0 , respectively) and such that˛.
Now, we can rephrase Theorem 4.1 providing a necessary condition for a pair .G; W / to be a Poincaré duality pair. The next result was proved in [7]. Theorem 4.7. Let .G; S/ be a PD n -pair. Then, only one of the statements is true: (i) S consists of only one subgroup S with OEG W S D 2.
(ii) S consists of two copies of G.
(iii) OEG W S D 1 for all S 2 S.
By using this result and Theorem 2.6 we can characterize the types of PD n -pairs with the notation of Dicks-Dunwoody.
Theorem 4.8. Let .G; W / be a PD n -pair. Then, only one of the statements is true: (i) W consists of exactly two elements and the G-action in W is transitive.
(ii) W consists of exactly two elements and the G-action in W is trivial.
(iii) OEG W G w D 1; for all w 2 W , and W is infinite.
Proof. We have that W D in W . Let S D fS i D G w i j i 2 I g be the family of isotropy subgroups. Since .G; W / is a PD n -pair we have that .G; S/ is a Poincaré duality pair (according to Bieri-Eckmann). By Theorem 4.7 we have three types of PD n -pairs .G; S/.
In case (i), we have S D fS g D fG w g, with OEG W S D 2. Hence W D G=G w has two elements and the G-action in W is transitive.
In case (ii), we have S D fS 1 ; S 2 g D fG w 1 ; G w 2 g D fG; Gg and, since W D G=G w i , it follows that G=G w i D G=G D f1g and so, G=G w i has one element for i D 1; 2. Therefore W has exactly two elements. Since G w i D G for i D 1; 2 the G-action in W is trivial.
Finally, in case (iii), we have that OEG W S i D OEG W G w i D 1. Since, for w 2 W , there exist g 2 G and w k 2 E such that G w D gG w k g 1 we have OEG W G w D 1; for all w 2 W . Hence, we conclude that W is infinite, Example 4.9. The only connected one-manifold-with-boundary is a line segment X D AB , with boundary consisiting of the two endpoints A and B an X is its own universal covering. We have G D 1 .X / D f1g and W D fA; Bg. The G-action in W is trivial and .G; W / is a PD 1 -pair of type (ii). Now, if X is an annulus, i.e., a region bounded by two concentric circles or a Möbius band, then the universal covering e X of X is an infinity band delimited by two lines r and s which are the components of the boundary of e X . We have G D 1 .X/ D Z, W D fr; sg. In the first case the G-action in W is trivial and .G; W / is a PD 2 -pair of type (ii). In the case of the Möbius band, the G-action in W is transitive and .G; W / is a PD 2 -pair of type (i).
Example 4.10. Consider X a torus with one hole. We have G D 1 .X / D hti hsi. If W consists of the components of boundary of the universal covering of X , by Theorem 3.10, the G-action in W is transitive and for all w 2 W we have W D G=G w where G w D< t st 1 r 1 >. In this case .G; W / is a PD 2 -pair of type (iii), because W is infinite, and by Theorem 4.6, E.G; W / D 1. But, if S D< s > and W D G=S then, by considering [1, Example 2.2(c)] and the proof of Theorem 2.6 above, it follows that E.G; W / D 1. Hence, by Theorem 4.6, in this case, .G; W / is not a PD n -pair.