topological abelian groups and equivariant homology

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Topological Abelian Groups andEquivariant HomologyMarcelo A. Aguilar a & Carlos Prieto aa Instituto de Matemáticas , Universidad Nacional , Autonoma deMéxico, MexicoPublished online: 07 Apr 2008.

To cite this article: Marcelo A. Aguilar & Carlos Prieto (2008) Topological Abelian Groups andEquivariant Homology, Communications in Algebra, 36:2, 434-454, DOI: 10.1080/00927870701716074

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Communications in Algebra®, 36: 434–454, 2008Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870701716074

TOPOLOGICAL ABELIAN GROUPS ANDEQUIVARIANT HOMOLOGY

Marcelo A. Aguilar and Carlos PrietoInstituto de Matemáticas, Universidad Nacional,Autonoma de México, Mexico

We prove an equivariant version of the Dold–Thom theorem by giving an explicitisomorphism between Bredon–Illman homology HG

∗ �X�L� and equivariant homotopicalhomology �∗�FG�X�L��, where G is a finite group and L is a G-module. We usethe homotopical definition to obtain several properties of this theory and we do somecalculations.

Key Words: Coefficient systems; Equivariant homology; Equivariant homotopy.

2000 Mathematics Subject Classification: 55N91; 55P91; 14F43.

INTRODUCTION

The presentation of homology using the Dold–Thom construction has beenvery useful in algebraic geometry. Lawson homology (see Friedlander and Lawson,1997, 1998; Lawson, 1989, 1995) was defined using this approach. In this article westudy the Dold–Thom–McCord theorem (see McCord, 1969) in the equivariant case.

Let G be a finite group. If L is a G-module, then one can define a coefficientsystem L on the category of canonical orbits of G by L�G/H� = LH , where LH

is the subgroup of fixed points of L under H ⊂ G. One then has an ordinaryequivariant homology theory HG

∗ �−� L�, called Bredon–Illman homology, whoseassociated coefficient system is precisely L. Let X be a (pointed) G-space andlet F�X�L� be the topological abelian group generated by the points of X, withcoefficients in L. Consider the subgroup FG�X� L� of equivariant elements, that is,the elements

∑lxx in F�X�L� such that lgx = g · lx. Then one can associate to X the

homotopy groups �q�FG�X� L��, and one has that, if X is a G-CW-complex, then

HGq �X� L� is isomorphic to �q�F

G�X� L��. When G is the trivial group, HG∗ �−� L� is

singular homology and this statement is the classical Dold–Thom theorem (Doldand Thom, 1958), which was extended to the equivariant case by Lima-Filho (1997)(when L = � with trivial G-action) and by dos Santos (2003) (when L is anyG-module). Both the original result and its equivariant generalization were proved

Received February 6, 2006; Revised October 5, 2006. Communicated by R. H. Villarreal.Address correspondence to Carlos Prieto, Instituto de Matemáticas, Universidad Nacional

Autónoma de México, Circuito Exterior, Ciudad Universitaria, México 04510, D.F. México; Fax: +52-55-5616-0348; E-mail: [email protected]

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TOPOLOGICAL GROUPS AND EQUIVARIANT HOMOLOGY 435

by showing that the homotopical definition satisfies the axioms of an ordinary or anequivariant homology theory, and then using a uniqueness theorem for homologytheories.

In this article, we prove the equivariant Dold–Thom theorem by givingan explicit isomorphism HG

q �X�L� −→ �q�FG�X� L�� for all q and any X of the

homotopy type of a G-CW-complex (Theorem 1.2). This isomorphism is constructedin Section 2 in two steps as follows. Let F��∗�X�� L� be the (reduced) singular chaincomplex of X with coefficients in the G-module L. Since X is a G-space, we havean action of G on the singular simplexes of X, which we denote by � �→ g · �.Let FG��∗�X�� L� be the subcomplex of equivariant chains, that is, chains

∑l��

such that lg·� = g · l�. Using the theory of simplicial sets, we give an isomorphismbetween the homology of this chain complex H∗�FG��∗�X�� L�� and �∗�FG�X� L��.Then we show that both the chain complex FG��∗�X�� L� and Illman’s chaincomplex (see Section 2), which defines HG

∗ �X� L� have the same universal property(Propositions 2.10 and 2.11) so that they are canonically isomorphic. For a differentapproach to the nonequivariant Dold–Thom theorem see Friedlander and Mazur(1994).

In Section 1, we state the main theorem and prove that the theory �G∗ �X� L� =

�∗�FG�X� L�� is additive. In Section 3 we study the theory �G∗ �X� L� in the

general context of equivariant homology theories and coefficient systems. Toeach equivariant homology theory hG

∗ �−� one can associate the G-module hG0 �G�.

We show (Theorem 3.5) that there is an isomorphism between the group ofnatural transformations Nat�hG

∗ ��G∗ �−� L��, and the group of G-homomorphisms

HomG�hG0 �G�� L�. We also show an analogous result for the classical equivariant

homology theory of Eilenberg and Steenrod.Finally, in Section 4 we show that there is some interesting information in the

groups �G0 �X� L�, and using the transfer for ramified covering maps, we calculate

��2q �X� L� for some �2-spaces X.

In this article we shall work in the category of compactly generated weakHausdorff spaces (see, e.g., May, 1999).

1. EQUIVARIANT McCORD’S TOPOLOGICAL GROUPSAND EQUIVARIANT HOMOLOGY

In McCord (1969), for a pointed topological space X and an abelian group L,McCord introduced topological groups F�X�L� consisting of functions u � X −→ L

such that u�∗� = 0 and u�x� = 0 for all but a finite number of elements x ∈ X (seeAguilar et al., 2002, for further details). If G is a finite group that acts continuouslyon X leaving the base point fixed, and additively on L (on the left), then F�X�L�

has a natural (left) action of G given by defining �g · u��x� = gu�g−1x�. This turnsF�X�L� into a topological ��G-module.

Definition 1.1. Let G be a finite group, X a pointed G-space, and L a ��G-module. Define

FG�X� L� = u ∈ F�X�L� � u�gx� = gu�x� for all x ∈ X� g ∈ G��

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436 AGUILAR AND PRIETO

In other words, FG�X� L� consists of the functions u ∈ F�X�L� that are G-functions,and coincides with the subspace F�X�L�G of fixed points under the G-action onF�X�L� given above.

Given a ��G-module L, there is a covariant coefficient system L called thesystem of invariants of L, which is given by taking the fixed point subgroups LH ofL under all subgroups H ⊂ G (see 3.2 3 below).

If X is a pointed G-space, then we denote by �q�X� the set of singularq-simplexes in X, which has an obvious G-action. We may define a chain complexby taking as q-chains the elements of FG��q�X�� L� (here �q�X� is taken with thediscrete topology; see next section). The boundary operator is given as the restrictionof the boundary operator of the singular chain complex of X. We show below(Theorem 2.9) that this chain complex is naturally isomorphic to Illman’s chaincomplex (Illman, 1975), which defines the Bredon–Illman equivariant homology ofX with coefficients in L, HG

q �X�L�.The following theorem of the article is the following.

Theorem 1.2. Let X be a pointed G-space of the same homotopy type of a G-CW-complex. Then there is an isomorphism HG

q �X� L� −→ �q�FG�X� L�� given by sending

a homology class �u represented by a cycle u = ∑l�� all of whose faces are zero, to

the map u � � q� q� −→ �FG�X� L�� ∗� such that u�t� = ∑l��t�.

In particular, when G is the trivial group and L = �, this will give a new proofof the classical Dold–Thom theorem, since SP�X ≈ F�X�� F�X��.

Definition 1.3. Define the (reduced) equivariant homology theory �G∗ with

coefficients in the coefficient system L by

�Gq �X� L� = �q�F

G�X� L��

As usual, �Gq �X� L� = �G

q �X+� L�, where X+ = X ∗� has the obvious extended

action.

We devote the next section to the proof of Theorem 1.2. Meanwhile, we makesome further considerations on our equivariant homology groups.

We shall now prove that our equivariant homology theory �G∗ is additive.

To prove that, we need the following concept. Let X�, � ∈ �, be a family of pointedG-spaces and let L be a ��G-module. Then we have (algebraically) the direct sumF = ⊕

�∈� FG�X�� L�. In order to furnish it with a convenient topology, take

Fn = �u�� ∈ F � #� ∈ � � u� �= 0� ≤ n��

Then obviously Fn ⊂ Fn+1, and⋃

n Fn = F . For each n, there is a surjection∐

�1��n∈��X�1

× L�× · · · × �X�n× L� � Fn�

Furnish Fn with the identification topology and F with the topology of the unionof the Fns. One clearly has the following lemma.

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Lemma 1.4. There is an isomorphism of topological groups⊕�∈�

FG�X�� L� FG( ∨

�∈�X�� L

)induced by the inclusions X� ↪→

∨X�.

Proof. The inverse is given by the restrictions FG�∨

X�� L� −→ FG�X�� L�, u �→u � X�

. �

Since obviously �q

(⊕� F

G�X�� L�) ⊕

� �q

(FG�X�� L�

), as a consequence, we

have the next.

Theorem 1.5. There is an isomorphism �Gq �∨

� X�� L� ⊕

� �Gq �X�� L�.

A more general case is as follows.

Definition 1.6. A G-space X is said to be G-0-connected (or G-path connected), ifgiven any two points x� y ∈ X, then there exists a G-path �� g� � x G y, that is, anelement g ∈ G and an ordinary path � from x to gy. The relation G is clearly anequivalence relation, and the equivalence classes are called the G-path componentsof X (see Rhodes, 1966).

Assume that a G-space X is locally 0-connected, then, since every G-pathcomponent X� of X is a topological sum of ordinary path components, there isa decomposition X = ∐

� X�. Given that �∐

� X��+ = ∨

� X+� , a consequence of the

additivity (1.4) is the following.

Corollary 1.7. Let X be a locally 0-connected G-space. There is an isomorphism�G

q �X� L� ⊕

� �Gq �X�� L�, where the G-spaces X� denote the G-path components

of X.

2. PROOF OF THE MAIN THEOREM

In this section we prove Theorem 1.2. We use the techniques of simplicial setsfor this.

We denote by the category whose objects are the sets n = 0� 1� 2� � � � � n�and whose morphisms f ∈ �m� n� are monotonic functions f � m −→ n. Recall thata simplicial set is a contravariant functor K � −→ �et; we denote the set K�n�simply by Kn. Let �q be the simplicial set �qn = �−� q�. We write �K� for thegeometrical realization given by

�K� = ⊔n

�Kn × n�/ ∼�

where n = �t0� t1� � � � � tn� ∈ �n+1 � ti ≥ 0� i = 0� 1� 2� � � � � n� t0 + t1 + · · · + tn = 1�is the standard n-simplex, and the equivalence relation is given by �fK�� t� ∼�� f#�t��, � ∈ Kn, t ∈ m. Here f# denotes the map affinely induced by f in thestandard simplices. Denote the elements of �K� by �� t, � ∈ Kn and t ∈ n.

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We say that a simplicial set K is pointed, if it is provided with a morphism(natural transformation) �0 −→ K. This means that each set Kn has a base pointand that for each f � m −→ n, the induced function fK � Kn −→ Km is base-pointpreserving.

Definition 2.1. Given a pointed simplicial set K and an abelian group L, wedefine the simplicial abelian group F�K�L� by F�K�L�n = F�Kn� L� (as mentionedin Section 1, where Kn has the discrete topology). The homomorphism induced byf � m −→ n is fK

∗ � F�Kn� L� −→ F�Km� L�.

The proof of the following uses results of Milnor (see May, 1992).

Lemma 2.2. The geometric realization �F�K�L�� is an abelian topological group suchthat �v� t+ �v′� t = �v+ v′� t.

Proof. Consider the projections pi � F�K� L�× F�K�L� −→ F�K�L�, i = 1� 2, andthe induced maps �pi� � �F�K�L�× F�K�L�� −→ �F�K�L��, and define � � �F�K�L�×F�K�L�� −→ �F�K�L�� × �F�K�L�� by

��v� v′�� t = ��p1��v� v′�� t� �p2��v� v′�� t�= ��p1�v� v

′�� t� �p2�v� v′�� t�

= ��v� t� �v′� t��

By May (1992, 14.3), � is a homeomorphism. The group structure + in �F�K�L�� isthen given by the diagram

where � � F�K� L�× F�K�L� −→ F�K�L� is the simplicial group structure. �

Proposition 2.3. The topological groups F��K�� L�� and �F�K�L�� are naturallyisomorphic.

Proof. Take

� � F��K�� L� −→ �F�K�L��

given by

��u� = ∑��t∈�K�

�u�� t�� t�

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where u � �K� −→ L, � ∈ Kn, and t ∈ n (thus u�� t� ∈ F�Kn� L�). Using Lemma 2.2one shows that � is a homomorphism. Thus we only need to check that ��Fk��K��L� iscontinuous. Consider the diagram

where the map on the top is the product of the maps given by the next diagram.

where the top map is given by �l� �� t� �→ �l�� t�.To see that � is an isomorphism of topological groups, we define its inverse � �

�F�K�L�� −→ F��K�� L� as follows. Take v ∈ F�Kn� L�; then ��v� t = ∑�∈Kn

v�� t.To see that � is well defined, take v = ∑r

i=1 li�i; then fK∗ �v� =

∑ri=1 lif

K��i�. Thus

�[fK∗ �v�� t

] = r∑i=1

[fK��i�� t

] = r∑i=1

��i� f#�t� = ��v� f#�t��

To see that � is continuous, consider the diagram

where the top arrow given by �∑

i li� �i�� t� �→ �l1��1� t� � � � � is obviously continuous.Moreover, � is a homomorphism. Namely, given �v� t� �v′� t′ ∈ �F�K�L��, by

Lemma 2.2, there exist unique elements, w�w′� t′′ such that �v� t = �w� t′′, �v′� t′ =�w′� t′′. Thus

��v� t+ �v′� t′ = ��w� t′′+ �w′� t′′�

= ��w + w′� t′′

= ∑�

�w + w′�� t′′

= ��w� t′′+ ��w′� t′′ = ��v� t+ ��v′� t′�

In generators, we have that ��l�� t�=��l�� t= l�� t, thus � � � is the identity. Onthe other hand, ��v� t=��

∑�∈Kn

v�� t�� = ∑�∈Kn

�v�� t = �∑

�∈Knv�� t =

�v� t, where the next to the last equality follows by Lemma 2.2. �

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Definition 2.4. Let G be a finite group. A (pointed) G-simplicial set is a (pointed)simplicial set K such that G acts on each Kn and the action of every g ∈ G

determines a (pointed) isomorphism of K. In other words, it is a functor K � −→G-�et∗.

Given a pointed G-simplicial set K and a ��G-module L, then F�K�L� inheritsan action of G, as also do �K�, F��K�� L�, and �F�K�L��. By the naturality of theisomorphism of Proposition 2.3 we obtain the following corollary.

Corollary 2.5. Let K be a G-simplicial set. Then the topological groups F��K�� L� and�F�K�L�� are G-isomorphic.

Let K be a G-simplicial set and H ⊂ G be a subgroup. Define KH as thesimplicial set such that �KH�n = �Kn�

H (write this set as KHn ). Then KH is a simplicial

subset of K and one easily verifies that �KH � = �K�H and �FH�K�L�� = �F�K�L��H .Thus we have the following corollary.

Corollary 2.6. Let K be a G-simplicial set. Then the topological groups FH��K�� L�,�F�K�L��H , and �FH�K�L�� are isomorphic.

Let � be a simplicial abelian group. Recall that the q-homotopy group of �is defined by

�q�� = Hq�N��

where N��q = �q ∩ ker d0 ∩ · · · ∩ ker dq−1 and �q = �−1�qdq; here di is the ith faceoperator of �. On the other hand, � can be seen as a chain complex, with � � �q −→�q−1 given by

∑qi=0�−1�idi.

We have the following result (cf. May, 1992, 22.1).

Proposition 2.7. The canonical inclusion of chain complexes N�� ↪→ � induces anisomorphism in homology.

Proof. The chain complex � is filtered by chain complexes �p, where

�pq = u ∈ �q �di�u� = 0� 0 ≤ i < minq� p��

The canonical inclusion ip � �p+1 ↪→ �p is a chain homotopy equivalence withinverse rp � �p −→ �p+1 given by rp�u� = u− spdp�u�, where sp is the pthdegeneracy operator of �. Obviously, rp � ip = 1�p+1 ; conversely, ip � rp is chainhomotopic to 1�p via the chain homotopy hp � �p

q −→ �pq+1 given by

hp�u� ={0 if q < p

�−1�psp�u� if q ≥ p� �

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Let G be a finite group and X a pointed G-space and let � �X� denote thesingular simplicial set given for each q by

�q�X� = � � q −→ X � � is a map��

Then, in fact, � �X� is a pointed simplicial G-set with the usual simplicial structure.On the other hand, let �G

q �X� be the G-singular set given by

�Gq �X� = T � q ×G/H −→ X �T is an equivariant map and H ⊂ G��

q ∈ � (here q has trivial G-action).Let L be a ��G-module. Define F ��G

q �X�� L� = v � �Gq �X�+ −→ L � v ∈

F��Gq �X�+� L�� v�T� ∈ LH if T � q ×G/H −→ X�. One easily sees that these groups

are exactly Illman’s groups CGq �X� L� (Illman, 1975, Def. 3.3). As Illman does, we

may declare that the generator lT is related to l′T ′ if there exists a G-function � �

G/H −→ G/H ′ such that the following diagram commutes

and l′ = �∗�l� ∈ LH ′(see 3.2 3. below). Divide the group F ��G

q �X�� L� by thesubgroup generated by the differences lT − l′T ′ where either lT is related to l′T ′ orl′T ′ is related to lT , as well as by all elements lT such that T � q ×G/H −→ X

is constant with value the base point x0, to obtain the group F ′��Gq �X�� L�. The

following proposition is clear.

Proposition 2.8. The simplicial group FG�� X�� L� and the graded groupF ′��G�X�� L� are chain complexes.

In fact, the chain complex F ′��G�X�� L� is identical to Illman’s chain complexSG�X� x0� L� (cf. Illman, 1975, p. 15). Then we have the following theorem.

Theorem 2.9. The chain complexes F ′��G�X�� L� and FG�� X�� L� are isomorphic.

The proof of this theorem requires some preparation.Let S be a pointed G-set with G-fixed base point �0. For each � ∈ S, let ��

LG� −→ FG�S� L� be given by ��l� =∑n

i=1�gil��gi��, where �g1� � � � � �gn� = G/G�.Then ��0

= 0 and �� = �g� � �g, where �g�l� = gl.We have the following universal property of FG�S� L�.

Proposition 2.10. If A is an abelian group and there is a family of homomorphisms�� L

G� −→ A satisfying ��0 = 0 and �� = �g� � �g, then there is a unique � �

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FG�S� L� −→ A such that � � �� = ��. Thus we have

Proof. Take u ∈ FG�S� L�. Write u as a sum∑n

i=1 li�i, where no two of theelements �i are equal. Select one of them, for instance, � = �1. Since u�g�1� = gl1,g�1 has to be some other �i. So the full orbit of � appears in the sum, namely,∑k

i=1�gil��gi��, where �g1� � � � � �gk� = G/G� is a subsum of u. Consider now u′ =u−∑k

i=1�gil��gi�� ∈ FG�S� L� and repeat the process until getting zero. Thus werewrite u = ∑

j ��j�lj� and define ��u� = ∑

j ��j �lj�. This is obviously well definedand has the desired property.

Observe that since u is G-equivariant, the sum∑k

i=1�gil��gi�� can be presentedby different pairs �l� ��, namely,

∑ki=1�gil��gi�� =

∑ki=1�gigl��gig��, where �g′1� � � � �

�g′k� = G/Gg� is also presented by the pair �gl� g��, g ∈ G. But since �� = �g� � �g,we have that �g�j �glj� = ��j

�lj�, so the value of ��u� does not change. �

Let X be a pointed G-space. Then the groups F ′��Gq �X�� L� have the same

universal property for S = �q�X�. Namely, take the homomorphisms �� LG� −→

F ′��Gq �X�� L� given by ��l� = �lT�, where T� �

q ×G/G� −→ X is defined byT��t� �g� = g��t�. Then ��0 = 0 and �� = �g� � �g. Then the universal property isgiven by the following proposition.

Proposition 2.11. If A is an abelian group and there is a family of homomorphisms�� L

G� −→ A satisfying ��0 = 0 and �� = �g� � �g, then there is a unique � �F ′��G�X�� L� −→ A such that � � �� = ��. Thus we have

Proof. Given a G-map T � q ×G/H −→ X, define �T � q −→ X by �T �t� =T�t� �e�, where e ∈ G is the neutral element.

Define � � F ′��G�X�� L� −→ A by ��lT = ��T �∑n

i=1 gil�, where G�T/H = �g1H�

� � � � �gnH�. This is well defined; if lT is related to l′T ′, we analyze two cases.We assume first that H ⊂ H ′ and that � � G/H −→ G/H ′ is the quotient

map. Then T ′ � �id× �� = T , thus T�t� �gH� = T ′�t� �gH ′�, and l′ = �∗�l� =∑r

j=1 h′jl,

where H ′/H = �h′1H� � � � � �h

′r H�. Notice that therefore �T ′ = �T , so that we have

H ⊂ H ′ ⊂ G�T. Let G�T

/H ′ = �g′1H ′ � � � � � �g′mH ′�. Therefore, G�T/H = g′1�H

′/H� ∪· · · ∪ g′m�H

′/H�. Hence

��lT = ��T

( r∑j=1

(g′1h

′jl)+ · · · +

r∑j=1

(g′mh

′jl))

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On the other hand,

��l′T ′ = ��T

( m∑i=1

�g′il′�)�

but l′ = ∑ri=1 h

′il. Thus ��lT = ��l′T ′.

Now assume that H ′ = g−10 Hg0 and � = �g0

� G/H −→ G/H ′ is given by righttranslation with g0. Then �∗ � LH −→ LH ′

is given by left translation with g−10 ,

namely, �∗�l� = g−10 l. Hence l′ = g−1

0 l and �T ′ = g−10 �T . Moreover, G�T ′ = g−1

0 G�Tg0.

Thus, if G�T/H = �g1H� � � � � �gnH�, then G�T ′ /H

′ = �g−10 g1g0H ′ � � � � � �g−1

0 gng0H ′�,and so

��l′T ′ = ��T ′

( n∑i=1

g−10 gig0l

′)= �g−1

0 �T

( n∑i=1

g−10 gil

)

= ��T g0

( n∑i=1

g−10 gil

)= ��T

( n∑i=1

gil

)= ��lT�

Obviously, � has the desired properties. �

Proof of Theorem 2.9. The isomorphism F ′��Gq �X�� L� −→ FG��q�X�� L� in the

previous corollary, as provided by the universal property, is given by

�lT �−→n∑

i=1

�gil��gi�T ��

where T � q ×G/H −→ X, l ∈ LH , and G/H = �g1H� � � � � �gnH�. One easilyverifies that this is a chain map. �

The following proposition is a generalization to the equivariant case of atheorem of Milnor (see May, 1992, 16.6).

Proposition 2.12. Let X be a pointed G-space of the same homotopy type of a G-CW-complex. Then � � �� X�� −→ X given by �� t = ��t� is a G-homotopy equivalence.

Proof. Let H ⊂ G be any subgroup. Note first, as already mentioned before, thatthe identity induces a homeomorphism �KH � ≈ �K�H for any simplicial G-set K. Onthe other hand, one also has a canonical isomorphism of simplicial sets � �XH� � �X�H . We have that Milnor’s map � � �� X�� −→ X, being natural, is a G-map. Onthe other hand, again by the naturality, it restricts to �H � �� XH�� −→ XH , whichby a theorem of Milnor is a homotopy equivalence. Therefore, one has the followingcommutative triangle

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where the vertical arrow is a homeomorphism, as mentioned above. Hence, �H is ahomotopy equivalence for every H ⊂ G. By a result of Bredon (1967, II(5.5)), then� is a G-homotopy equivalence. �

Proof of Theorem 1.2. We shall give an isomorphism

HG�X� L� Hq�FG��q�X�� L�� −→ �q�F

G�X� L�� = �Gq �X� L��

Here the left-hand side is the Bredon–Illman (reduced) homology of X, and the firstisomorphism follows from the natural isomorphism of Theorem 2.9.

To construct the arrow, we shall give several isomorphisms as depicted in thefollowing diagram, where H ⊂ G is any subgroup:

By Proposition 2.7, i∗ is an isomorphism. In particular, this shows that every cyclein HG�X� L� is represented by a chain u, all of whose faces are zero. We shall callthis a special chain.

The homomorphism � , which is given by ��u��t = �u� t, where u is a specialq-chain and t ∈ q, is an isomorphism, as follows from May (1992, 16.6).

In order to define �, we must express ��u� as a map � � � �q� �q� −→�� FH�� X�� L�� ∗�. By the Yoneda lemma, � is the unique map such that ��q� =��u�, where �q = id � q −→ q. The homomorphism �, defined by ��f� s =��f��s�, for f ∈ �qn and s ∈ n, is given by the adjunction between the realizationfunctor and the singular complex functor (see May, 1992, 16.1).

That �∗ is an isomorphism follows from Proposition 2.3. Finally, thehomomorphism �∗ is an isomorphism by 2.12.

Chasing along this diagram and using the homeomorphism � �q� −→ q

given by �f� t �→ f#�t�, one obtains that the isomorphism maps a homology class�u ∈ Hq�F

H�� X�� L�� represented by a special chain u = ∑� l��, to the map u �

� q� q� −→ �FH�X� L�� ∗� given by u�t� = ∑l��t�. �

3. EQUIVARIANT HOMOLOGY AND OTHER COEFFICIENT SYSTEMS

In this section we recall the general concept of a (covariant) coefficient systemfor a group G and give explicit examples of ordinary equivariant homology theorieswith particular systems as coefficients.

Definition 3.1. Let G be a finite group. A covariant coefficient system for G, M , isa covariant functor from the category of homogeneous sets G/H , H a subgroup ofG, and G-functions � � G/H −→ G/K, to the category of abelian groups. We denotethe induced homomorphisms by �∗ � M�G/H� −→ M�G/K�. There is, of course, acategory CoeffsysG of covariant coefficient systems for G.

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Example 3.2. Let L be a ��G-module. We have the following associatedcovariant coefficient systems:

1. To start with, consider the constant coefficient system, denoted again byL, with value the group L given by L�G/H� = L for all H ⊂ G and for � � G/H −→G/K, H ⊂ K ⊂ G, by �∗ = 1L. This is realized by the theory

hGq �X� L� = Hq�X/G�L��

where the second term is singular homology with coefficients in L.

2. Another useful example is the following. Let L be given by definingL�G/H� = LH , the quotient of L by the subgroup generated by the elements ofthe form l− h · l, l ∈ L, h ∈ H , and for a � G/H −→ G/K, a∗ � LH −→ LK is thequotient homomorphism. We call it the system of coinvariants of L. If, in particular,L has trivial action, then this coefficient system coincides with the constant systemgiven in 1. The coefficient system L is realized by a theory constructed by Eilenbergand Steenrod (1952) (see also Brown, 1982) as follows. Consider the usual singularcomplex S∗�X� on which G acts (here G acts on X on the right). Take the complexS∗�X�⊗��G L. Define the classical ordinary equivariant homology theory by

�Gq �X� L� = Hq�S∗�X�⊗��G L��

If X = G/H , then �Gq �G/H�L� = 0 if q > 0 and �G

0 �G/H�L� = S0�G/H�⊗��G L =��G/H⊗��G L LH .

3. The other example that we consider is L, that we call the system ofinvariants of L.1 It is defined as follows. Take L�G/H� = LH = l ∈ L �h · l =l for all h ∈ H�. For H ⊂ K ⊂ G and � � G/H −→ G/K the quotient function, take�∗�l� =

∑k∈�K/H kl, where �K/H denotes a set of representatives in K of the cosets

of H in K. For any equivariant map � � G/H −→ G/K one has that H ⊂ aK =aKa−1 for some a ∈ G, and so � is the composite of the quotient map G/H −→G/aK and the obvious bijection G/aK ≈ G/K. Hence we define a∗ � LH −→ LK asthe composite of LH −→ L

aK given by l �→ ∑k∈�aK/H kl followed by the isomorphism

LaK −→ LK given by l �→ a−1l. As shown in the main theorem, our theory

�Gq �X� L� = �q�F

G�X� L��

realizes L. Observe that if L has trivial G-action, then L defines the semiconstantcoefficient system, namely, L�G/H� = L for all H ⊂ G and for � � G/H −→ G/K,define �∗ � L −→ L by �∗�l� = ��K�/�H��l.

As shown in Greenlees and May (1992), the system of invariants is rightadjoint to the forgetful functor from the category of coefficient systems to thecategory of ��G-modules. Similarly, one can show that the system of coinvariantsis left adjoint to the same forgetful functor. One has the following proposition.

1Notice that the system of invariants of L is denoted by L in Lima-Filho (1997) and dos Santos(2003).

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Proposition 3.3. There are natural bijections

CoeffsysG�L�M� HomG�L�M�G��

CoeffsysG�M�L� HomG�M�G�� L��

As shown by Illman (1975), given any (covariant) coefficient system, thereis an ordinary equivariant homology theory HG

∗ �−�M� with M as coefficients.Moreover, given any natural transformation � � M −→ N of coefficient systems,there is an extension of it to a natural transformation � � HG

∗ �−�M� −→ HG∗ �−� N�

of (ordinary) homology theories. On the other hand, Willson (1975) constructs forany equivariant homology theory hG a (natural) spectral sequence such that E2

pq HG

p �X� hGq �·��, where hG

q �·� is the coefficient system given by G/H �→ hGq �G/H�, that

converges to hGp+q�X�. Clearly, if h

G is ordinary, then this spectral sequence collapses.This shows that if � � hG −→ kG is a natural transformation of ordinary homologytheories such that it induces the zero transformation in coefficients hG�·� −→ kG�·�,then � itself has to be zero. We thus have the following proposition.

Proposition 3.4. There is an isomorphism between natural transformations ofcoefficient systems and natural transformations of ordinary homology theories. Moreprecisely, we have

CoeffsysG�M�N� Nat�HG∗ �−�M��HG

∗ �−� N��

By this and the adjunction result 3.3 above, we have the following theorem.

Theorem 3.5. There are isomorphisms

Hom��G�L� hG0 �G�� Nat��G

∗ �−� L�� hG∗ ��

Hom��G�hG0 �G�� L� Nat�hG

∗ ��G∗ �−� L��

where hG∗ represents any ordinary equivariant homology theory, �G

∗ �−� L� the classicalequivariant homology theory, and �G

∗ �−� L� the homology theory that we defined in 1.3.

4. SOME COMPUTATIONS OF THE EQUIVARIANTHOMOLOGY GROUPS ��G

∗ �−� L�

In this section, we compute some equivariant homology groups, especially indimension 0, for the theory defined in 1.3.

Theorem 4.1. Let X be a 0-connected space with a free G-action, and let L be a��G-module. Then

�G0 �X� L� LG�

where LG = L/�l− g · l� is the quotient of L divided by the subgroup generated by theelements l− g · l, l ∈ L, g ∈ G.

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Proof. First observe that if the G-action on L is trivial, then the transfer of p �

X −→ X/G yields an isomorphism tp � F�X/G+� L�

−→ FG�X+� L� (see Aguilar andPrieto, 2006a, Thm. 6.2), that implies the result.

For the general case, let xi� be a set of representatives in X of the orbits (inX/G), and define

�G � FG�X+� L� −→ LG by �G�u� = ∑i

u�xi��

where l ∈ LG denotes the class of l ∈ L. One easily verifies that �G is independent ofthe choice of the representatives xi. We shall show below that �G is continuous, butfor the time being, we assume it is.

Let � � L −→ FG�X+� L� be given by ��l� = lx0, where

lx0�x� ={g · l if x = gx0�

0 if x � orbit �x0��

and x0 is one of the representatives of the orbits taken above. We then have acommutative diagram

Namely, if we take h ∈ G, we have to prove that both lx0 and ˜�hl�x0 lie in the samepath component of FG�X+� L�. Clearly, ˜�hl�x0 = ˜l�h−1x0�. Let � � I −→ X be a pathfrom x0 to h−1x0, and define � � I −→ FG�X+� L� by ��t� = ˜l��t�. Then ��0� = lx0,and ��1� = ˜l�h−1x0� = ˜�hl�x0.

In order to prove that � is continuous, note first that lx = ∑g∈G�g · l��gx�,

where

�lx��x′� ={l if x′ = x�

0 if x′ �= x�

and t �→ �g · l��g��t�� is continuous because the action of G on X is continuous.Thus, � is continuous and � is well defined.

Consider �G��l� = �G�lx0 = l. Thus �G � � = 1LGand hence � is injective.

To see that � is also surjective, take any u ∈ FG�X+� L� and call li = u�xi� foreach representative xi of the orbits in X. Then u = ∑

i lixi and hence lxi�, l ∈ L, is aset of generators of FG�X+� L�. Since � is obviously a homomorphism, it is enoughto show that every generator �lxi of �0�F

G�X+� L�� is in the image of �. Let now� � I −→ X be a path from x0 to xi, then ��t� = ˜l��t� is a path from ��l� = lx0 tolxi.

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To finish the proof we have to show that �G is indeed continuous. Let � �L −→ LG be the quotient homomorphism and call q = p∗�FG�X+�L� � F

G�X+� L� −→F�X/G+� L�. Take u ∈ FG�X+� L�; then

��∗q�u��x = �

( ∑x′∈p−1�x

u�x′�)= �

(∑g∈G

u�gx�

)= �

(∑g∈G

g · u�x�)

= ∑g∈G

��g · u�x�� = �G�u�x��

hence the image of �∗ � q lies in F�X/G+� �G�LG� where one can divide by �G�.Call �∗ � F�X/G+� �G�LG� −→ F�X/G+� LG� the homomorphism given by dividingthe values of the elements by �G�. Then �G is the composite

FG�X+� L��∗��∗�q−−−−→ F�X/G+� LG�

�−−−−→ LG�

where � � F�X/G+� LG� −→ LG is the augmentation given by

��v� = ∑x∈X/G

v�x��

Since all maps in the composite are continuous, so is �G too. �

Theorem 4.2. Let X be a G-0-connected G-space with a single orbit type and let Lhave trivial G-action. Then �G

0 �X� L� L.

Proof. The result follows from the same arguments of 4.1, but, on the one hand,since L has trivial G-action, G-paths are good enough, and on the other, instead ofdividing by the order of G, we divide by the order of any orbit. �

As a consequence of 1.7 and 4.2, we have the next result.

Corollary 4.3. Let X be a locally path-connected G-space and let X�, � ∈ �, be theG-path components of X such that each G-path component has a single orbit type. Then

�G0 �X� L�

⊕�∈�

L�� L� = L� ∀��

In what follows, we analyze the groups �G0 �X� L� in some special cases.

Let X be a G-space with (at least) one fixed point x0. The inclusion i � 0 ↪→X+ that sends one point to + and the other to x0 is an equivariant embedding whoseimage is a retract with the obvious retraction r � X+ −→ 0. Thus we have that r∗ �FG�X+� L� −→ F�0� L� = L is a split epimorphism. Thus FG�X+� L� L× ker r∗.Thus we have the following proposition.

Proposition 4.4. Let X be a G-space with a fixed point x0 under the G-action, andlet L have a trivial G-action. Then �G

0 �X� L� L× �0�ker r∗�.

In what follows we analyze the 0-connectedness of ker r∗. In case that theG-0-connected G-space X has one fixed point x0 (thus it is 0-connected) and G acts

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freely in the complement, let xi� be a set of representatives of the orbits of X and let�i � I −→ X be a path from x0 to xi for each i (�0 is the constant path). Assume thatu ∈ ker r∗; this means that r∗�u� =

∑x∈X u�x� = 0. For each x ∈ X, x = gxi for some

i and some unique g ∈ G. Let �x � I −→ X be given by �x�t� = g�i�t� and define� � I −→ FG�X+� L� by ��t� = ∑

x∈X u�x��x�t�. Then ��t� is G-invariant. Namely,∑gx∈X u�gx��gx�t� =

∑x∈X u�x�g�x�t�, since u�gx� = u�x� and �gx�t� = g�x�t�. On the

other hand, ��0� = ∑x∈X u�x�x0 = 0, ��1� = ∑

x∈X u�x�x = u; moreover, for any t ∈I , r∗��t�� =

∑y∈X�

∑x∈X u�x��x�t��y� =

∑x∈X u�x� = 0 and hence ��t� ∈ ker r∗ for

every t. Hence we have the following proposition.

Proposition 4.5. Let X be G-0-connected G-space with one fixed point x0 and suchthat G acts freely in the complement. Then ker r∗ is 0-connected and so �G

0 �X� L� Lif L has trivial G-action.

It is quite straightforward to verify that the previous proof holds also if theG-0-connected G-space X has exactly one fixed point x0 and for any other point xthe isotropy group Gx is a fixed subgroup H ⊂ G. Thus we have the following too.

Proposition 4.6. Let X be G-0-connected G-space with one fixed point x0 and suchthat Gx = H ⊂ G for all x �= x0, for some fixed subgroup H . Then ker r∗ is 0-connectedand so �G

0 �X� L� L if L has trivial G-action.

The following proposition deals with the transfer for the ramified coveringmap X −→ X/G studied in Aguilar and Prieto (2006a) and it will be useful below.It generalizes Theorem 6.2 therein.

Proposition 4.7. Let G act on a space X such that the fixed point set XG coincideswith the base point ∗ and the action in the complement of XG is free. Then the transferfor p � X −→ X/G induces an isomorphism

� � Hn�X/G� −→ �Gn �X��

Its inverse is given by 1�G� · p∗.

Proof. In the McCord topological groups F�X/G�� and FG�X�� every elementis zero in the base point. Thus in the diagram

where u is G-invariant, u exists if and only if u exists. Since off the fixedpoint ∗ the multiplicity function of the �G�-fold ramified covering map p �X −→ X/G is equal to 1 (except in ∗, where it is �G�, but u�∗� = 0 = u�∗�),the transfer � � F�X/G�� −→ FG�X�� is given by ��u� = p � u and thus it isan isomorphism. Since �p∗�u��x� ∈ �G��, we can divide by �G�, which yields acontinuous homomorphism. Thus the inverse is given by u �→ 1

�G� · u. �

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The following theorem shows, in particular, that there is nontrivialinformation contained in the equivariant homology groups of dimension 0.

Theorem 4.8. Let X be a G-CW-complex such that the fixed point set XG is finite,say XG = x1� x2� � � � � � xk�, and the action on the complement of XG is free. Then

�G0 �X�

⊕k−1

��G� and � � Hn�X/G� −→ �G

n �X�� sif n ≥ 1�

where ��G� is the cyclic group of order �G�.

Proof. Let X denote the G-space obtained by collapsing XG in X to one (fixed)point, and consider the diagram of cofiber sequences of G-maps

Applying equivariant homology in dimensions 0 and 1 we obtain

By 4.7, the transfer � in the middle is an isomorphism and so, by the five lemma, thetransfer � on the left is an isomorphism too, thus the second assertion in the casen = 1 follows. Moreover, the last epimorphism on the bottom right splits. Hence,the diagram transforms into the following:

From this, the first assertion follows easily. For the second if n > 1, applyequivariant homology in dimension n, to obtain, by the pullback property of thetransfer and Aguilar and Prieto (2006b, Theorem 2.11),

From here the second part follows. �

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The following example is an interesting application of the previous theorem.

Example 4.9. Let G = �2 act on 1 by complex conjugation. Then 0 =−1� 1� ⊂ 1 is the fixed point set of the action and we have

��20 �1� �2 and ��2

n �1� = 0 if n ≥ 1�

More generally than the previous example we have the following.

Example 4.10. Let G = �2 act on n ⊂ �n+1, with n > 1, by changing the signof the last coordinate. Then n−1 ⊂ n is the fixed point set of this action and it isconnected, and we have the G-cofiber sequence

n−1 ↪→ n� n ∨ n�

where �2 acts on the wedge by interchanging the summands. Passing this sequenceto the orbit spaces we have a commutative diagram

that yields a commutative diagram of homology groups with exact sequences on topand bottom

In the case k = n− 1, the diagram translates into

In order to verify that, indeed, the vertical arrow on the left-hand side ismultiplication by 2, it is enough to note that F�2�n ∨ n�� ⊂ F�n ∨ n�� F�n��⊕ F�n�� corresponds to the diagonal, that is, F�2�n ∨ n�� F�n��. Since the orbit map induces the sum of the factors in F�n��⊕F�n�� −→ F�n��, its restriction to the diagonal (modulo the obviousisomorphism) yields multiplication by 2. On the other hand, ��2

n−1�n ∨ n�

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Hn−1�n� = 0. Hence the commutativity of the diagram implies that the question

mark is also multiplication by ±2 and so

��2n−1�

n� �2�

In the case k = n, the (extended) top row becomes 0 −→ 0 −→ ��2n �n� −→

�·2−→ �, hence ��2

n �n� = 0. Moreover, in the case 0 ≤ k ≤ n− 2 or k > n, the toprow of the diagram converts into 0 −→ 0 −→ ��2

k �n� −→ 0, thus

��2k �n� = 0 if k �= n− 1�

The following two are other examples of a computation of ��20 �X� and

��21 �X�.

Example 4.11. Let X consist of a 2-sphere on which �/2 acts rotating it 180�

around its (horizontal) axis. Glue to its poles two symmetric arcs, which areexchanged by the group. This �2-space has two fixed points (the poles) and theaction is free elsewhere. Call X the quotient of X by identifying the poles. SeeFigure 1.

The equivariant cofiber sequence 0 ↪→ X � X and that of its orbit spaces0 ↪→ X/�2 � X/�2 yield a commutative diagram with exact rows

This diagram translates into

(One easily shows that H1�X/�2� �⊕ �.) Thus we conclude that

��21 �X� � and ��2

0 �X� �2�

Figure 1

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Figure 2

By 4.8 we may easily see that ��22 �X� ��2

2 �X� �, and ��2n �X� ��2

n �X� = 0,n > 2.

Example 4.12. Set T = 1 × 1 and let �2 act on T by �−1� · �� = �� . ThenT�2 = −1� 1�× 1 = 1 1. Moreover, T/T�2 ≈ X ∨ X, where X is the result ofidentifying in a 2-sphere the two poles in one point, and �2 acts interchanging thesummands (see Figure 2).

Using similar techniques as in the previous examples, one can prove that

��21 �T� �⊕ �2 and ��2

0 �T� �2�

Once more these groups differ from the corresponding nonequivariant groups of theorbit spaces.

ACKNOWLEDGMENT

Carlos Prieto was partially supported by PAPIIT grant IN110902 and byCONACYT grant 43724.

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topological abelian groups and equivariant homology

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