Math. J. Okayama Univ. 56 (2014), 91–115

EQUIVARIANT STABLE HOMOTOPY THEORY FOR
PROPER ACTIONS OF DISCRETE GROUPS
Noé Bárcenas

Abstract. Following ideas of Graeme Segal [Segal(1973)], [Segal(1968)],
Christian Schlichtkrull [Schlichtkrull(2007)] and Kazuhisa Shimakawa[Shimakawa(1989)] we construct equivariant stable homotopy groups forproper equivariant CW complexes with an action of a discrete group. 1

Contents
1. Introduction91
2. Γ-spaces, configuration spaces and equivariant stable homotopy
groups for finite groups
92
2.1. Equivariant stable homotopy for finite groups
92
2.2. Γ-spaces
96
2.3. Introducing G-equivariance100
2.4. The associated contravariant Or(G)-space103
3. Equivariant Homology Theories
105
3.1. Axiomatic description of Equivariant Homology Theories
105
4. Definition and verification of the axioms
106
4.1. G-homological structure
106
4.2. Induction Structure108
5. The Bivariant Theory and the Bivariant Burnside Module forinfinite groups.
109
References
114

1. Introduction
In this work we extend the notion of equivariant stable homotopy groups
to the context of proper actions of discrete groups on G-CW complexes.
This is done by using an adequate generalization of Γ-spaces.
This paper is organized as follows: in section 1, after a quick review of
equivariant algebraic topology, we come over to the definition of equivariant
homotopy groups for actions of finite groups on G-CW complexes. We recall
facts related to Γ-spaces in section 2, and we introduce adequate equivariant
Mathematics Subject Classification. 55Q91,55N91 Secondary: 22F99 .
Key words and phrases. proper actions, equivariant homotopy theory, configuration
spaces.
91

92

NOÉ BÁRCENAS

generalizations of them. We also prove in this section that equivariant homotopy groups generalize naively graded equivariant homotopy groups forfinite groups.
In section 3, we recall the axiomatic description of an Equivariant Homology Theory and prove in section 4.1 the main theorem of this note, whichstates that the definition of equivariant stable homotopy groups provide anequivariant homology theory. In section 5 we extend the notion of equivariant stable homotopy theory to a bivariant theory. We finally state someproblems and results related to stable maps between classifying spaces.
This work was supported by the SFB 878 “Groups, Geometry and Actions” at the University of Münster, the Hausdorff Center for Mathematics
in Bonn and Wolfgang Lück’s Leibnizpreis.
2. Γ-spaces, configuration spaces and equivariant stablehomotopy groups for finite groups
2.1. Equivariant stable homotopy for finite groups. For completeness,
we include the following elementary facts about G-CW complexes, which will
be needed later:
Definition 1. Let G be a discrete group. A G-CW complex structure on
the pair (X, A) consists of a filtration of the G-space X = ∪Xn φ = X−1 ⊂X0 = A ⊂ . . . ⊂ Xn ⊂ Xn+1 ⊂ X for which every space is inductively
obtained from the previous one by attaching cells in pushout diagrams ofthe type` n−1/ Xn−1
× G/HiiS`i

D n × G/H

i


/ Xn

A G-CW complex is called proper if all isotropy groups are finite. In
particular, any G-CW complex is proper if the acting group is finite.
For any G-CW pair, the inclusion A → X is a G-cofibration and (X, A)
is a G-equivariant neighborhood deformation pair.
We will need for technical purposes the following lemma about homotopy
classes of maps between G-CW complexes:
Lemma 2.1 (Lemma 1.1 in [Lück and Oliver(2001)]). Let F be a familyof subgroups of G, which is closed under taking subgroups and conjugation.
′
Let f : Y → Y be a map such that the restriction to fixed points in F is a
homotopy equivalence. Then, for any G-CW complex all of which isotropy′
subgroups are in F, the map f∗ : [X, Y ]G → [X, Y ]G is a bijection.

EQUIVARIANT STABLE HOMOTOPY THEORY

93

Paralell to G-CW complexes, we will use also spaces over the orbit category.
We refer the reader to [Davis and Lück(1998)] for further reference andfor the proof of the results in this section. All spaces have the compactly
generated topology, in the sense of [McCord(1969)].
Definition 2. Let C be a small category. A covariant (contravariant) Cspace over the category C is a covariant (contravariant) functor C −→ Spaces
to the category of compactly generated spaces.
Example 1. Let F be a family of subgroups of the discrete group G closed under intersection and conjugation. The orbit categoy Or(G, F) has as objects
are homogeneous spaces G/H for H ∈ F, a morphism is a G-equivariant
map F : G/H −→ G/K. If X is a G-space, we define the contravariant
Or(G, F)-space associated to X to be the functor G/H 7−→ X H . The covariant Or(G, F)-space associated to X is the functor G/H 7→ G/H ×G X.
We will deote by FIN the family of finite subgroups of G, and we will use
the notation Or(G) for the family of all subgroups of G.
We present two useful constructions for spaces over a category. They are
an instance of ends and coends in category theory. Well known constructions
like geometric realizations and mapping spaces give examples of coends.
Definition 3. Let X be a contravariant C-space over C and let Y be acovariant C-space over C. Their tensor product X ⊗C Y is the space definedbyaX(C) × Y (C)/ ∼C∈Obj(C)where ∼ is the equivalence relation generated by (xφ, y) ∼ (x, φy).
Definition 4. Let X and Y be C-spaces of the same variance. Theirmap space MorC (X, Y ) is the space of natural transformations between
the functors X and Y , which is topologized as subspace of the productΠC∈Ob(C) Map(X(C), Y (C))
In the case of discrete groups the orbit category provides a convenient
tool for the study of transformation groups. There exists a pair of adjoint
functors between a certain category of Or(G)-CW complexes and G-CW
complexes.
Definition 5. Let C be a small category. A contravariant, free C-CW complex is a contravariant C-space together with a filtration∅ = X−1 ⊂ X0 ⊂ . . . = ∪n Xn

94NOÉ BÁRCENAS

such that X has the colimit topology with respect to this filtration and each
Xn is obtained from the previous one by a pushout of the form`n−1
/ Xn−1i∈In morC (?, ci ) × S
`




/ Xn

n
i∈In morC (?, ci ) × D

where In is an index family and ci is an object in C.
Definition 6. Let ∇ be the covariant Or(G)-space given by sending G/H
to itself. Given a contravariant Or(G, F)-space X, we define the associatedleft G-space byX̂ = X ⊗Or(G,F ) ∇The left action of an element g ∈ G is given by id⊗Lg , where Lg : G/H →G/H is the map of covariant Or(G, F)-spaces given by left multiplication
with g. The construction ˆ is functorial and is refered to as the coalescencefunctor.
The following results are stated and proved on [Davis and Lück(1998)],
page 249 and 250:
Lemma 2.2. The map f : X(G/1) → X̂ given by the composition
X(G/1) ∼= X(G/1) × {1} ⊂ X(G/1) × G/1 → X̂is a G-homeomorphism.
Lemma 2.3. The coalescence functor ˆ and M apG ( , X) constructed in
Example 1 and Definition 6 are adjoint, i.e. for a contravariant Or(G, F)space X and a left G-space Y there is a natural homeomorphism
M apG (X̂, Y ) → MorOr(G,F ) (X, M apG (, Y ))

Theorem 2.4. Let Y , X be left Or(G, F)-spaces.
(1) The adjoint of the identity on M apG ( , Y ) under the adjunction of
proposition 2.2 is a natural homeomorphism.
(2) Given a contravariant Or(G, F)-space, the adjoint of the identity
on X̂ under the adjunction of proposition 2.2 is a natural map ofOr(G, F)-spaces
S(X) : X → M apG (, X̂)It is an isomorphism if and only if for each H ∈ F, the projection
G/1 → G/H induces an homeomorphism X(G/H) → X(G/1)H .
This condition is satisfied if X is a free Or(G, F)-CW complex.

EQUIVARIANT STABLE HOMOTOPY THEORY

95(3) If Y is a left G-CW complex with isotropy in the family F, thenM apG ( , Y ) is a free Or(G, F)- CW complex.
In the case of a finite group, equivariant stable homotopy is an RO(G)graded equivariant homology theory, in the sense of [May(1996)].
Definition 7. Let G be a finite group and X a G-CW complex. For any
representation W , form the one-point compactification S W and define theset of arbitrary pointed maps ΩW S W ∧ X = Map(S W , S W ∧ X+ ), on which
G acts by conjugation. The equivariant stable homotopy group in degree
V = V1 − V2 , where Vi are finite dimensional real representations, is defined
to be the abelian group constructed as the colimit of the system of homotopy
classes of maps
πVG (X) = colimW [S V1 , ΩW S W ⊕V2 ∧ X+ ]G
where the systems runs along a complete G-universe, that is a Hilbert
space containing as subspaces all irreducible representations, where the trivial representation appears infinitely often. Given any finite group G and any
G-CW complex X, one defines the equivariant infinite loop space associatedto X, QG (X) as the colimit of the directed diagram associating a representation W the G space ΩW S W ∧ X = Map(S W , S W ∧ X+ ), and where theorder is given by the inclusion of representations as direct summand.
In particular, for any integer n, the n-th-equivariant stable homotopy
groups of an G-CW complex X, πnH (X) are defined as[S n , QG (X)]Gn≥0[S 0 , QG (S −n ∧ X+ )]G

n<0

where S n is the n-dimensional sphere with the trivial G-action.
Finite dimensional representations are involved in this definition in a crucial way and this is the main handicap to extend equivariant homotopy tomore general settings. In fact, pathological examples from group theory[Ol′ shanskiı̆(1982)] provide finitely generated, discrete groups for which all
finite dimensional representations over R, or C are trivial. Precisely:
Example 2. Let G be a finitely presented group. G is said to be residuallyfinite if for every element g 6= 1, there exists a homomorphism ϕ to a finitegroup mapping g to an element different from 1. The maximal residuallyfinite quotient Gmrf of G is the quotient by the normal subgroup consistingof the intersection of all subgroups of finite index. In symbols:
Gmrf =

G∩(G:H)<∞ H

96NOÉ BÁRCENASThis is a residually finite group, characterized by the property that every
group homomorphism to a residually finite group factorizes trough the quotient map. Recall [Malcev(1940)], that if G is a finitely generated subgroupof Gln (F ) for some field, then G is residually finite. This means that in thissituation, every finite dimensional representation of G is induced from one ofGmrf . An example of Olshanskii, [Ol′ shanskiı̆(1982)] gives for every primep > 1075 a finitely generated, infinite group all of whose proper subgroups
are finite of order p. That means that G does not contain proper subgroupsof finite index, hence Gmrf = {e} and consequently every finite dimensional
representation of G is trivial.
2.2. Γ-spaces. We use in a fundamental way a generalization of certain
configuration spaces with labels in (unpointed) spaces [Segal(1973)], seealso [Schlichtkrull(2007)] in order to represent equivariant stable homotopy
theory. This construction goes back to Graeme Segal und lies in a closeinteraction with infinite loop space Theory and Γ-spaces in particular. In
this section we recall the needed definitions and facts related to these constructions.
Let Γ be the category whose objects are finite sets, and where a morphismθ : S → T consists of a map which sends elements s ∈ S to subsets θ(s) ⊂ T
in such a way that for two different elements s, s′ ∈ S one has θ(s)∩θ(s′ ) = ∅.
Such maps correspond to a map between power sets P (S) → P (T ) which
sends disjoint unions in S to disjoint unions in T .
The simplicial category ∆ is the category whose objects are the the setsn = {1, 2, . . . , n} for n ∈ N0 , and whose morphisms consist of the orderpreserving maps ϕ : m → n. Note that there is a canonical functor ∆ → Γ.
It is the identity on objects, and on morphisms it is given by sending ϕ :m → n to the map θ(ϕ) with θ(ϕ)(i) = {j ∈ n | ϕ(i − 1) < j < ϕ(i)}.
Let T OP denote the category of compactly generated weak Hausdorff
spaces.
Definition 8. A Γ-space is a functor A : Γop → T OP with the property
that(1) A(0) is a point2; andQ(2) for each n > 0 the canonical map A(n) → ni=1 A(1), induced by
the i maps κi : 1 → n, κi (1) = {i}, is a homotopy equivalence.
From the definition it follows that the spaces A(S) are pointed, the basepoints being the image of A(0) under the map induced by the unique morphims S → 0 in Γ, which sends every subset of S to the empty set. Throughprecomposition with the functor ∆op → Γop one can regard any Γ-space as
2Segal in his original article [Segal(1974)] requires A(0) to be contractible

EQUIVARIANT STABLE HOMOTOPY THEORY

97

a simplicial space, and we define |A| to be its fat (topological) realization.
Here and in the following realization always stands for the so-called fat realization (as described in [Segal(1974)], appendix A), i.e., without taking
degeneracies.
For a Γ-space A we can define its delooping BA as the Γ-space given by
BA(S) = |A(S × )|. The latter can be iterated, and we define B n A =B(B n−1 A) for n ∈ N. By construction of B n A(1), it naturally comesequipped with a map S 1 ∧ B n−1 A → B n A. Hence the sequence of spaces
BAn (1) together with the maps S 1 ∧B n A → B n+1 A yield a spectrum, which
we shall denote A. By construction A is an Ω-spectrum in the sense that theadjoints of the structure maps BAn → ΩBAn+1 are homotopy equivalences
for n ≥ 1, see Theorem in [Puppe(1974)]. In particular, A defines a (naively
graded) cohomology theory on the category of CW -complexes.
We state now the universal property of Γ-spaces, the group completion
theorem:Lemma 2.5. Let A be a Γ-space such that π0 (A(1)) contains a cofinal freeabelian monoid. Denote by KA the contravariant functor defined on the
category of compact spaces by X 7→ [X, ΩBA(1)]. Then the natural transformation [X, A(1)] → KA , induced as adjoint to the map ΣA(1) → BA(1)is universal among transformations Θ : [ , A(1)] → F , where F is a representable abelian-group valued functor on compact spaces and Θ is a transformation of monoid valued functors.
Proof. See [Segal(1974), Proposition 4.1].

A relevant example for Γ-spaces comes from categories with sums. Thisis an instance of a more general construction stated in the following
Definition 9. Let CAT be the category of small categories. A Γ-category
is a functor C : Γop → CAT , which satisfies(1) C(0) is equivalent to the category with one object and one morphism;andQ(2) for each n > 0 the canonical functor C(n) → ni=1 C(1), induced by
the i maps κi : 1 → n, κi (1) = {i}, is an equivalence of categories.
Example 3. Let C be a topological category where finite sums exist. For
every object n in Γop , consider the category C(n) of functors P(n) → C
which send disjoint unions to finite sums, and where a morphism is an
isomorphism of functors.
In the previous example, notice that
• C(0) is equivalent to the category with one object and one morphism

98

NOÉ BÁRCENAS• For each n, there is a functor pn : C(n) → C(1) × . . . × C(1) induced
n−timesby the morphisms in Γ, ik : 1 → n which gives an equivalence of
categories.
Hence, this determines in a natural way a Γ-category. We obtain straightforwardly:
Lemma 2.6 (Corollary 2.2 in [Segal(1974)]). If C is a Γ-category, then thefunctor |C| : S 7→ |C(S)| is a Γ-space.
We turn our attention to topological categories. All spaces to be considered have the compactly generated topology.
We focus in the following construction, which goes back to Graeme Segal[Segal(1974)]:
Definition 10. Let Y be an (unpointed) space. The category of finite setslabelled on Y , CY has the following description. Objects are pairs (S, ϕ),where S is a finite set and ϕ : S → Y is an injective function. Morphisms
from (S, ϕ) to (T, ψ) consist of an isomorphism Θ : S → T inducing an
isomorphism ϕ∗ (S) → ϕ∗ (T ) 3. This is a topological category, where thespace of morphisms carries the discrete topology, and the space of objects is
topologized as the configuration space C∗ (Y ), see [Segal(1973)], that is, asa subspace of the symmetric producta
Sp(Y) =
Yk /Σkkwhere Σk is the symmetric group in k letters.

The category of finite sets labelled in a space admits a symmetric monoidal
structure determined
essentially by the disjoint union of finite sets and the‘S
TSfact that Y≈ Y × Y T if the spaces are given the compactly generated
topology.
Associated to this structure there is a Γ-space MC (Y ) such that ΩBMC(Y ) (1)is homotopy equivalent to Ω∞ Σ∞ Y+ . If Y is the singleton space, then theconstruction above is related to the Barrat-Priddy-Quillen-Segal theorem[Segal(1974)]. We remit the reader to the article [Schlichtkrull(2007)] for afurther, more recent reference of these facts, as well as their relationship to
structured ring spectra. We point out that altough the construction does
not take pointed spaces as input, the space of objects of the topologicalcategory of sets on Y has a basis point: the empty configuration.
In order to deal with the cohomological aspects of these constructions, andtheir generalizations to the equivariant setting, we need some facts related
3Segal in his original paper, [Segal(1974)] requires the morphisms to be injective maps

θ : S → T such that θ∗ (ψ) = ϕ

EQUIVARIANT STABLE HOMOTOPY THEORY

99

to the notion of a homotopy fibration. The author thanks an anonymousreferee for the suggestion of methods related to homotopy fibrations.
Given a pointed map p : X → Y between compact generated spaces andy ∈ Y , the homotopy fiber at y, hofib(p)y is defined as the space of maps
beginning on the base point and ending on a preimage of y under p. In
symbols,{(x, α) | α : I → X

p(α(0)) = y, α(1) = {•}}

Definition 11. A map f : E → X is a homotopy fibration with fiber F if
the homotopy fiber at every point x ∈ X is homotopy equivalent to F .
Homotopy fibrations are well behaved with respect to (homotopy) pushouts
and sequentially directed (homotopy) colimits, as the following results show,
Lemmas 2 and 3 in [Puppe(1974)], pages 3 and 4 .
Lemma 2.7.

(1) Let
X1 o

f˜1

p1

X0

f˜2

p0



A1 o

p2



f1

/ X2

A0

f2


/ A2

be a diagram where the maps X0 → fi ∗ hofib(pi )b are homotopy equivalences for all points b. Suppose that the map XZ → AZ between the
row-wise mapping cylinders is a homotopy fibration with fiber F , andthe canonical pairs of inclusions Xi → XZ and Ai → AZ are homotopy fibrations. If the projections XZ → X and AZ → A of the
double mapping cylinders to the pushouts are homotopy equivalences(which is the case if at least one horizontal map in each row is a
cofibration), then the induced map between the row-wise colimits isa homotopy fibration with fiber F .
(2) Suppose that in the diagram
X0

/ X1

/ X2

p0

p1

p2



A0



/ A1


/ A2

/ ...

/ ...

the induced map XT → AT between the row-wise telescopes is a
homotopy fibration with fiber F and the canonical pairs of inclusions
Xi → XT , Ai → AT are homotopy fibrations with fiber F . If the
projections of the row-wise telescopes to the row-wise colimits are
homotopy equivalences, (e.g if all horizontal maps are cofibrations),

100

NOÉ BÁRCENAS

then the induced map between the row-wise colimits is a homotopy
fibration with fiber F .
Recall that a semisimplicial object in the category of spaces is a simplicial
space with no degeneracies and the fat geometric realization is the space obtained by ignoring the degeneracy maps in the identification of the quotient
space giving the geometric realization. As a consequence of the previous
results, Puppe [Puppe(1974)] obtains:
Theorem 2.8. Let p : X • → A• be a semisimplicial map between semisimplicial objects in the category of compactly generated spaces which is given
levelwise by homotopy fibrations with fiber F . Then, the map between thefat geometric realizations is a homotopy fibration with fiber F .
2.3. Introducing G-equivariance. We implement equivariance on these
constructions. In the case of finite groups, two definitions of equivariant
Γ-spaces are available and both of them are equivalent [Shimakawa(1991)].
For the purposes of studying proper actions of discrete (infinite) groups on
G-CW complexes, the lack of finite G-sets and finite dimensional respresentations makes the usual techniques unavailable. The same applies formethods and results coming from the context of RO(G)-graded homology
theories, much more suitable for compact Lie groups. The definition below seems to be the most adequate to address proper actions of discretegroups on G-CW complexes. We stress that this approach is not adequate
to handle actions of lie groups (even compact) and remit the reader to[Blumberg(2006)], Appendix C for more on this point.
Definition 12. Let G−T OP be the category of compactly generated, weak
Hausdorff G-spaces. A G-equivariant Γ-space is a contravariant functor
A : Γop → G − T OP which satisfies(1) A(0) is a point; and(2) Qfor each n > 0 and any subgroup H ⊂ G the canonical map A(n)H →n
Hi=1 A(1) , induced by the n maps κi : 1 → n, κi (1) = {i}, is a
homotopy equivalence.

Note, if AG : Γ → G − T OP is a G-equivariant Γ-space then the spaces
BAnG (1) come equipped with continuous G-actions and the maps S 1 ∧B n AG →B 1+n AG are G-equivariant. Thus the corresponding spectrum AG has the
structure of a (naive) G-spectrum.
We need the following definitions for our construction of an equivariant
Γ-space.
Definition 13 (Categories associated to a group). Let G be a discretegroup. The transport category EG has as objects the element of the group,with exactly one morhism between every pair of elements. If H is a subgroup

EQUIVARIANT STABLE HOMOTOPY THEORY

101

of G, the groupoid category BH is defined as the category with only one
object and morphisms all of H.
Remark (Relevant category equivalences). There is a equivalence of categories EG/H ∼= BH, which is given by the composition of functors given by
the inclusion EH/H → EG/H and the the equivalence BH ∼= EH/H.
Definition 14 (Configuration Model). Let Y be a proper G-CW complex.
The category of G-sets on Y is the category of functors from the transport
category to the category of sets labelled on Y . The Γ-space of G-sets on
Y is the Γ-space associated to the symmetric monoidal structure of CY . In
symbols
MCY (n̄) =| Fun(EG, CY (n̄)) |there is a simplicial action of G on this space determined by the action on the
category of functors which is described as follows. Let f : EG → CY (n̄) be
such a functor. Denote by f (h) = (S, ϕ(h) : S → X) the object associatedto the object h. g · f is the object in the functor category which assigns to
the object h the object (S, g · ϕ(g−1 (h)), where g · ϕ stands for the action
on the space Y . Note that this Γ-space also admits deloopings, denoted by
B and that ΩBMCY (1) is an infinite loop space with an action of G.
Lemma 2.9. The Γ-space MCX,G (1) is an equivariant Γ-space.
Proof. Let H be any subgroup. Consider the map
H

CY (n)→nY

CY (1)H

i=1

induced by assigning an additive functor the direct sum of its values at 1.
Notice that the target of the map can be identified with the realization of thecategory of functors from EG/H to CYn , while the source can be thought of as
the realization of the category of functors from EG/H to a category of direct
sum decompositions in CY . Choosing a model for the direct sum for eachn-tuple of objects in CY yields a functor from CYn to CY (n) which can be used
to show that the canonical functor CY (n) → CY (1)n is an equivalence. Hencefor any H ⊂ G the induced functor Fun(EG/H, CY (n)) → Fun(EG/H, CYn )is an equivalence of topological categories, and hence yields a homotopy
equivalence after realization.

We examinate the equivariant homotopy type of the Γ-space ΩBMCX,G (1).
Lemma 2.10. Let H be a finite subgroup of G. Let X be a space with trivial
action by H. Then, for every G-CW complex Y , there is an isomorphism[X × G/H, ΩBMC (1)]G ∼= [X, QH (Y+ |H )]HY102NOÉ BÁRCENAS

Proof. We first recall that EG/H is naturally equivalent to B(H) We will
study the set [X, | Fun(BH, CY ) |]H . We note that giving a functor out of
the groupoid category amounts to give an object of CY with a conveniently
defined action of the finite group H. That is, there is a group homomorphism
H → Auto(S, ϕ). It follows that we can identify such a functor with a finite
H-set and a map to Y -defined on the finite set- which is compatible withthe action. That is to say, the map takes values into the H-fixed point set.
We shall call such objects a finite H-set on Y . The geometric realization
has the homotopy type ofaBAutoS,ϕS,ϕwhere S, ϕ denote the isomorphism classes of finite H-sets on Y . Given an
H-set S, we` look at its decomposition into orbits of the type H/Ki andwrite S = ni H/Ki , where ni H/Ki denotes the disjoint union of the orbit
H/Ki iterated ni times. We consider now an H- universe, V ∞ = ⊕N C(H).
Due to an equivariant version of the Whitney embedding theorems, see[Wasserman(1969)], the space of equivariant embeddings EmbH (H/K, V ∞ )is a model for the classifying space of the Weyl group of K in H, in symbols
EWK,H ≃H EmbH (H/K, V ∞ )we choose such a fixed homotopy equivalence for the rest of the proof. On
the other hand, for every map ϕ : S → Y , evaluation at the identity coset
gives a map
Y
ϕ Y Ki
Map(ni H/Ki , V ∞ ) ≈Map(H/Ki , V ∞ ) −→Yiiwhich we denote by ϕi . By means of this we can identify the spacesaaa(Y K )nBAutoS,ϕ ≃Emb(nH/K, V ∞ )
×
n

S,ϕ

K

(WK,H )n ≀Σn`Now, up to homotopy, given an embedding of ni H/Ki in V ∞ , we canchoose finite dimensional, WKi ,H -invariant subspaces Vi such that ni H/Ki ⊂Ui ⊂ Vi and a tubular neighborhood U = ∪Ui suitable for constructing acollapse map Si : S Vi → S Vi sending the tubular neigborhood homeomorphically into Vi and collapsing the exterior to ∞. We now form the WK,H
-equivariant map τi : S Vi → S Vi ∧ Y+Ki defined by
x 7→

(

∞
S(v) ∧ ϕi (eH)

And we pass to the quotientv ∈ V i − Ū or v = ∞v ∈ UiEQUIVARIANT STABLE HOMOTOPY THEORY

103

EmbH (H/K, V ∞ ) × Y K → Emb(H/K, V ∞ ) × Y K
WK,HThis defines a map to the equivariant stable WK,H -maps as follows, where
the product is meant to contain all subgroups K ⊂ H.
Y{{∗}, EWK,H| Fun(BH, CY ) |→K

× Y K }WK,H

WH,K

To resume, we get a map

αY[X, | Fun(BH, CY ) |]H →[X, QWH,K (EWK,HK

× Y K )]

WK,H

The map α determines a transformation of contravariant functors between
a monoid valued and a group valued functor, thus factoring out trough thegroup completion. As both functors of X are representable, the universal
property of the group completion implies that the map α̃ in the following
diagram is an isomorphism[X, | Fun(BH, CY ) |]H/Q

K [X, QWK,H (EWK.H

gg3
ggggg
g
g
g
g
ggggg α̃ggggg

× Y K )] ∼= [X, QH Y ]H

WK,H[X, ΩBMCY ]HWhere the right upper isomorphism is obtained by a splitting result for finitegroups, for instance Theorem 2.1 p. 206 in [May(1996)].

We obtain from proposition 2.10
Corollary 1. For any finite subgroup H of G, and any number n ∈ N ∪ {0},there exist group isomorphisms[S n × G/H, ΩBMC ]G ∼= π H (Y )
Y

n

H
(Y )
π0 (MapG (G/H, ΩBn MCY )G ) ∼= π−n2.4. The associated contravariant Or(G)-space. Although the configuration model introduced in 14 is useful for some purposes, particularly theproof of exactness, lemma 4.2 below, the configuration model is not functorial.
We overcome this difficulty by describing an Or(G)-space which does depend functorially on X and which represents the same homotopy type withrespect to spaces with finite isotropy. Related constructions have been considered in [Shimakawa(1991), Section 1, Page 243].

104

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Let set be the category of finite sets, which has a symmetric monoidal
structure coming from disjoint union. For any discrete group G, define theequivariant Γ-space | setG | as the functor which assigns to an object n in
Γ the geometric realization of the category of functors from the transport
category of G to the category set(n) of disjoint unions of finite sets indexed
by n and coherent set isomorphisms.
In Symbols:| setG |=| Fun(EG, set(n)) |as described in example 1, this G- space has an associated Or(G)-space,which we denote by set? .
The following is an ad-hoc adaptation of [Shimakawa(1989)].
Definition 15. Let X be a proper G-CW complex.
The associated Or(G)-Γ- space associated to X is the contravariant Or(G)space given by assigning to an orbit G/H the functor which to an object
G/H assigns the functor defined on Γ
given by the geometric realization of the topological category ΣH
X whosespace of objects is the spacen

X H+ ∧ setH (n)and where a morphism between the objects x(n) ∧ y(n) and x(m) ∧ y(m) isa morphism in Γ, θ : n → m for which x(θ(n)) ∧ y(θ(n)) = x(m) ∧ y(m).
Lemma 2.11. Let X and Y be proper G-CW complexes. There exists anatural isomorphism?
∧ set? )]Or(G,F IN ) −→ [Y ? , Q? (X+ )]Or(G,F IN )[Y ? , ΩB(X+

Proof. The map is given as the composition of two functors, the Γ-category{e}functors CY (n) → ΣY (n) which assigns to the object (S, ϕ) the object(S, ϕ(S)). The second functor involved in the definition of the left hand side
map is the associated space functor Z 7→ Z ? . After this composition, it iseasy to verify that the obtained Or(G)-space asigns to an object (G/H) thecategory
Fun(EG, ΣY )H , which is equivalent to Fun(BH, ΣY ). Using an analogous
argument to proposition 2.10 for ΣX , one checks that for any object G/Hin Or(G, FIN ), the group completions agree with QH (Y+ ) up to weak homotopy equivalence.
EQUIVARIANT STABLE HOMOTOPY THEORY

105

3. Equivariant Homology Theories
3.1. Axiomatic description of Equivariant Homology Theories. In
this section, we define equivariant stable homotopy groups for proper GCW complexes and prove that this definition yields an equivariant homology
theory on the category of proper G-CW complexes.
We recall the axioms of an Equivariant Homology Theory, as introduced
in [Lück and Reich(2005)].
Definition 16. Let G be a group and fix an associative ring with unit R.
A G-homology theory with values in R-modules is a collection of covariant
functors HnG indexed by the integer numbers Z from the category of G-CW
G (A) =
pairs together with natural transformations ∂nG : Hn (X, A) → Hn−1
Hn−1 (A, ∅), such that the following axioms are satisfied:(1) If f0 and f1 are G-homotopic maps (X, A) → (Y, B) of G-CW pairs,
then HnG (f0 ) = HnG (f1 ) for all n.
(2) Given a pair (X, A) of G-CW complexes, there is a long exact sequence...

HG
n+1 (j)→G∂n+1

HG (i)n
G (X, A) → HG (A) →HnG (X)
Hn+1
n

Hn (j)→

GG
Hn−1 (i)∂nG(A) → . . .
HnG (X, A) →Hn−1

where i : A → X and j : X → (X, A) are the inclusions.
(3) Let (X, A) be a G-CW pair and f : A → B be a cellular map. The
canonical map (F, f ) : (X, A) → (X∪f B, B) induces an isomorphism∼=

HnG (X, A) → HnG (X ∪f B, B)(4) Let {Xi |`i ∈ I} be a family of G-CW -complexes and denote byji : Xi → i∈I Xi the inclusion map. Then the map
a∼=⊕i∈I HnG (ji ) : ⊕i∈I HnG (Xi ) → HnG ( Xi )iis bijective for each n ∈ Z.
Let α : H → G be a group homomorphism and X be an H-CW
complex. The induced space indα X, is defined to be the G-CW
complex defined as the quotient space obtained from G × X by theright H-action given by (g, x) · h = (gα(h), h−1 x).

An equivariant homology theory consists of a family of G-homology theories H∗G together with natural group homomorphisms
indα : HnH (X, A) −→ HnG (indα (X, A))106

NOÉ BÁRCENAS

associated with a homomorphism α : H → G which satisfy the following
conditions(1) indα is an isomorphism whenever ker α acts freely on X.
(2) For any n, ∂nG ◦ indα = indα ◦ ∂nH .
(3) For any group homomorphism β : G → K such that ker β ◦ α acts
freely on X, one has
indα◦β = HnK (f1 ◦ indβ ◦ indα ) : HnH (X, A) → HnK (indβ◦α (X, A))
where f1 : indβ indα → indβ◦α is the canonical G-homeomorphism.
(4) For any n ∈ Z, any g ∈ G, the homomorphism
indc( g):G→G HnG (X, A) → HnG (ind)c(g):G→G (X, A))agrees with the map HnG (f2 ), where f2 : (X, A) → indc(g):G→G (X, A)
sends x to (1, g −1 x) and c(g) is the conjugation isomorphism in G.
4. Definition and verification of the axioms
4.1. G-homological structure. We introduce now the equivariant stablehomotopy groups between proper G-CW pairs
Definition 17. Let (X, A) be a proper G-CW pair. The equivariant stablehomotopy groups of (X, A) are defined as(
πn (ΩBMCX∪ConeA,G (1)), n ≥ 0
πnG (X, A) =π1 (ΩB1−n MCX∪ConeA,G (1)), n < 0
The main result of the rest of this section is
Theorem 4.1. Definition 17 gives an example of a functor satifying exactness and excision. By means of proposition 2.11, it can be associated with
an Equivariant Homology Theory.
Remark. It is known, [Lück and Reich(2005)], proposition 6.8 in page 94,
that in order to provide a Homology Theory, it suffices to construct afunctor from the category of small groupoids to the category of spectra
sending equivalences to weak equivalences. Whereas this approach can beused for the equivariant Homology Theory associated to algebraic K-theoryand (topological) K-Homology, as for example in [Davis and Lück(1998)], it
presents some difficulties when trying to define equivariant stable homotopy
theory. Our proof of exactness uses the concrete model from definition 14.
The following lemma deals with the exactness property:

EQUIVARIANT STABLE HOMOTOPY THEORY

107

Lemma 4.2. Let (X, A) be a G-CW pair. The inclusions j : X → X ∪A
Cone(A) and i : A → X induce a sequence of abelian groupsj∗i∗π∗G (X) → π∗G (X, A)π∗G (A) →which is exact in the middle.
Proof. The question is reduced to analize the behaviour of the map
MCX,G (1) → MCX∪ConeA,G (1)induced by the map j. Using Remark 2.3, and Lemma 2.1, as well as Theorem 2.8, the question is reduced to prove that for any subgroup H ⊂ G,
any natural number n, the natural transformation j∗ , defined between thesimplicial objects ChnH in the category of compactly generated spaces whose
objects are given by length n chains of natural transformations
f1 : BH/ CX

f2 : BH



/ CX

...



...

fn : BH/ CX

between functors fi , induces a homotopy fibration with fiber A.
We will examinate the behaviour of the natural transformation inducedby j when restricted to ChnH , beginning with Ch0H , the space of 0-simplices.
We will introduce the following notation
Given an equivalence class of a functor f : BH → set, a model for f is a
finite set f ({•}) = S ⊂ N together with an group homomorphism H → Σ|S|
implementing an action given by the functor. We call a model S irreducible if
it has no proper subfunctors. Let CSH (X) be the full subcategory of functors`
equivalent to those defined by finite disjoint unions ki=1 f , in the sense`kthat they admit a model given by the disjoint unionof`` i S, where S is ∗anirreducible model for the functor, and a map
ϕi : i S → Objects CX H
to the space of objects.
Specializing to {•}, the functor admiting a model S with only one point
H (X) consists of the category of finite
and trivial action, the category C{•}sets labelled in X H . Topologized as subspace of the configuration space,
H . Thethe space of functors given by an irreducible model agrees with X+

108

NOÉ BÁRCENAS

natural transformation induced by j agrees up to homeomorphism with the
H → (X ∪ ConeAH ) , which is homotopy equivalent to the quomap X+
A+
H → X/AH . Lemma 2.7 allows to conclude that the natural
tient map X+
transformation induced by j induces a homotopy fibration with fiber AH at
H (X). In particular, for any
the level of morphism spaces in the category C{•}natural number, the map defined on the reducible functors given as n-th disjoint union of maps defined in {•}, up to homeomorphism the map induced
H ) → C n (X ∪ ConeAH ), is a homotopy
on the configuration spaces C n (X+A
fibration:
H (X) conFor any irreducible model S, the irreducible functors in C{•}sist of a point in the configuration space C |S|(X H ) and a permutation of
the elements induced by the action on the model S. Hence, the naturaltransformation induced by j restricts to the induced map C |S|(X H ) →C |S| (X ∪A ConeAH ) equivariantly with respect to the actions permuting
the coordinates labelled on S.
Since the space of objects decomposes as the wedge of CSH (X) for all
equivalence classes of models S, the result follows for ChH
0 . Since morphisms
are discrete, Theorem 2.8 finishes the proof`Lemma 4.3. Let X = Xi . Then MCX,G (1) ≃ ∨i MCXi (1)`Proof. This boils down to the fact that C∗ ( i Xi ) ≈ ∨i C∗ (Xi )Lemma 4.4. Let (X, A) be a proper G-CW pair and f : A → B be acellular map. The canonical map (F, f ) : (X, A) → (X ∪f B, B) induces anisomorphism∼=
πnG (X, A) → πnG (X ∪f B, B)
Proof. The result is equivalent to the fact that for any G-CW complexes,
and any subcomplexes X1 , X2 such that X = X1 ∪X2 , the groups πnG (X, X1 )and πnG (X1 , X1 ∩ X2 ) are isomorphic. But this follows since the inclusion(X1 , X1 ∩ X2 ) → (X, X1 ) induces an equivariant homotopy equivalence.

4.2. Induction Structure. The following proposition resumes the results
concerning induction structure.
Lemma 4.5. Let X be an H-CW complex and α : H → G be an homomorphism of discrete groups. Suppose that ker α is a finite group and acts freely
on the H- space X. Then there exists a map
MCX,H → MCindα X,Ginducing an isomorphism at the level of homotopy groups.

EQUIVARIANT STABLE HOMOTOPY THEORY

109

Proof. We will construct the map as a functor indα between the relevant
categories. Let f : EH → CX be a functor and`denote by f (e) = S, ϕ : S →f (e) the object in CX given
X the value at the object e ∈ H. Denote by|ker α|by the set Sker α obtained as the iterated disjoint union` of copies of S, as
many times as the cardinality of ker α, and the function ϕ. Since ker α acts
freely, the map i : X → indα X is injective,
and hence the constant functor`
EG → Cind X with value Sker α , i ◦ ker α ϕ is well defined. We will analyze
the behaviour of the functor ind α. Let K ≤ H be any subgroup. Using
remark 2.3, the behaviour of the map on the K, respectively α(K)- fixed
point set reduces to the study of the functor categories Fun(EH/K, CX ),Fun(EG/α(K), Cind α X ), and ultimately to the behaviour of the functor indαin the functor categories Fun(BK, CX ), Fun(Bα(K), Cindα X ). That is thefunctor which assigns to a functor f defined` `on the unique object with value
S, ϕ : S → X, the object with value i◦ S, ϕ, and where an automorphism
given by k ∈ K gives the block isomorphism given by the iterated union of
the isomorphisms. We will use the usual fibering Theorem [Quillen(1973)],
Theorem A in page 93 to conclude the (weak) homotopy equivalence of thesespaces.
Let f ∈ Fun(Bα(K), Cindα X ) be a functor. Given Any pair of objects
c1 , c2 in Fun(BK, CX ) with isomorphisms indα ci → f is isomorphic to f via
an isomorphism induced from one of finite sets. The freeness of the actionof ker α implies that there exists isomorphisms c1 → c2 . Hence, the functor
induces a homotopy equivalence.

Remark. The previous functor defines a kind of “transfer map”
Fun(EH/K, CX ) → Fun(EG/α(K), Cind α X )in the sense that it gives a map in the opposite direction to the one induced by the projection H/K → G/α(K). Induction structures are closely
related to collapse maps or appropriate transfer constructions in RO(G)graded cohomology built up out of them. See [Wirthmüller(1974)], and also
[Lewis et al.(1986)Lewis, May, Steinberger, and McClure]. Although the hypotesis of the finiteness of ker α seems to be annoying, the reason for thislies perhaps in this fact. Transfer Maps are usually defined under the assumption of compact fibers.
5. The Bivariant Theory and the Bivariant Burnside Module
for infinite groups.
A naive spectrum E with an action of a group G yields a Z-graded GCohomology Theory, defined by110NOÉ BÁRCENASn
HG(X, A) = π∗ (MapG (X/A, E(n + ∗))We extend the covariant notion given in the previous section to a bifunctor
defined in the category of proper G-CW complexes, depending covariantly
and contravariantly, consisting in the covariant part of an equivariant homology theory and in the contravariant part of an equivariant cohomology
theory. We will describe a multiplicative structure in the cohomologicalpart.
Definition 18. Let (X, A) and (Y, B) be proper G-CW complexes. We
define the stable bivariant theory to be the groups−nωG((X, A); (Y, B)) = πn (MapG (X/A, ΩBMCY ∪ConeB )(1)), n ≤ 0

respectivelyn((X, A); (Y, B)) = π1 (MapG (X/A, ΩB1−n MCY ∪ConeB )(1)), n < 0ωG

We have straightforwardly from theorem 4.1, and from the fact that equivariant Γ-spaces as introduced in definition 8 produce Z-graded equivariant
cohomology theories:
Lemma 5.1. The stable bivariant theory restricts to a Z-graded equivariant
homology, respectively, cohomology theory for a given proper G-CW complex
X, respectively Y .
The Z-graded equivariant cohomoloogy theories associated to equivariant
Γ-spaces in the sense of definition 8 come with an (internal) multiplicative
structure.
``
Let s∈S Xs , t∈T Yt be objects in the Γ-category of finite sets anddisjoint union. By means of an isomorphism α : N × N ∼= N, the set
Xs × Yt ⊂ N × N can be handled as a finite set in N. Hence, for every
pair
we get a functor α∗ : setS × setT → setS×T given by` of finite` sets S, T `( s∈S Xs , t∈T Yt ) 7→ (s,t)∈S×T α∗ (Xs × Yt ).
For any pair of groups G, H, this extends to a functor α∗ : MCX,G (S) ×
MCX,H (T ) → MCX,G×H (S × T ).
After geometric realization, this gives a G×H- equivariant map MCX,G (S)×
MCX,H (T ) → MCX,G×H (S×T ). Since the restriction of the map to MCX,G (S)∨MCX,H (T ) is constant with value on the basis point (the product with theempty set is empty), the map α∗ determines one β∗ : MCX,G (S)∧MCX,H (T ) →MCX,G×H (S × T ). Notice that both the target and the source of this map
have two compatible structures of Γ-spaces, namely, those which are given

EQUIVARIANT STABLE HOMOTOPY THEORY

111

by letting S, respectively T , run over the category of finite sets. After considering the separate realization of this Γ-spaces as simplicial spaces we geta map BMCX,G (1) ∧ BMCX,H (1) → B2 MCX,G×H (1). And at the level of loopβ∗spaces, Ω2 (BMCX,G (1) ∧ BMCX,H (1)) → Ω2 B2 MCX,G×H (1).
Now define the pairing
αX : ΩBMCX,G (1) ∧ ΩBMCX,H (1) →Ω2 |β∗ |Ω2 (BMCX,G (1) ∧ BMCX,H (1)) → Ω2 B2 MCX,G×H (1) ≃ ΩBMCX,G×H (1)This map does not depend on the choice of the isomorphism α as a consequence of the following
Lemma 5.2. Let α : N∞ → N∞ be an injective map. Then, the inducedα
map α∗ : MCX,G (1) → MCX,G (1) defined by composition with set → set is
G-homotopic to the identity.
Proof. There is a natural transformation of the functor α∗ to the identity.
Notice that there is more than one homotopy equivalence Ω2 B2 MCX,G (1) →ΩBMCX,G (1). The following technical result denies a role of this choice inour discussion.
Lemma 5.3. Let A be any equivariant Γ-space. Then, for any n and 0 ≤k ≤ n, the maps ikn : Ωn+1 Bn+1 A(1) → Ωn Bn A(1) induced by the inclusions{1} → {1, . . . , n} differ up to weak G-equivariant stable equivalence.
Proof. Note that all ikn differ by permutation of the coordinates of Bn A as asimplicial set and a switch of the looping. Write σ∗ : Ωn Bn A → Ωn Bn A forthe map induced by such a permutation σ ∈ Σn ⊂ Σn+1 . It suffices to showthat all σ∗ are homotopic to the identity. Consider for this the following
diagramΩBA(1)

ϕ/ Ωn+1 Bn+1 A(1) oinnΩn Bn A(1)
σ∗

1×σ∗

id



ΩBA(1)ϕ/ Ωn+1 Bn+1 A(1) o



in
n

Ωn Bn A(1)where ϕ = i0n ◦ . . . ◦ inn is induced by identifying A(S) with A(S, 1, . . . , 1).
All maps are weak G-homotopy equivalences. Hence, (1 × σ∗ ) and (σ)∗ areweakly G-homotopic to the identity.
112

NOÉ BÁRCENAS∗ ((X, A), (Y, B)) given by
Consider the pairing α(X,A);(Y,B) defined on ωG
the composite mapsp+qpq((X, A), (Y, B))((X, A), (Y, B)) → ωG×G((X, A), (Y, B)) × ωG
ωG
∆∗

p+q
((X, A), (Y, B))→ ωG

Where the first map is the pairing αX∪ConeA , and ∆∗ is induced by restriction to the diagonal subcategory EG of EG × G. Then, we have thefollowing:
Lemma 5.4. For any discrete group G and any proper G-CW pair (Y, B) the∗ ((X, A)(Y, B))
pairings α(X,A);(Y,B) define a structure of graded ring on ωG
We will now introduce a bivariant Burnside module. This has been
described for finite groups by [Lewis et al.(1982)Lewis, May, and McClure]and generalized for pairs of a finite and a compact Lie group in [Lee(1994)].
We recall for this the notion of the classifying space for a family of subgroups.
Definition 19. Let F be a collection of subgroups in a discrete group G
closed under conjugation and intersection. A model for the classifying space
for the family F is a G-CW complex X whose isotropy subgroups all lie in F
and for every subgroup H ∈ F, the fixed point set X H is either contractible
if H ∈ F or empty otherwise.
Particularly relevant is the classifying space for proper actions, the classifying space for the family FIN of finite subgroups, denoted by EG.
The classifying space for proper actions always exists, is unique up to
G-homotopy and admits several models [Lück(2005)]. The following list
includes some examples. We remit to [Lück(2005)] for further discussion.
• If G is a finite group, then the singleton space is a model for EG.
• Let G be a group acting properly and cocompactly on a Cat(0) space
X. Then X is a model for EG.
• Let G be a Coxeter group. The Davis complex is a model for EG.
• Let G be a mapping class group of a surface. The Teichmüller space
is a model for EG.
• Let G be a Gromov hyperbolic group. The Rips complex is a modelfor the classifying space for EG.
Remark. The classifying space for proper actions, EG can be characterized
by the universal property of being a proper G-CW-complex EG such thatfor any proper G-CW complex X, there exists up to G-homotopy a uniqueG-equivariant map
X → EGEQUIVARIANT STABLE HOMOTOPY THEORY

113

Definition 20. Given a pair of discrete groups Π, G , the classifying spacefor proper, Π-free actions is the classifying space for the family of finite
subgroups H ⊂ G × Π for which H ∩ e × Π = {e}. We will denote this spaceas
EF IN ,Π (G × Π)
this is a G-space via the projection G × Π → Π
Definition 21. [ Bivariant Burnside module for infinite groups] Let G andΠ be discrete groups. The bivariant Burnside module A(G, Π) is defined as
the 0-th equivariant cohomology associated to EF IN ,Π (G × Π) evaluated in
the classifying space for proper actions EG. In symbols:∗A(G, Π) = ωG
(EG, EF IN ,Π (G × Π)) = [EG, ΩBACEF IN ,Π (G×Π) (1)]

Problem 1. Let G be a group accepting a finite model for EG. Relate thebivariant Burnside ring to the set of stable maps between classifying spaces[BG, Ω∞ Σ∞ (Bπ+ )]When both G and Π are finite groups, with the additional hypothesis that G is a finite p-group, a quite complete description was given by[Lewis et al.(1982)Lewis, May, and McClure]. Some progress has been madeas to give an explicit decomposition of the set of stable maps between the
p-completed classifying spaces.
The following result was proven in [Lee(1994)], Theorem 1.2 page 722:
Theorem 5.5. Let Π be a virtually torsion-free group vith finite virtual
cohomological dimension and Q a finite p-group. For each conjugacy classof subgroup of Q, the group WQ (H) acts on the set of Π-conjugacy classesof homomorphisms ϕ : H → Π. For each representative (ϕ) of the WQ (H)orbit, define∆ϕ = {(u, ϕ(u) | u ∈ H} ⊂ Q × H

Nϕ = NQ×π (∆ϕ)

Wϕ = (N ϕ/∆ϕ)Then there is an equivalence map of spectraΨ : ∨ϕ BWϕ −→ F [BQ+ , BΠ+ ]which is a stable homotopy equivalence after completion at p, where F denotes the function spectrum.
The relation to the bivariant Burnside ring is as follows: Since Π is avirtually torsion free group it may be assumed that there exists a finite′group quotient K = Γ/Γ . Then, there exists a finite model for the classifying space for proper Γ- actions denoted by X in [Lee(1994)]. In this case,114

NOÉ BÁRCENAS

the space EQ×K (K, FIN ) × X is homotopy equivalent to BΠ. After theΠapplication of the splitting result cited in [May(1996)], page 206 Theorem
2.1, and a consequence of the Segal Conjecture, Theorem C , page 198 in[Carlsson(1984)], the result follows.
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Noé Bárcenas
Hausdorff Center for Mathematics
Mathematisches Institut
Endenicherallee, 62,
D-53115, Germany
Current Address:
Centro de Ciencias Matematicas
UNAM, Campus Morelia
Apartado Postal 61-3 (Xangari)
C.P. 58089, Morelia, Michoacan, MEXICO
e-mail address: barcenas@math.uni-bonn.de, barcenas@matmor.unam.mx(Received August 2, 2011 )(Revised March 7, 2013 )