the ordinary ro c -graded bredon cohomology of a point · eilenberg-steenrod axioms for cohomology...
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The Ordinary RO(C2)-Graded Bredon Cohomology of aPoint
Tiago Duarte Guerreiro
Thesis to obtain the Master of Science Degree in
Mathematics and Applications
Examination Committee
Supervisor: Prof. Pedro Ferreira dos SantosCo-supervisor: Prof. Wojciech Chacholski
Members of the Committee: Chairperson: Prof. Pedro Manuel Agostinho ResendeProf. Gustavo Rui Gonçalves Fernandes de Oliveira Granja
May 2015
ii
Para a minha avo.
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Acknowledgments
This project would not have been possible without the invaluable help and support of my advisor,
Pedro Ferreira dos Santos, who always found the time and patience to walk me through the course of
this work. I would also like to thank my co-advisor, Wojciech Chacholski, who helped me in my first steps
in Algebraic Topology. I would also like to thank Gustavo Granja for reading my thesis and giving me
valuable suggestions which made everything much clearer. It was a pleasure to work with them.
To all my friends who supported me, specially Ricardo, from whom I have probably received more
constant support than from anyone else during the whole process of this work and, of course, to Feder-
ica.
To my family, who always trusted me and to whom I owe most of what I am.
I would like to mention my University, Instituto Superior Tecnico, which I was lucky enough to be part
of for the last years of my studies. It opened me an infinite number of doors and it felt like a second
home, if not the first.
This project was supported by an FCT grant, ref PTDC/MAT-GEO/0675/2012, for which I thank Mar-
garida Mendes Lopes and Pedro Ferreira dos Santos.
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Resumo
Seja C2 o grupo cıclico com dois elementos. Calculamos a estrutura de grupo da cohomologia
ordinaria de Bredon graduada em RO(C2) de um ponto para qualquer functor de Mackey constante.
Os ingredientes principais para tal sao o uso de sucessoes de cofibracao adequadas juntamente com
a versao equivariante da Dualidade de Spanier-Whitehead para um ponto.
Quando M provem de um anel comutativo, existe uma estrutura de anel em H∗,∗(X;M) que tem
origem num produto cup generalizado. Calculamos explicitamente essa estrutura para os functores
de Mackey constantes R, Z and Z2, usando aplicacoes entre os aneis de cohomologia equivariante e
cohomologia singular e o conceito de espectro-G que representa teorias de cohomologia de Bredon
graduadas em RO(G).
No decurso deste trabalho introduzimos varias nocoes em teoria em homotopia equivariante, nomeada-
mente os conceitos centrais de complexo-CW-G, functores de Mackey e espectros-G.
Palavras-chave: Complexo-CW-G, Functor de Mackey, Espectro-G, Cohomologia Equivari-
ante.
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Abstract
Let C2 be the cyclic group with two elements. We compute the group structure of the ordinary
RO(C2)-graded Bredon cohomology of a point for any constant Mackey functor. The main ingredients
to do so are suitable cofiber sequences together with equivariant Spanier-Whitehead Duality for a point.
When M comes from a commutative ring, there is a ring structure arising from the existence of a
generalized cup product in H∗,∗(X;M). We compute this structure explicitly for the constant Mackey
functors R, Z and Z2 using existing ring maps between equivariant cohomology and singular cohomol-
ogy rings and the notion of G-spectra which represent RO(G)-graded Bredon cohomology theories.
Throughout this work we introduce various notions in equivariant homotopy theory, namely the central
concepts of G-CW-complex, Mackey functors and G-spectra.
Keywords: G-CW-complex, Mackey functor, G-spectra, Equivariant Cohomology.
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Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Preliminaries 4
2.1 The equivariant homotopy category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 G-maps and G-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Group representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.3 Recovering the non-equivariant setting . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.4 G-Homotopy theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Bredon Homology and Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 G-CW-complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 Relations between Bredon and non-equivariant homologies . . . . . . . . . . . . . 18
2.3 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Non-equivariant spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.2 Equivariant spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Mackey Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 The RO(G)-grading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5.1 Representability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Computation of the cohomology ring of a point 29
3.1 RO(G)-graded cohomology of a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.1 The group structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.2 The forgetful map to singular cohomology . . . . . . . . . . . . . . . . . . . . . . . 33
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3.1.3 The multiplicative structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Bibliography 43
xii
List of Tables
2.1 The bookkeping diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 Group structure of H∗,∗(pt;R). The first index runs horizontally and the second index runs
vertically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Group structure of H∗,∗(pt;Z). The first index runs horizontally and the second index runs
vertically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Group structure of H∗,∗(pt;Z2). The first index runs horizontally and the second index
runs vertically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
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Chapter 1
Introduction
1.1 Motivation
In [Bredon, 1967], Bredon initiated the study of equivariant cohomology theory, extending the usual
Eilenberg-Steenrod axioms for cohomology to account for equivariance coming from the action of a
finite group G. This is achieved using the language of coefficient systems which are functors from the
category of orbits of G to the category of abelian groups.
These axioms, however, assume that cohomology functors are graded in Z although equivariant
cohomology theories are more naturally graded by the free abelian group generated by isomorphism
classes of irreducible representations of G. This group is usually called RO(G) for historical reasons
even though RO(G) is actually the group of equivalence classes of formal sums of representations.
In fact, it is proven in [May, 1996] that if the coefficient system of Bredon cohomology extends to a
Mackey functor, or, in other words, is the underlying coefficient system of a Mackey functor, then Bredon
cohomology theory extends to an RO(G)-graded cohomology theory.
This extension comes, however, at a price. It is significantly harder to make explicit computations
even for very simple spaces like a point and the existing literature is often incomplete. As a result, many
calculations that ought to be known are not and many standard tools from the non-equivariant world are
yet to be generalized to this setting.
This thesis is an attempt to make a self-contained exposition of basic G-equivariant homotopy theory
and present some important explicit computations in the case when G = C2.
1.2 Organization of the thesis
Throughout this work, a topological space means a compactly generated weak Hausdorff space and
we use X and Y to denote topological spaces with an action by a finite group G, unless otherwise
stated. By a map between topological spaces we mean a continuous map. For any prime p, the cyclic
group with p elements is written Cp to denote the group of equivariance and Zp otherwise. We are
particularly interested in studying equivariant cohomology in the case G = C2. We denote the constant
1
Mackey functor, defined in example (2.17), by M , and we will be specially interested in the cases when
M = R, Z and Z2. Singular cohomology in degree p of a space X with coefficients in an abelian group
M is written as Hpsing(X;M) and similarly for singular homology.
This work is organized in two main parts, the ”Preliminaries”, where we present the concepts and
results needed for the second part, the ”Computation of the cohomology ring of a point”.
Chapter 2 starts by giving an overview and tries to lay the foundations of the world of equivariant
homotopy theory. There, we generalize the notion of homotopy groups using representation spheres,
SV , where V is a finite-dimensional G-representation, which are spheres with an action, and that of
Eilenberg-MacLane spaces. We state and prove a panoply of adjunctions which are going to be essential
for our purposes. In particular, we prove that there is an adjunction,
G/H+ ∧ − : H − Top G− Top : U,
where, for K a subgroup of G, K − Top denotes the category of K-spaces and K-maps. We use this
to obtain a natural isomorphism, in Proposition (2.16), between the equivariant cohomology of C2+ ∧X
and singular cohomology of X, where X is a pointed C2-space.
In section (2.2), we introduce the main objects of equivariant Bredon cohomology, theG-CW-complexes.
These are built in much the same way as usual CW-complexes, only now we attach equivariant cells,
G/H × Dn, where H runs over the subgroups of G. We are then in a position to speak about the or-
dinary Z-graded Bredon cohomology and homology and give some examples. We also outline a proof
of a theorem by Lima-Filho, see [Lawson et al., 2003], that relates ordinary Z-graded Bredon homology
with singular homology in subsection (2.2.3).
We define and give examples of Mackey functors in section (2.4). Of particular importance for us is
the constant Mackey functor M described in example (2.17).
The RO(G)-graded Bredon cohomology theories, introduced in section (2.5), are representable
through a generalization of spectra, the genuine G-spectra, which we explain in section (2.3). Later
on, dos Santos, [dos Santos, 2003], identified a simple model for the spaces that represent these the-
ories generalizing the Dold-Thom theorem that we outline in this section. He proved, in particular, that
there is a natural equivalence,
HGV (X;M) ∼=
[SV ,M ⊗X
]G,
where V is a finite-dimensional G-representation and M ⊗X is a specific G-space with the property of
being an equivariant infinite loop-space. This will also be helpful in establishing an equivariant version
of Spanier-Whitehead Duality for a point, which we heavily use in computations.
Finally, in chapter 3, we proceed to the computations of the cohomology of a point. Subsection
(3.1.1) deals with the computation of the group structure of Hp,q(pt;M). There we make extensive use
of the cofiber sequence S(Rq,q)+ → D(Rq,q)+ → Sq,q of pointed G-spaces which is sequence (2.1) in
the text. The induced long exact sequence in homology allows us to reduce the problem of computing
the cohomology group of a point to that of computing the homology or cohomology of some projective
space. The results are summarized in tables (3.1), (3.2) and (3.3) and in theorem (3.1).
An important tool will be the forgetful map to singular cohomology, described in section (3.1.2). For
2
any pointed G-CW-complex X and constant Mackey functor M , the map,
[X,M ⊗ SV
]G
ϕ−→[C2+ ∧X,M ⊗ SV
]G,
induced by the folding map ∇ : C2+ ∧X → X, can be seen as a forgetful map to singular cohomology.
We include examples and a straightforward proof of the important fact that this map is multiplicative with
respect to the cup product in H∗,∗(X;M).
This work ends in section (3.1.3) with the computation of the cohomology ring of a point, H∗,∗(pt;M),
whenM is each of the above mentioned Mackey functors. As a crucial step we compute the cohomology
ring of C2 and use it to compute that of a point. It is there that we use the forgetful map to singular
cohomology and the machinery of G-spectra to get our final results.
3
Chapter 2
Preliminaries
2.1 The equivariant homotopy category
2.1.1 G-maps and G-spaces
Let G be a topological group with unit e and X be a topological space. A left G-action on X is a
continuous map
ρ : G×X −→ X
(g, x) 7−→ g · x
such that
1. e · x = x
2. g(h · x) = (gh) · x.
for all g, h ∈ G and x ∈ X.
A G-space is a topological space together with a left G-action. We usually drop the dot and write
ρ(g, x) = gx. A right G-action is a map X × G → X such that xe = x and (xh)g = x(hg). We usually
work with left actions since if (x, g) 7→ xg is a right action then, (g, x) 7→ xg−1 is a left action.
A (continuous) map f : X → Y between G-spaces is called a G-map or a G-equivariant map if
f(gx) = gf(x) for all g ∈ G, x ∈ X. G-spaces and G-maps form a category denoted by G − Top.
The morphisms in that category are denoted by homG(X,Y ) or homG−Top(X,Y ) and an isomorphism
between objects is called a G-homeomorphism.
A based G-space is a G-space with a basepoint fixed by G and a map between based G-spaces is
a G-map that preserves the basepoint. Notice that it is possible to turn any unbased G-space X into a
based one by letting X+ = X∐∗ where ∗ is fixed by G. Based G-spaces and based G-maps form a
category denoted by G− Top∗. If (X,x0) and (Y, y0) are G-spaces we define their smash product as
X ∧ Y = (X × Y )/(X × y0 ∪ Y × x0).
4
The orbit space of a G-space X, denoted X/G, consists of equivalence classes of the form x ∼ gx
where g ∈ G and x ∈ X. The orbit space can be endowed with the quotient topology. See [tom Dieck,
1987].
Example 2.1. 1. Any topological group G is a G-space.
2. Let H < G. The group multiplication H ×G→ G is a left H-action.
3. If H / G, then G acts on H by conjugation.
4. G/H is the orbit space of the right action G × H → G. It is itself a G-space with action given by
(g′, gH) 7→ g′gH.
5. If X and Y are G-spaces, then X × Y is a G-space where the action is componentwise. Such an
action is called diagonal. If X and Y are based then X ∧ Y is a G-space with action induced by
the diagonal action.
6. The space of continuous maps between G-spaces X and Y endowed with the compact open
topology, homTop(X,Y ), is a G-space under conjugation: Let f : X → Y . Define (gf)(x) =
gf(g−1x). If X and Y are based we denote by F (X,Y ) the subspace of homTop(X,Y ) consisting
of based maps.
7. Suppose C2 acts on Sn via the antipodal involution. The orbit space of this action is RPn.
An action of G in X is called free if, for all x ∈ X, gx = x implies that g is the identity element of G. It
is trivial if all the points of X are fixed by the action.
2.1.2 Group representations
Let G be a group. A representation of G, or G-representation, on a vector space V over the field k
is a group homomorphism,
ρ : G→ GL(V ).
Thus, defining a representation of G on V is the same as defining a linear map,
ρ(g) : V → V g ∈ G,
such that ρ(e) = idV and ρ(g1g2) = ρ(g1)ρ(g2) for g1, g2 ∈ G and e the identity of G. Another equivalent
way to see a representation is as an action on V ,
ρ : G× V −→ V
(g, v) 7−→ g · v,
where g acts linearly.
5
If V has the extra structure of an inner product, a G-representation is orthogonal if ρ(G) is a sub-
group of the orthogonal automorphisms of V , O(V ). An abuse of notation is frequent and the represen-
tation is sometimes denoted by (ρ, V ) or just by V when ρ is clear from the context. The dimension of
the representation is the dimension of V over k, and it is written dim(ρ). The representation is faithful if
ker(ρ) = e and it is called trivial if ρ(g) = idV for every g ∈ G.
Example 2.2. Let G = C2 and V be the vector space R2 over R. Choose the canonical basis for R2. We
have, for example, the representation given by
ρ(e) =
1 0
0 1
, ρ(g) =
−1 0
0 1
This is a two-dimensional faithful representation of C2 on R2.
Example 2.3. Let G be the additive group of integers, (Z,+), and ρ a representation of G on V . Then
ρ(0) = idV and, for every r ∈ Z, ρ(r) = ρ(1)r. The representation is then completely determined by the
invertible linear map ρ(1) : V → V .
It is a basic problem of representation theory to classify all the representations of a group. In order
to do that it is necessary to have an adequate notion of when two representations are essentially the
same. This leads us to define isomorphism of representations.
Two representations (ρ1, V1) and (ρ2, V2) are isomorphic if there is a linear isomorphism ϕ : V1 → V2
that interwines the actions of G, that is, such that
ϕ ρ1(g) = ρ2(g) ϕ, ∀ g ∈ G.
Let (ρ, V ) be a representation of the group G. A linear subspace W of V is G-invariant if ρ(g)(W ) ⊂
W , for every g ∈ G. In that case we can define a representation (ρW ,W ) by ρW (g)(w) = ρ(g)(w), for
every g ∈ G and w ∈ W and we call W a subrepresentation of V . A subrepresentation W of V is
called proper if W 6= 0 and W 6= V . Given two representations, (ρ1, V1) and (ρ2, V2), we can form a new
representation, (ρ1 ⊕ ρ2, V1 ⊕ V2) by
ρ1 ⊕ ρ2 : G→ GL(V1 ⊕ V2)
g 7→ ρ1(g)⊕ ρ2(g),
where V1 ⊕ V2 denotes the direct sum of the vector spaces V1 and V2.
A nonzero representation is irreducible if it has no proper subrepresentations and it is completely
reducible if it is a direct sum of irreducible subrepresentations. A central result in representation theory
for finite groups is Maschke’s theorem:
Theorem 2.1. Let G be a finite group and k a field whose characteristic does not divide the order of G.
Then every representation V is completely reducible.
Example 2.4. There are only two irreducible realC2-representations up to isomorphism. The one dimen-
sional trivial representation denoted by 1 or R1,0, and the one dimensional sign representation denoted
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by σ or R1,1. Any C2-representation, V , is thus isomorphic to a direct sum of these two. In other words,
V =(R1,0
)p ⊕ (R1,1)q. We simplify notation and write V = Rp+q,q.
To any representation V of the group G we can associate its representation sphere. The represen-
tation sphere of the representation V , written SV , is the one-point compactification of V . The action of
G on V extends to SV acting trivially on∞.
Given an orthogonal representation V , we define the unit disk in V as
D(V ) = v ∈ V | ‖v‖ ≤ 1
and the unit sphere in V as
S(V ) = v ∈ V | ‖v‖ = 1.
There is an important induced cofiber sequence of based G-spaces,
S(V )+ → D(V )+ → SV , (2.1)
arising from the homeomorphism
D(V )/S(V ) ∼= SV .
For more details see [Masulli, 2011].
Following the previous example, we write SV = Sp+q,q when V = Rp+q,q is a C2-representation. The
first index, p + q, is called the topological dimension of the sphere and the second index, q, is called
the weight of the representation.
2.1.3 Recovering the non-equivariant setting
In this section we explore an important tool when working within the equivariant world: It is often
possible to translate equivariant information into non-equivariant data. Roughly, the idea is to associate
to an equivariant map a set of maps between fixed point sets.
Let H < G be a subgroup of G. The H-fixed point set of X is the set
XH = x ∈ X |hx = x, ∀h ∈ H.
Example 2.5. The normalizer ofH inG, NG(H), acts naturally onXH . In fact, if x ∈ XH and g ∈ NG(H)
then g−1hg ∈ H for every h ∈ H. So, g−1hgx = x and hgx = gx, which implies that gx is also in XH .
Call NG(H)/H the Weyl group of H in G and denote it by WG(H). Since H acts trivially on XH the
action of the normalizer on XH induces an action by the Weyl group.
It is interesting to note that a G-map f : X → Y takes fixed point sets to fixed point sets. If H is
a subgroup of G and x ∈ XH , then hf(x) = f(hx) = f(x) and so f(x) ∈ Y H . Therefore we have a
fixed point functor , (−)H : G − Top → Top, that takes a G-map f : X → Y to the continuous map
fH : XH → Y H .
7
Recall from example (2.1) that we can see homTop(X,Y ) as a G-space under conjugation. There is
an equality of G-spaces,
homG−Top(X,Y ) = homTop(X,Y )G.
Proposition 2.2. Let X be a locally compact Hausdorff G-space and Y an Hausdorff G-space. Then,
for any G-space B, there is a natural homeomorphism,
homG−Top(X × Y,B)ψ→ homG−Top(Y,homTop(X,B)),
where, for f : X × Y → B we define ψ(f)(y)(x) = f(x, y) with x ∈ X and y ∈ Y .
Proposition 2.3. Let X be a G-space and K a space with the trivial G-action. Then, the following
natural homeomorphisms hold:
1. homG−Top(K,X) ∼= homTop(K,XG).
2. homG−Top(X,K) ∼= homTop(X/G,K).
The details of the previous propositions can be seen in [May, 1996].
Next, we construct the so called induced and coinduced spaces. Let H be a subgroup of G and Y
an H-space. We define the induced G-space as the space
G×H Y = (G× Y )/ ∼
where (gh, y) ∼ (g, hy) for all g ∈ G, h ∈ H and y ∈ Y . The G-action on G ×H Y is defined by
γ · [g, y] 7→ [γg, y]. Similarly, the coinduced G-space is a subspace of homTop(G, Y ) and is
mapH(G, Y ) = f : G→ Y | f(gh−1) = hf(g), h ∈ H, g ∈ G.
The G-action on mapH(G, Y ) is defined by (γf)(g) = f(γ−1g). Analogous constructions can be made
for based spaces. It is usual to write G+ ∧H Y for the induced based G-space and map∗H(G+, Y ) for the
coinduced based G-space.
Lemma 2.4. If X is a G-space and H is a subgroup of G, then there are G-homeomorphims:
1. G×H X ∼= G/H ×X
2. mapH(G,X) ∼= homTop(G/H,X).
Proof. For the first isomorphism we consider the two maps
G×H X → G/H ×X
([g, x]) 7→ ([g] , gx)
8
and
G/H ×X → G×H X
([g] , x) 7→ ([g, g−1x
]).
These are well defined G-maps that are inverses of each other and the result follows. For the second
isomorphism we consider the two maps
mapH(G,X)→ homTop(G/H,X)
(f : G→ X) 7→ (f : G/H → X)
[g] 7→ gf(g)
and
homTop(G/H,X)→ mapH(G,X)
(f : G/H → X) 7→ (f : G→ X)
g 7→ g−1f([g]).
Again these two maps are well defined G-maps that are inverses of each other.
Recall that, given a pair of functors F : C → D and G : D → C between categories C and D, we say
that F and G are adjoint functors and, in particular, F is left adjoint to G and G is right adjoint to F , if
there are natural transformations η : 1C → G F and ε : F G→ 1D, such that,
1F = εF Fη 1G = Gε ηG.
We call η and ε the unit and counit of the adjunction, respectively. This definition is equivalent as requiring
the existence of isomorphisms,
homD(FC,D) ∼= homC(C,GD),
natural in C ∈ C and D ∈ D. See [Awodey, 2010].
Both constructions of induced and coinduced spaces are functorial and give adjoints to the forgetful
functor U : G-Top→ H-Top.
Proposition 2.5. Let H be a subgroup of G and U be the forgetful functor from G-Top to H-Top. The
induced space functor, G×H − is left adjoint to U and the coinduced space functor mapH(G,−) is right
adjoint to U .
Proof. The unit and counit for the first adjunction are the H-map given by η : Y → G ×H Y, y 7→ [e, y]
and the G-map defined by ε : G×H X → X, [g, x] 7→ gx, respectively. In fact, the compositions,
G×H Y G×H (U(G×H Y )) G×H Y
[g, y] [e, [g, y]] [g, y]
G×HηY εG×HY
9
and,
U(X) U(G×H (U(X))) U(X)
x [e, x] x
ηU(X) Uε
are the identity maps idG×HY and idU(X), respectively. For the second adjunction, the unit and counit
are the G-maps η : X → mapH(G,X), x 7→ (fx(g) = x, ∀g ∈ G) and the H-map ε : mapH(G, Y ) →
Y, f 7→ f(e), respectively. The verification is analogous to the previous one and is omitted.
We can rewrite the previous proposition in the more useful way:
Proposition 2.6. Let H be a subgroup of G, X be a G-space and Y an H-space. Then we have the
following natural homeomorphisms
1. homG−Top(G×H Y,X) ∼= homH−Top(Y,X),
2. homG−Top(X,mapH(G, Y )) ∼= homH−Top(X,Y ).
Corollary 2.7. Let H be a subgroup of G. The functor G/H × − : Top → G − Top is left adjoint to the
fixed point functor (−)H : G− Top→ Top.
Proof. The following isomorphisms are all natural:
homG−Top(G/H ×K,X) ∼= homG−Top(G×H K,X) Lemma (2.4).
∼= homH−Top(K,X) Proposition (2.6).
∼= homTop(K,XH). Proposition (2.3).
The arguments used apply in the based case and, in particular, G/H+ ∧ − is left adjoint to (−)H .
This adjunction allows us to regard equivariant homotopy groups of a pointed space X, defined in the
next section, as non-equivariant homotopy groups of the H-fixed points of X.
2.1.4 G-Homotopy theory
In this subsection we introduce the basic notions of equivariant homotopy theory and define, in
particular, equivariant homotopy groups.
Let I = [0, 1] be the unit interval with the trivial action by G. Two G-maps f, g : X → Y are said to be
G-homotopic if there is a continuous G-map,
F : X × I → Y,
such that F (x, 0) = f(x) and F (x, 1) = g(x), for every x ∈ X. Here X × I has the diagonal action. As in
the non-equivariant case, G-homotopy is an equivalence relation. That being so, we define [X,Y ]G to be
the set of G-homotopy classes of G-maps X → Y and let [f ] denote the G-homotopy class represented
by f : X → Y . The homotopy category hG-Top has G-spaces as objects and G-homotopy classes of
10
G-maps as morphisms. A G-space is also an H-space for every H < G, and we write [X,Y ]H for the
set of homotopy classes of H-maps X → Y .
Similarly, we can define G-homotopies between based G-spaces. A based G-homotopy between
based G-maps, f, g : X → Y is a continuous based G-map,
F : X ∧ I+ → Y,
such that F (x, 0) = f(x) and F (x, 1) = g(x), for every x ∈ X. Based G-homotopy is also an equivalence
relation and we define [X,Y ]∗G to be the set of based G-homotopy classes of based G-maps X → Y .
The homotopy category hG-Top∗ is analogously defined. When the context is clear we use [X,Y ]G to
denote [X,Y ]∗G.
We now turn to one of the most important notions in equivariant homotopy theory, that of equivariant
homotopy group. Let X be a pointed G-space and H < G. How do we define πn(X) equivariantly? In
other words, how do we encode the homotopic information about X taking the G-action into consider-
ation? A natural place to start is from considering the homotopy classes of based G-maps Sn → X,
where Sn has the trivial G-action. This, however, is not enough to fully explore the equivariance since
it only captures the homotopy of XG. Thus, we look instead to homotopy classes of based G-maps
G/H+ ∧ Sn → X. Note that homG−Top∗(G/H+ ∧ Sn, X) ∼= homTop∗(Sn, XH).
Let n ∈ N and X a G-space. For each H < G, we define the component H of the nth equivariant
homotopy group of X, πHn (X), by
πHn (X) = [G/H+ ∧ Sn, X]∗G .
Proposition 2.8. Let X be a G-space, and H a subgroup of G. Then,
πHn (X) ∼= πn(XH).
The computation of the equivariant homotopy groups of a G-space consists of the computation of
the homotopy groups of its H-fixed points where H runs over the subgroups of G.
A G-map f : X → Y is said to be a weak G-equivalence if it induces isomorphisms
f∗ : πHn (X)∼=−→ πHn (Y )
for all H < G and n ≥ 0.
Notice that f : X → Y is a weak G-equivalence if and only if fH : XH → Y H is a weak equivalence
for all H < G. If such a map exists we say that X and Y are weakly G-equivalent and write X ' Y .
The notion of equivariant homotopy group can be further generalized by allowing spheres with
non-trivial G-actions, namely, by using non-trivial representation spheres. Let V be an orthogonal G-
11
representation. In this case we define the V th equivariant homotopy group of X as
πHV (X) =[G/H+ ∧ SV , X
]∗G.
Notice that, from lemma (2.4) and proposition (2.6) it follows that πHV (X) ∼=[SV , X
]∗H
.
2.2 Bredon Homology and Cohomology
2.2.1 G-CW-complexes
We would like to extend the definition of CW-complex to one that accounts for the presence of equiv-
ariance. Such an extension, while meaningful, should also be restricting enough in order that analogs
of the basic theorems on CW-complexes hold.
Let Dn = x ∈ Rn| ‖x‖ ≤ 1 be the unit disk in Rn and Sn−1 its boundary, both with trivial G-action.
Set D0 as a point and S−1 = ∅ and call an equivariant n-cell of type G/H to the G-space G/H ×Dn.
The construction of a G-CW-complex is done inductively much like that of non-equivariant CW-
complexes. The main difference is that orbit spaces, G/H with H < G, play the role of 0-cells. These
are the equivariant ‘points’.
Start with a disjoint union of orbits, G/H, where H runs over the subgroups of G, as the 0-skeleton,
X0. Then, each skeleton Xn+1 is obtained from the previous one, Xn, by attaching equivariant cells,
G/Hα ×Dn+1 along the boundaries.
More precisely, a G-CW-complex X is a G-space together with a filtration Xn |n ≥ 0 such that:
1. X0 is a disjoint union of orbit spaces, G/Hα, where Hα < G.
2. Xn+1 is obtained from Xn by attaching equivariant (n + 1)-cells, G/Hα × Dn+1, via attaching
G-maps, ϕα : G/Hα × Sn → Xn. So,
Xn+1 =
(Xn
∐ϕα
G/Hα ×Dn+1
)/ ∼
where (gHα, x) ∼ ϕ(gHα, x) for (gHα, x) ∈ G/Hα × Sn.
3. X = ∪nXn and X carries the weak topology with respect to the family Xn. This means that
A ⊂ X is closed if and only if A ∩Xn is closed in Xn for every n.
Example 2.6. Let X be the the n-sphere with antipodal action by the group C2, S(Rn+1,n+1). We give
X the structure of a C2-CW-complex.
• X0 = C2 ×D0. Call its elements (e, x0) and (g, x0).
• To obtain X1 we attach C2 ×D1 to X0 via ϕ : C2 × S0 → X0. Let S0 = 0, 1. Then, we make the
choices, (e, 1) 7→ (e, x0) and (e, 0) 7→ (g, x0). By equivariance, we get that ϕ(g, 1) = g · ϕ(e, 1) =
(g, x0) and ϕ(g, 0) = g · ϕ(e, 0) = (e, x0). The space X1 is G-homeomorphic to S(R2,2).
12
• We obtain X2 by attaching an equivariant 2-cell, C2 × D2 via a map ϕ : C2 × S1 → X1. Again,
ϕ(g, x) = g · ϕ(e, x), for every x ∈ S1. After identifying X1 with S(R2,2) we can regard ϕ(e,−) as
the identity of the underlying set of S(R2,2).
• We carry on like that until we attach an equivariant n-cell to Xn−1.
Then,
X =(C2 ×D0
)⋃(C2 ×D1
)⋃· · ·⋃
(C2 ×Dn) .
Example 2.7. Let X = Sn,n. We give X a C2-CW-complex structure. The action of C2 on X fixes the
north and south poles of Sn.
• X0 = C2/C2 ×D0∐C2/C2 ×D0. These are the two disjoint fixed 0-cells and we call them 0 and
∞.
• X1 is obtained from X0 by attaching to it a free 1-cell, C2 × D1 via ϕ : C2 × S0 → X0 such that
ϕ(e, 0) = ϕ(g, 0) = 0 and ϕ(e, 1) = ϕ(g, 1) =∞. We can identify X1 with S1,1.
• X2 is obtained from X1 by attaching to it a free 2-cell, C2 × D2 via ϕ : C2 × S1 → X1 such that
x 7→ ϕ(e, x) is a homeomorphism. This space is G-homeomorphic to a sphere of dimension 2 in
which C2 acts by rotating it by 180 denoted by S2,2.
• This process is carried until we attach a free n-cell, C2 ×Dn to Xn−1.
Then,
X =(C2/C2 ×D0
∐C2/C2 ×D0
)⋃(C2 ×D1
)⋃· · ·⋃
(C2 ×Dn) .
2.2.2 Definitions and examples
In [Bredon, 1967], Bredon describes the first equivariant cohomology theories. These theories are
defined for G-CW-complexes. The slogan here is ‘Orbits are equivariant points’.
The orbit category, denoted OG, is a category consisting of:
• Objects: The G-spaces G/H where H is a subgroup of G.
• Morphisms: G-maps between the objects.
Example 2.8. Let G = Cp where p is prime. We can represent OG through the diagram,
Cp
τ
eπ
where τ : Cp → Cp is multiplication by the generator and π : Cp → e is the projection. These satisfy
πτ = π and τp = id.
Recall that outside our equivariant world, the coefficients of the ordinary homology and cohomology
theories are defined to be H0(pt) or H0(pt), in the unreduced case. Such groups constitute the first
13
distinctions between different homology and cohomology theories. Actually, these groups determine the
value of the (co)homology theory on any finite CW-complex. This is essentially because points constitute
the building blocks of a space with a CW structure.
In the equivariant setting, things get more complicated as the building blocks are now orbit spaces
G/H, for H < G. So, the coefficient system has to contain a family of groups, one for each orbit
space. Also, it is necessary to say how these blocks are related and that requires defining induced
homomorphisms between the family of groups corresponding to equivariant maps G/H → G/K.
More precisely, a contravariant coefficient system for G is a functor M : OopG −→ Ab and a
covariant coefficient system for G is a functor N : OG −→ Ab.
Example 2.9. Let X be a G-space.
• We have the contravariant constant coefficient system,
A : G/H 7→ A
f : G/K → G/H 7→ id : A→ A
where A is a fixed abelian group.
• Let H < K < G. The covariant constant coefficient system defined as,
A : G/H A
f : G/H → G/K f∗ : A A
r [K : H] r,
where A is an abelian group and [K : H] the index of H in K.
• Homotopy and homology groups are examples of contravariant coefficient systems:
πn(X)(G/H) := πn(XH).
and
Hn(X)(G/H) := Hn(XH).
The contravariant coefficient systems for a group G form an abelian category that we denote by
CG := Fun(OopG , Ab). See [Bredon, 1967] for more details. This means that it is possible to compute
kernels and cokernels and, in particular, to have chain complexes and homology.
Non-equivariantly, the cellular cohomology of a CW-complex X is defined to be the cohomology of
the cellular chain complex,
Cn(X) = Hn(Xn, Xn−1;Z),
where Xn denotes the n-skeleton of X.
14
Let X be a G-CW-complex. Based on the previous construction, we define contravariant coefficient
systems, Cn(X), called the cellular chains on X by,
Cn(X) := Hn(Xn, Xn−1;Z) : OG → Ab
G/H 7→ Hn((Xn)H , (Xn−1)H ;Z).
For any H < G, the H-fixed point set of the n-skeleton of X, is a CW-complex. Thus, for each H, we
have a long exact sequence of the triple ((Xn)H , (Xn−1)H , (Xn−2)H) with connecting homomorphism,
· · · → Hn((Xn)H , (Xn−1)H ;Z)→ Hn−1((Xn)H , (Xn−1)H ;Z)→ · · · .
This produces a natural transformation d : Cn(X) → Cn−1(X), such that the components, dH , H < G
satisfy d2H = 0 and, so, d2 = 0. We can, therefore, conclude that C∗(X) is a chain complex in CG. Given
M ∈ CG, homOG(Cn(X),M) is the group of natural transformations Cn(X) → M and C∗G(X;M) :=
homOG(C∗(X),M) is a cochain complex in Ab.
The ordinary Z-graded Bredon cohomology of a G-CW-complex X with coefficients in the con-
travariant coefficient systemM , denotedH∗G(X;M), is the cohomology of the cochain complexC∗G(X;M).
The coboundary map δ : CnG(X;M) → Cn+1G (X;M) is given by the composition f 7→ f d, for every
f ∈ CnG(X;M) with d : Cn+1(X)→ Cn(X) defined in the previous paragraph.
Example 2.10. We calculate the Bredon cohomology of X = C2+ ∧Sn where G = C2 acts by swapping
the summands. The first step is to give X a G-CW-structure. Let
• X0 = C2/C2 ×D0. This is a fixed 0-cell.
• Xi = X0, ∀i < n.
• Xn = C2 ×Dn. We attach a free n-cell via the constant map, ϕ : C2 × Sn−1 → X0.
Next, we calculate the values of the chain complex C∗(X) for each orbit:
Cn(X)(C2) = Hn(Xn, Xn−1;Z) = Z⊕ Z.
Cn(X)(C2/C2) = Hn((Xn)C2 , (Xn−1)C2 ;Z) = 0.
Ci(X)(C2) = Ci(X)(C2/C2) = Hi(Xi, Xi−1;Z) = 0 ∀1 ≤ i < n.
C0(X)(C2) = H0(X0, ∅) = Z.
C0(X)(C2/C2) = H0((X0)C2 , ∅) = Z.
For each 0 ≤ i ≤ n, we have to compute homOC2(Ci(X),M) where we choose M to be the constant
15
coefficient system M = Z, given by the identity on orbits, Z id→ Z.
homOC2(Cn(X),Z) = Z⊕ Z;
homOC2(Ci(X),Z) = 0, ∀1 ≤ i < n;
homOC2(C0(X),Z) = Z.
We end up with a chain complex in Ab of the form:
0→ Z→ 0→ · · · → 0→ Z⊕ Z→ 0→ · · · .
Therefore, with such coefficient system, we have that HnC2
(X;M) = Z⊕ Z and H0C2
(X;M) = Z are the
only non-trivial Bredon cohomology groups for X.
The ordinary Z-graded Bredon homology of a G-CW-complex X with coefficients in the covariant
coefficient system N , denoted HG∗ (X;N), is the homology of the chain complex of abelian groups,
CG∗ (X;N), which in degree n is
Cn(X)⊗OG N =⊕
G/H∈OG
Cn(X)(G/H)⊗Z N(G/H)/ ∼ .
Where the equivalence relation ∼ is induced by (f∗m,n) ∼ (m, f∗n) for all f : G/H → G/K and
m ∈ Cn(X)(G/K), n ∈ N(G/H). The boundary map, CGn (X;N)→ CGn−1(X;N), is defined as ∂ = d⊗1
where d : Cn(X)→ Cn−1(X) is the map defined above.
Example 2.11. Let X = S1,1 be the one point compactification of R with an involution. This space has a
C2-CW-structure given by example (2.7). We calculate the Bredon homology groups of X with covariant
constant coefficient system Z from example (2.9). Recall that Cn(X)(G/H) = Hn((Xn)H , (Xn−1)H ;Z).
Then,
C1(X)(C2) = H1(X1, X0;Z) = Z⊕ Z.
C1(X)(C2/C2) = H1((X1)C2 , (X0)C2 ;Z) = 0.
C0(X)(C2) = H0(X0, ∅) = Z⊕ Z.
C0(X)(C2/C2) = H0((X0)C2 , ∅) = Z⊕ Z.
The natural transformation d : C1(X)→ C0(X) corresponds to the middle horizontal maps in the follow-
ing commutative diagram,
0 Z⊕ Z Z⊕ Z 0
0 0 Z⊕ Z 0.
0 id
The first row corresponds to the cellular chain complex associated to the free orbit, C∗(X)(C2), and the
second row to the cellular chain complex of the fixed point, C∗(X)(pt). The vertical maps are induced by
16
the equivariant map π : C2 → pt. There is additional structure coming from the action of the generator
τ ∈ C2 on C∗(X)(C2).
From example (2.9), the covariant constant coefficient system Z takes the projection π : C2 → pt to
π∗ which is multiplication by 2 in Z and takes τ : C2 → C2 to the identity map in Z.
To get the chain complex related to Bredon homology in degree zero we tensor the right vertical
groups with the covariant constant coefficient system Z and get
C0(X)⊗ Z = C0(X)(pt)⊗ Z(pt)⊕
C0(X)(C2)⊗ Z(C2)/ ∼
= (Z⊕ Z)⊗ Z⊕
(Z⊕ Z)⊗ Z/ ∼
= Z⊕ Z⊕
Z⊕ Z/ ∼
Notice that, in degree zero, τ∗ : Z ⊕ Z → Z ⊕ Z is the identity map in Z ⊕ Z because the action on the
0-cells is trivial. The same holds for π∗ : Z⊕Z→ Z⊕Z. This makes the relation induced by τ trivial and
that induced by π be
π∗(m1,m2)⊗ 1 ∼ (m1,m2)⊗ π∗(1), (m1,m2) ∈ C0(X)(pt) ∼= Z⊕ Z
in the first two components, or, more explicitly,
(m1,m2, 0, 0) ∼ (0, 0, 2m1, 2m2).
So an element of the quotient is represented by an element of the form (0, 0, c, d) for c, d ∈ Z and, finally,
Z⊕ Z⊕
Z⊕ Z/ ∼ ∼= Z⊕ Z.
To get the chain complex in degree one we tensor the left vertical groups with the constant coefficient
system Z and get
C1(X)⊗ Z = C1(X)(pt)⊗ Z(pt)⊕
C1(X)(C2)⊗ Z(C2)/ ∼
∼= 0⊗ Z⊕
(Z⊕ Z)⊗ Z/ ∼
∼= Z⊕ Z/ ∼ .
We have that π∗ : C1(X)(pt) → C1(X)(C2) is the zero map since C1(X)(pt) is trivial and τ∗ swaps
the summands. So we obtain the relation,
τ∗(m3,m4)⊗ 1 ∼ (m3,m4)⊗ τ∗(1) (m3,m4) ∈ C1(X)(C2) ∼= Z⊕ Z
or, in other words,
(m4,m3) ∼ (m3,m4).
17
So we have an isomorphism between Z⊕Z/ ∼ and Z. The only thing missing to calculate the 0th Bredon
homology of X is to calculate the image of the map
· · · → 0→ Z ∼= C1(X)⊗ Z ∂−→ C0(X)⊗ Z ∼= Z⊕ Z→ 0→ · · · .
Notice that the equivariance is captured within this chain complex. We have indeed two equivariant
0-cells and one equivariant 1-cell. The boundary map, ∂, sends each non-equivariant 1-cell to the
difference of the vertices in its boundaries and so the image of ∂ is equal to the image of 2∆− where ∆−
is the anti-diagonal map. Since any element of Z⊕Z may be uniquely written as a sum (m, 0) + (n,−n),
we conclude that HG0 (S1,1;Z) ∼= Z⊕ Z2 and that HG
1 (S1,1;Z) ∼= 0.
Equivariant Bredon cohomology and homology theories have reduced versions and those can be
defined analogously to the non-equivariant case. Consider the constant map ε : X → pt for a G-CW-
complex X and let M be a contravariant coefficient system. The kernel of the induced map in homology
ε∗ : HG∗ (X;M) → HG
∗ (pt;M) is the equivarint reduced Bredon homology of the pointed G-space X
and is denoted by HG∗ (X;M).
If X is not pointed, then HG∗ (X+;M) = HG
∗ (X;M). Equivariant reduced Bredon cohomology is
defined similarly and we refer to [Bredon, 1967] for the details.
2.2.3 Relations between Bredon and non-equivariant homologies
There is an important long exact sequence that relates Bredon homology with singular homology.
The goal of this section is to explain the ingredients in the proof of the following theorem and give some
applications.
Theorem 2.9 ([Lawson et al., 2003]). Let G = C2 and let X be a G-CW-complex. Then, there is a long
exact sequence,
· · · → Hn(X/G;Z)→ HGn (X;Z)→ Hn(XG;Z2)→ Hn−1(X/G;Z)→ · · · .
There is an analogous long exact sequence in reduced homology for pointed G-spaces which can
be obtained in a similar way to what we explain next. See [Lawson et al., 2003] for more details. Let X
be a topological space and denote by Z[X] the free abelian group on X. This is a topological group with
topology induced by that of X. If, moreover, X is a G-space, then Z[X] has a natural G-action given by
g(∑
nixi
)=∑
nigxi, ni ∈ Z, xi ∈ X.
Let τ be the generator of G. Call
Z[X]av = c+ τc|c ∈ Z[X]
the subgroup of averaged cycles of Z[X]G. This is a proper subgroup of Z[X]G and, so, we have an
18
exact sequence,
0→ (Z[X])av→ Z[X]G → Z[X]G/ (Z[X])
av → 0. (2.2)
H. Blaine Lawson, Jr., Paulo Lima-Filho and Marie-Louise Michelsohn proved the following lemma in
[Lawson et al., 2003].
Lemma 2.10. 1. Let X be a compact topological C2-space. Then, there is a natural homeomorphism
Z[X]av ∼= Z[X/C2], where the latter denotes the free abelian group on X/C2.
2. Sequence (2.2) is a fiber sequence.
Moreover, there is a natural homeomorphism,
Z[X]G/Z[X]av ∼= Z2[XG].
The main idea used in the proof of theorem (2.9) is that homotopy groups of the G-fixed points of Z[X]
compute Bredon homology. See [Lima-Filho, 1997] for further details.
Corollary 2.11. Let X be a G-space with a free G-action. Then, Bredon homology reduces to non-
equivariant homology of the orbit space. More concretely, we have an isomorphism,
HGi (X;Z) ∼= Hi(X/G;Z).
Proof. Immediate from theorem (2.9) since XG = ∅.
Example 2.12. Consider the n-sphere in Rn+1 with the action given by the antipodal map, S(Rn+1,n+1).
Then,
HC2i (S(Rn+1,n+1);Z) ∼= Hi(RPn;Z) ∼=
Z if i = 0 or i = n and n is odd,
Z2 if 0 < i < n and i odd,
0 otherwise.
Corollary 2.12. Let n > 0. Then,
HC2i (Sn,n;Z) ∼=
Z if i = n and n is even,
Z2 if 0 ≤ i < n and i even,
0 otherwise.
Proof. We use the reduced version of the long exact sequence from theorem (2.9) with X = Sn,n and
get,
0→ Hi(Sn,n/G;Z)→ HG
i (Sn,n;Z)→ Hi((Sn,n)
G;Z2)→ Hi−1(Sn,n/G;Z)→ · · · .
Let i ≥ 0. The first map Hi(Sn,n/G;Z) → HG
i (Sn,n;Z) is injective because G = C2 only fixes two
19
points of Sn,n and, so, Hi+1((Sn,n)G
;Z2) ∼= 0. Moreover, we have that
Hi((Sn,n)
G;Z2) ∼= Hi(pt
∐pt;Z2) ∼=
Z2 i = 0,
0 otherwise.
Therefore there are isomorphisms,
HGi (Sn,n;Z) ∼= Hi(S
n,n/G;Z),
for every i ≥ 1. On the other hand, we have isomorphisms,
Hi(Sn,n/G;Z) ∼= Hi(ΣRPn−1;Z) ∼=
Z if i = n and n is even,
Z2 if 1 < i < n and i is even,
0 otherwise.
It follows that
HG0 (Sn,n;Z) ∼= H0((Sn,n)
G;Z2) ∼= Z2.
which altogether gives us the result.
2.3 Spectra
2.3.1 Non-equivariant spectra
We define a prespectrum E to be a collection of based topological spaces En, for n ∈ N, together
with structure maps,
σ : ΣEn → En+1, ∀n ∈ N.
Call En the nth level of E. A map of prespectra , f : E → E′, is a sequence of maps fn : En → E′n
that commute with the structure maps, i.e., such that the diagram
ΣEn En+1
ΣE′n E′n+1
Σfn fn
commutes. This forms the category of prespectra, denoted by P.
A spectrum is a prespectrum such that the adjoint maps
σ : En → ΩEn+1
are homeomorphisms. A map of spectra f : E → E′ is a map between E and E′ regarded as prespectra.
We obtain the category of spectra which we denote by S.
20
Example 2.13. Let X be a based space. The suspension prespectrum of X consists of spaces
Xn = ΣnX together with the identifications σ : Σ(ΣnX) → Σn+1X. If X = S0 this is called the sphere
prespectrum and is denoted by S.
Example 2.14. Let G be an abelian group. The Eilenberg-MacLane prespectrum is defined by putting
Xn = K(G,n), where K(G,n) is the usual Eilenberg-MacLane space with homotopy concentrated in
degree n. It can be proven that K(G,n) and ΩK(G,n+ 1) are weakly equivalent and it follows that there
exists an Ω-spectrum HG such that (HG)n is a K(G,n) for all n.
2.3.2 Equivariant spectra
We want to extend the notions of spectrum and prespectrum to our equivariant world. One way to
do so, while exploiting all the structure arising from equivariance, is to index spectra on representation
spheres, which come already equipped with G-actions. More formally, we want to index in a G-universe.
A G-universe, U , is a countably infinite dimensional representation of G with an inner product such
that
1. U contains the trivial representation.
2. U contains countably many copies of each finite dimensional subrepresentation of iteself.
U is usually constructed as a direct sum of (Vi)∞ :=
⊕k≥1 Vi, where each Vi is an irreducible represen-
tation. A G-universe is trivial if it contains only the trivial representation and complete if it contains all
finite dimensional irreducible representations of G.
Given aG-universe U , let V be a finite dimensional subrepresentation of U . We define the suspension
and loop functors by
ΣV (−) = SV ∧ − and ΩV (−) = F (SV ,−).
Recall from example (2.1) that SV ∧X and F (X,Y ) denote the smash product with diagonal action and
the space of pointed maps X → Y with the action given by conjugation, respectively.
A G-prespectrum indexed on U is a collection of based G-spaces EV for each finite dimensional
subrepresentation of U together with basepoint-preserving G-maps,
σV,W : ΣW−V EV → EW ,
whenever V ⊂ W ⊂ U , where W − V denotes the orthogonal complement of V in W . We require that
σV,V = idV and the commutativity of the transitivity diagram,
ΣW−UEU ∼= ΣW−V ΣV−UEU ΣW−V EV
EW ,
ΣW−V σU,V
σU,W
σV,W
for U ⊂ V ⊂W ⊂ U .
21
A G-spectrum indexed on U is a G-prespectrum indexed on U where the adjoint structure maps
σU,V : EU → ΩV−UEV
are homeomorphisms.
A map of G-spectra (or G-prespectra) , f : D → E, is a collection of maps, fV : DV → EV , such that,
ΣW−VDV DW
ΣW−V EV EW
ΣW−V fV
σV,W
fW
σV,W
commutes for every V ⊂ W ⊂ U . The category of G-spectra and the category of G-prespectra
indexed on U are denoted by GSU and GPU respectively.
A G-spectrum indexed on a trivial universe is called naive and one indexed in a complete universe is
called genuine. Notice that naive G-spectra are equivalent to ordinary spectra such that the spaces are
based G-spaces and the structure maps are equivariant. If, moreover, G is trivial, G-spectra, reduces to
non-equivariant spectra.
Example 2.15. Let X be a G-space and define EV = SV ∧X. Together with the obvious structure maps
these form the suspension prespectrum of X.
Example 2.16. For each constant ”coefficient system” M there is an Eilenberg-MacLane G-spectrum
denoted by HM . To each finite-dimensional subrepresentation V it associates an equivariant Eilenberg-
MacLane space that we define in section (2.4).
The forgetful functor U : GSU → GPU has a left adjoint,
L : GPU → GSU ,
called the spectrification functor. See [May, 1996] for further details.
A map of spectra f : D → E is said to be a G-spectra weak equivalence , or weak G-equivalence
if the context is clear, if for each finite dimensional subrepresentation V of U , fV : DV → EV is a weak
G-equivalence.
2.4 Mackey Functors
Let G be a finite group and let G− Set denote the category of finite G-sets and G-maps. A Mackey
functor for G is a pair of functors (M∗,M∗) from G− Set to Ab satisfying the following axioms:
1. M∗ is a contravariant functor and M∗ and is a covariant functor.
2. M∗(S) = M∗(S) for all G-sets S. For a morphism ϕ ∈ G − Set we use the notation M∗(ϕ) = ϕ∗
and M∗(ϕ) = ϕ∗ and these are called the restriction and transfer maps respectively.
22
3. For every pullback square in G− Set
U S
T V
γ
δ
α
β
it is required that α∗ β∗ = δ∗ γ∗.
4. Both M∗ and M∗ take disjoint unions to direct sums.
In the case G = Cp, the description of Mackey functors over G can be significantly simplified and we
include it for convenience. This is due to the fact that the orbit category of G is very simple, as shown in
example (2.8).
Lemma 2.13 ([Shulman, 2010]). A Mackey functor M over G = Cp for any prime p consists of abelian
groups M(Cp) and M(e) together with the corresponding restriction and transfer maps such that
M(e) M(Cp)π∗
π∗M(Cp) M(Cp)
τ∗
τ∗
satisfy
1. Contravariant functoriality: (τ∗)p = id and τ∗π∗ = π∗.
2. Covariant functoriality: (τ∗)p = id and π∗τ∗ = π∗.
3. τ∗τ∗ = id.
4. π∗π∗x =∑p−1i=0 (τ i)∗x, for all x ∈M(Cp).
A Mackey functor M for C2 consists of abelian groups M(C2) and M(e) together with the corre-
sponding restriction and transfer maps satisfying the conditions from lemma (2.13). We can specify it
as
M(C2)
τ∗
M(e)π∗
π∗
Example 2.17. For G = C2, the constant coefficient functor, Z, extends to a Mackey functor,
Z
id
Z×2
id
This is a Mackey functor since π∗π∗x = 2x = ex+ τ∗x, for every x ∈M(C2).
Example 2.18. More generally, such a Mackey functor exists over any finite group G and any abelian
groupM in place of Z with the restriction maps being the identity maps and the transfer mapsM(G/H)→
M(G/K) given by multiplication by the index of H in K, [K : H]. This is called the constant Mackey
functor for G with value M and denote it by M .
23
Example 2.19. The Burnside ring Mackey functor B associates to each orbit space G/H the burnside
ring of H. For G = C2 it is described by the diagram,
Z
id
Z⊕ Zπ∗
π∗
where π∗(a, b) = a + 2b and π∗(a) = (0, a). It is easy to see that this functor satisfies the conditions on
lemma (2.13) and so it is a Mackey functor. See [Shulman, 2010] for more details on this functor.
Denote by MG the category of Mackey functors over the finite group G. The morphisms in this
category are the natural transformations.
2.5 The RO(G)-grading
So far we have only seen equivariant cohomology theories graded on the integers. Even non-
equivariantly, the usual Eilenberg-Steenrod axioms assume the cohomology functors to be graded on
Z. It happens, however, that equivariant cohomology theories are more naturally given by a collection of
functors indexed on the free abelian group on isomorphism classes of irreducible representations of a
fixed group G. We refer the reader to [Shulman, 2010] and to [May, 1996] for many examples of this. In
fact, if we understand G-spheres to be representation spheres, SV , then we must adapt the suspension
axiom to allow suspension by such spheres and this force us to grade on representations.
Such a theory is usually called RO(G)-graded even though the underlying abelian group of RO(G)
actually consists of equivalence classes of formal sums of representations.
Fix a group of equivariance, G, and let M be a contravariant coefficient system. It is a result
due to Gaunce Lewis, Peter May and James MacClure that the ordinary Z-graded cohomology the-
ory, H∗G(−;M), extends to an RO(G)-graded cohomology theory if and only if M extends to a Mackey
functor, or in other words, M is the underlying contravariant coefficient system of a Mackey functor.
Mackey functors are, thus, the right coefficient systems to be used in RO(G)-graded cohomology
theories. In fact, every Mackey functor has an associated RO(G)-graded cohomology theory, denoted
V 7→ HVG (−;M), where V runs over a complete G-universe. This cohomology theory is uniquely char-
acterized by
• Hn(G/H;M) =
M(G/H) if n = 0,
0 otherwise.
• The restriction maps H0(G/K;M)→ H0(G/H;M) induced by i : G/H → G/K are the restriction
maps i∗ in the Mackey functor M .
There are transfer maps H0(G/H;M)→ H0(G/K;M) and these coincide with the transfer maps of
the Mackey functor.
Let us be more precise and define axiomatically a cohomology theory graded on representations.
Recall that we are thinking of RO(G) not as equivalence classes of representations but as the free
24
HαG(X;M) Hα+V
G (ΣVX;M)
Hα+WG (ΣWX;M) Hα+W
G (ΣVX;M)
ΣW
ΣV
Hid+f (id)
(f ∧ id)∗
HαG(X;M) Hα+V
G (ΣVX;M)
Hα+V+WG (ΣV+WX;M).
ΣV
ΣV+W
ΣW
Table 2.1: The bookkeping diagrams.
abelian group on isomorphism classes of irreducible representations of G. In other words, every α ∈
RO(G) is of the form α =∑aiVi, where ai are integers and Vi distinct irreducible representations. The
following definition is adapted from [Shulman, 2010].
An ordinary RO(G)-graded reduced cohomology theory consists of a collection of functors,
HαG(−;M) : hG− Top∗ → Ab satisfying the following axioms:
1. Weak Equivalence: if f : X → Y is weak G-equivalence then HαG(f) := f∗ : Hα
G(Y ;M) →
HαG(X;M) is an isomorphism.
2. Exactness: HαG(−;M) is exact on cofiber sequences. See sequence (2.3) below to see what this
means when G = C2.
3. Additivity: the inclusions ιi : Xi →∨iXi of based spaces induce an isomorphism, ι∗ : Hα
G(∨Xi;M)→∏
i HαG(Xi;M).
4. Suspension: for each α ∈ RO(G) and G-representation V , there is a natural isomorphism
ΣV : HαG(X;M)→ Hα+V
G (ΣVX;M),
covariant in V and contravariant in X, with Σ0 the identity natural transformation.
5. Dimension: HnG(G/H+;M) = 0 for every nonzero integer n and H0
G(G/H+;M) = M(G/H). The
restriction maps H0(G/K+;M) → H0(G/H+;M) induced by i : G/H → G/K are the maps i∗ in
the Mackey functor M .
6. Bookkeeping: Let f : SV → SW be the equivariant map induced by an orthogonal isomorphism
between representations V and W . Then the diagrams in table (2.1) commute.
Similar axioms characterize ordinary RO(G)-graded reduced Bredon homology , HGα (−;M). For
an unbased space X we define HαG(X;M) to be Hα
G(X+;M) and call it the ordinary RO(G)-graded
unreduced Bredon cohomology of X and similarly for homology. When the group of equivariance, G,
is clear we often write Hα(−;M) to mean HαG(−;M) and similarly for homology.
We set up some notation for our case of interest, when G = C2. Recall that we think of RO(G) as
formal sums of distinct irreducible representations with coefficients in Z. When G = C2, there are only
two such representations up to isomorphism, the trivial representation, 1, and the sign representation,
σ, both one dimensional. We can, therefore, make the identification RO(C2) ∼= 1 · Z ⊕ σ · Z ∼= Z ⊕ Z.
25
Let V = Rp+q,q be the C2-representation corresponding to (p, q). In this case, we use the following
convenient notation,
HV (X;M) = HRp+q,q
(X;M) = Hp+q,q(X;M),
and say that RO(C2)-graded cohomology is bi-graded. Moreover, let V = Rp+q,q and V ′ = Rp′+q′,q′
be two C2-representations. Then, V ⊕ V ′ ∼= Rp+q+p′+q′,q+q′ . Following this observation, if α = (r, s) ∈
RO(C2), we write Hα(X;M) = Hr,s(X;M) and Hα+V (X;M) = Hr+p+q,s+q(X;M).
When we say that RO(C2)-graded Bredon cohomology extends Z-graded Bredon cohomology we
mean that
Hp,0G (X;M) = Hp
G(X;M)
and analogously for homology.
Let
Σp+q,q(−) = Sp+q,q ∧ −.
This allows us to write the suspension axiom in the following form:
Suspension: For each α = (r, s) ∈ RO(C2) and C2-representation Rp+q,q, there is a natural isomor-
phism,
Σp+q,q : Hr,s(X;M)→ Hp+q+r,q+s(Σp+q,qX;M),
covariant in the representation and contravariant in X, with Σ0,0 the identity natural transformation.
The exactness in cofiber sequences can also be more explicitly written as:
Exactness: Consider the cofiber sequence A → X → X/A. For every fixed integer s, there is a long
exact sequence in bi-graded cohomology,
· · · → Hr,s(X/A;M)→ Hr,s(X;M)→ Hr,s(A;M)→ Hr+1,s(X/A;M)→ · · · . (2.3)
Define,
H∗,∗(X;M) =⊕
(r,s)∈Z×Z
Hr,s(X;M).
WhenM is a commutative ring, there is a cup product^: Hp,q(−;M)×Hp′,q′(−;M)→ Hp+p′,q+q′(−;M)
that endows H∗,∗(X;M) with a graded commutative ring structure. See [Dugger, 2005].
2.5.1 Representability
As in the non-equivariant case, the RO(G)-graded cohomology theories can be represented. Actu-
ally, G-spectra represent such theories. In this section we identify a model for the Eilenberg-MacLane
G-spectra and prove an equivariant version of Spanier-Whitehead duality. Most of the content of this
section is taken from [dos Santos, 2003].
Let M be a discrete abelian group, and (X,x0) a pointed G-space. We define M ⊗X as the set of
26
finite formal sums
M ⊗X :=
∑i
mixi | mi ∈M, xi ∈ X
/ m · x0 | m ∈M .
This is a G-space with action given by (g,∑mixi) 7→
∑imigxi. Notice that Z ⊗X+ is the pointed free
abelian group generated by X, Z[X]+.
In [dos Santos, 2003], dos Santos generalizes the equivariant Dold-Thom theorem, proved by Lima-
Filho, to the setting of ordinary RO(G)-graded Bredon cohomology. This result will be often used in our
computations in the next chapter. Let α ∈ RO(G). An equivariant Eilenberg-MacLane space of type
(M,α), denoted by K(M,α), is a classifying space for the functor HαG(−;M).
Theorem 2.14 (dos Santos). Let X be a based G-CW-complex and let V be a finite-dimensional G-
representation. Then, there is a natural equivalence,
HGV (X;M) ∼=
[SV ,M ⊗X
]G.
We will call this result the Equivariant Dold-Thom Theorem. As a consequence we see thatM⊗SV
is a K(M,V ) space. As mentioned in [dos Santos, 2003], this allows us to identify a simple model for
the Eilenberg-MacLane spectrum, HM . In fact, since any K(M,V ) space is a classifying space for the
cohomology functor HVG (−;M), it follows that, under the same conditions as in the previous theorem,
Corollary 2.15. There is a natural isomorphism,
HVG (X;M) ∼=
[X,M ⊗ SV
]G.
Corollary 2.16. There is a natural isomorphism,
HVG (C2+ ∧X;M) ∼= H
dim(V )sing (X;M),
where the right hand-side group is non-equivariant cohomology with coefficients in the abelian group M .
Proof. Let Xe denote the the space X with trivial action. There is a natural C2-homeomorphism, C2+ ∧
X → C2+ ∧ Xe given by [e, x] 7→ [e, x] and [τ, x] 7→ [τ, τx], where τ is the generator of C2. We have a
chain of natural isomorphisms,
HVG (C2+ ∧X;M) ∼=
[C2+ ∧X,M ⊗ SV
]C2
Corollary (2.15)
∼=[C2+ ∧Xe,M ⊗ SV
]C2
∼=[Xe,M ⊗ Sdim(V )
]Proposition (2.6)
∼= Hdim(V )sing (Xe;M) Corollary (2.15).
In the next chapter we will need to use an equivariant version of the Spanier-Whitehead Duality that
we introduce here.
27
Corollary 2.17 (Spanier-Whitehead Duality for a point). There is a natural isomorphism,
Hr,s(pt;M) ∼= H−r,−s(pt;M).
Proof. Let Rp+q,q be a C2-representation such that r + p+ q ≥ s+ q ≥ 0. Then,
Hr,s(pt;M) ∼= Hr+p+q,s+q(Sp+q,q;M)
∼= [Sp+q,q,M ⊗ Sp+q+r,q+s]G Corollary (2.15)
∼= Hp+q,q(Sp+q+r,q+s;M) Theorem (2.14)
∼= H−r,−s(pt;M).
28
Chapter 3
Computation of the cohomology ring
of a point
3.1 RO(G)-graded cohomology of a point
Throughout this chapter we consider G = C2. We compute the ordinary RO(G)-graded cohomology
of a point for the constant Mackey functors M = R, Z and Z2. The idea is to reduce these computa-
tions to those of the (Z-graded) singular cohomology or homology of some space, for each underlying
coefficient group.
In order to do so, we heavily use the suspension axiom present in both equivariant cohomology and
homology theories. For certain degrees, the use of equivariant Spanier-Whitehead duality is critical in
order to arrive at singular homology. Applying these tools, together with suitable cofiber sequences
allows us to compute almost all of the group structure for each Mackey functor.
Furthermore, for any G-space X and constant Mackey functor M , where M is a commutative ring,
there is a cup product turning H∗,∗(X;M) into a graded commutative ring. This ring structure is also
explicitly computed for each of the Mackey functors mentioned above. The main ingredient here is the
use of an existing forgetful map from RO(G)-graded cohomology to singular cohomology giving ring
homomorphisms H∗,∗(X;M)→ H∗,∗(C2+ ∧X;M).
3.1.1 The group structure
Let M be an abelian group. Recall from example (2.18) that the constant Mackey functor is given by,
M : OC2−→ Ab
G/H 7−→M,
where M takes τ : C2 → C2 to identity maps τ∗ : M → M and τ∗ : M → M and takes the projection
π : C2 → pt to the identity map π∗ : M(pt)→M(C2) and π∗ : M(C2)→M(pt) to multiplication by 2.
29
We start by computing the group structure of the ordinary RO(C2)-graded Bredon cohomology of a
point with the constant Mackey functor M , Hp,q(pt;M), for every p, q ∈ Z. We refer to H∗,q(pt;M) as
the positive cone if q runs over the non-negative integers and as the negative cone when q runs over the
negative integers.
If q = 0 the dimension axiom tells us that Hp,0(pt;M) = M if and only if p = 0.
Let q > 0. Then,
Hp−q,−q(pt;M) ∼= Hp−q,−q(S0,0;M)
∼= Hp,0(Sq,q;M)
∼= Hpsing(S
q,q/C2;M).
The last isomorphism holds since, for any pointed G-space X, H∗sing(X/G;M) is a Bredon cohomol-
ogy theory graded on the integers and, therefore, it is completely determined by the dimension axiom.
In fact we have that H∗sing((G/H)/G;M) ∼= H∗(G/H;M) ∼= M .
The space Sq,q is the unreduced suspension of the (q − 1)-sphere inside Rq,q, S(Rq,q). So it follows
that
Hpsing(S
q,q/C2;M) ∼= Hpsing(ΣRP q−1;M)
∼= Hp−1sing(RP
q−1;M),
and we conclude that the negative cone is the singular cohomology of the (q − 1)-dimensional real
projective space,
Hp−q,−q(pt;M) ∼= Hp−1sing(RP
q−1;M), q > 0.
We proceed with the calculation of the positive cone. This case is more difficult and, in particular, we
have to use equivariant Spanier-Whitehead duality for a point which is corollary (2.17) in the text.
Hp+q,+q(pt;M) ∼= Hp+q,+q(S0,0;M)
∼= H−p−q,−q(S0,0;M) Corollary (2.17)
∼= H−p,0(Sq,q;M).
By the exactness axiom for homology, the homology functor H∗,0(−;M) applied to the cofiber se-
quence S(Rq,q)+ → D(Rq,q)+ → Sq,q, which is an instance of sequence (2.1), yields a long exact
sequence,
· · · → H−p,0(S(Rq,q);M)→ H−p,0(D(Rq,q);M)→ H−p,0(Sq,q;M)→
→ H−p−1,0(S(Rq,q);M)→ H−p−1,0(D(Rq,q);M)→ · · · . (3.1)
The disc D(Rq,q) is weakly C2-equivalent to a point and by the weak equivalence axiom for homology,
30
for p 6= 0 and p 6= −1, we have H−p,0(D(Rq,q);M) ∼= H−p−1,0(D(Rq,q);M) ∼= 0. Hence,
H−p,0(Sq,q;M) ∼= H−p−1,0(S(Rq,q);M), whenever p 6= 0 and p 6= −1.
Since S(Rq,q) has a free C2-action, then, by corollary (2.11),
H−p−1,0(S(Rq,q);M) ∼= Hsing−p−1(S(Rq,q)/C2;M) ∼= Hsing
−p−1(RP q−1;M),
and, finally,
Hp+q,q(pt;M) ∼= Hsing−p−1(RP q−1;M), whenever p 6= 0 and p 6= −1.
By now we are able to compute all the cohomology groups Hp,q(pt;M) in the positive cone except
when p = q or p = q − 1.
Notice that, in order to get the remaining results, one has to compute H0,0(Sq,q;M) ∼= Hq,q(pt;M)
and H1,0(Sq,q;M) ∼= Hq−1,q(pt;M). This follows from equivariant Spanier-Whitehead duality and the
suspension axiom.
Sequence (3.1) yields the exact sequence,
0→ H1,0(Sq,q;M)→ H0,0(S(Rq,q);M)→ H0,0(D(Rq,q);M)→ H0,0(Sq,q;M)→ 0 (3.2)
and, as we shall see, its analysis for each abelian group M suffices to get the remaining cases.
It follows from the equivariant Dold-Thom theorem that the mapH0,0(S(Rq,q);M)→ H0,0(D(Rq,q);M)
in sequence (3.2) is the map M ×2−−→M . In fact, π0(X⊗M) is the set of connected components of X⊗M
and it is easy to see that, given x ∈ S(Rq,q), the element x ⊗m can be connected by a path to 0 ⊗ 2m
with 0 ∈ D(Rq,q) the origin.
Thus, we get the exact sequence,
0→ Hq−1,q(pt;M)→M×2−−→M → Hq,q(pt;M)→ 0,
and we see that Hq−1,q(pt;M) and Hq,q(pt;M) are isomorphic to the kernel and cokernel of M ×2−−→M ,
respectively. More concretely,
Hq−1,q(pt;M) ∼= M2, if q > 0,
where M2 denotes de 2-torsion subgroup of M and,
Hq,q(pt;M) ∼= M/2M, if q > 0.
We have thus proved the following theorem, which summarizes the group structure of the bigraded
cohomology of a point for any constant Mackey functor M ,
Theorem 3.1. Let M be a constant Mackey functor. Then, there are group isomorphisms,
1. Hp,0(pt;M) ∼= M if p = 0 and is trivial otherwise.
31
6 R54 R32 R10 R-1-2 R-3-4 R-5-6 R
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Table 3.1: Group structure of H∗,∗(pt;R). The first index runs horizontally and the second index runsvertically.
2. Hp,q(pt;M) ∼= Hp−q−1sing (RP−q−1;M), for q < 0.
3. Hp,q(pt;M) ∼= Hsingq−p−1(RP q−1;M), for q > 0, q 6= p, p+ 1. The remaining cases are
Hp,p+1(pt;M) ∼= M2 and Hp,p(pt;M) ∼= M/2M,
where p ≥ 0 and p > 0, respectively.
We use theorem (3.1) to provide explicit computations of the cohomology groups Hp,q(pt;M) where
M = R, Z and Z2.
The group structure of Hp,q(pt;R).
Hp,0(pt;R) ∼= R if p = 0 and trivial otherwise.
For q < 0 we have, Hp,q(pt;R) ∼= Hp−q−1sing (RP−q−1;R) ∼=
R if p = 0 and q is even,
0 otherwise.
For q > 0, Hp,q(pt;R) ∼= Hsingq−p−1(RP q−1;R) ∼=
R if p = 0 and q is even,
0 otherwise,where p 6= q and p 6= q − 1.
Moreover, Hp,p+1(pt;R) = Hp,p(pt;R) = 0, for every p > 0. The results are gathered in table (3.1).
The group structure of Hp,q(pt;Z).
Hp,0(pt;Z) ∼= Z if p = 0 and trivial otherwise.
For q < 0,
Hp,q(pt;Z) ∼= Hp−q−1sing (RP−q−1;Z) ∼=
Z if p = 0 and q is even,
Z2 if q + 1 < p < 0 and p 6≡ q mod 2 or p = 0 and q < −1 odd,
0 otherwise,
32
6 Z Z2 Z2 Z2
5 Z2 Z2 Z2
4 Z Z2 Z2
3 Z2 Z2
2 Z Z2
1 Z2
0 Z-1-2 Z-3 Z2
-4 Z2 Z-5 Z2 Z2
-6 Z2 Z2 Z-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Table 3.2: Group structure of H∗,∗(pt;Z). The first index runs horizontally and the second index runsvertically.
and, for q > 0,
Hp,q(pt;Z) ∼= Hsingq−p−1(RP q−1;Z) ∼=
Z if p = 0 and q is even,
Z2 if q − 1 > p > 0 and p ≡ q mod 2,
0 otherwise,
where p 6= q and p 6= q − 1.
The remaining cases are Hp,p+1(pt;Z) = 0 and Hp,p(pt;Z) ∼= Z2 for p > 0. The results are gathered
in table (3.2).
The group structure of Hp,q(pt;Z2).
Hp,0(pt;Z2) ∼= Z2 if p = 0 and trivial otherwise.
In this case, for q < 0, Hp,q(pt;Z2) ∼= Hp−q−1sing (RP−q−1;Z2) ∼=
Z2 if q + 2 ≤ p ≤ 0,
0 otherwise,
and, for q > 0, Hp,q(pt;Z2) ∼= Hsingq−p−1(RP q−1;Z2) ∼=
Z2 if q − 2 ≥ p ≥ 0,
0 otherwise,where p 6= q and
p 6= q − 1.
Moreover, Hp,p+1(pt;Z2) ∼= Hp,p(pt;Z2) ∼= Z2 for p > 0. The results are gathered in table (3.3).
3.1.2 The forgetful map to singular cohomology
In this section we introduce an important map that allows us, often, to relate RO(C2)-graded coho-
mology with singular cohomology. We will make extensive use of this map when computing the ring
structure of the cohomology of a point for different constant Mackey functors.
Let Rp+q,q be a C2-representation and let X be a pointed C2-space. Recall from corollary (2.16) that
there are natural isomorphisms,
Hp+q,q(C2+ ∧X;M)∼=−→ Hp+q(X;M). (3.3)
33
6 Z2 Z2 Z2 Z2 Z2 Z2 Z2
5 Z2 Z2 Z2 Z2 Z2 Z2
4 Z2 Z2 Z2 Z2 Z2
3 Z2 Z2 Z2 Z2
2 Z2 Z2 Z2
1 Z2 Z2
0 Z2
-1-2 Z2
-3 Z2 Z2
-4 Z2 Z2 Z2
-5 Z2 Z2 Z2 Z2
-6 Z2 Z2 Z2 Z2 Z2
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Table 3.3: Group structure of H∗,∗(pt;Z2). The first index runs horizontally and the second index runsvertically.
Using the suspension axiom in ordinary RO(C2)-graded Bredon cohomology, it is possible to extend the
natural isomorphism (3.3) for any integers r and s. More concretely, there are natural isomorphisms,
Hr,s(C2+ ∧X;M)∼=−→ Hr(X;M)
Therefore, we define the map
ϕ : Hr,s(X;M)→ Hr,s(C2+ ∧X;M),
induced by the folding map ∇ : C2+∧X → X, which restricts to the identity in X, and call it the forgetful
map to singular cohomology.
Consider the usual cofiber sequence, S(Rq,q)+ → D(Rq,q)+ → Sq,q. When q = 1 we have the cofiber
sequence C2+ ∧ S0,0 ∇−→ S0,0 → S1,1.
Applying H∗,q(−;M) we get the exact sequence in cohomology,
0→ H0,q(S1,1;M)→ H0,q(S0,0;M)ϕ−→ H0,q(C2+ ∧ S0,0;M)→ H1,q(S1,1;M)→ H1,q(S0,0;M)→ 0,
which after appropriate shiftings becomes,
0→ H−1,q−1(pt;M)→ H0,q(S0,0;M)ϕ−→ H0,q(C2+ ∧ S0,0;M)→ H0,q−1(pt;M)→ H1,q(pt;M)→ 0.
(3.4)
We use sequence (3.4) to compute a few examples of forgetful maps.
Example 3.1. We compute the forgetful map ϕ : H0,−2n(S0,0;M)→ H0,−2n(C2+ ∧S0,0;M), with n > 0.
When q = −2n, sequence (3.4) yields,
0→ H−1,−2n−1(pt;M)→ H0,−2n(S0,0;M)ϕ−→ H0,−2n(C2+ ∧ S0,0;M)→ H0,−2n−1(pt;M)→ 0.
Referring to tables (3.1), (3.2) and (3.3), and proposition (3.3) we can conclude the nature of the
34
forgetful maps in each case. For M = R, we get that ϕ is an isomorphism. For the Mackey functor Z,
we have the short exact sequence,
0→ Z ϕ−→ Z→ Z2 → 0,
and ϕ is the inclusion 2Z ⊂ Z. Finally, for Z2, we have
0→ Z2 → Z2ϕ−→ Z2 → Z2 → 0.
In this case, ϕ is the zero map. Actually, we will see later that, for q < 0, ϕ : H0,q(S0,0;Z2)→ H0,q(C2+ ∧
S0,0;Z2) is always the zero map as a consequence of the multiplicative structure of the forgetful map.
Example 3.2. We compute the forgetful map ϕ : H0,2n(S0,0;M)→ H0,2n(C2+ ∧ S0,0;M) with n ≥ 0.
When q = 2n, sequence (3.4) yields,
0→ H−1,2n−1(pt;M)→ H0,2n(S0,0;M)ϕ−→ H0,2n(C2+∧S0,0;M)→ H0,2n−1(pt;M)→ H1,2n(pt;M)→ 0.
Again, we just have to refer to tables (3.1), (3.2) and (3.3) and proposition (3.3) to understand what the
forgetful maps ϕ : H0,2n(S0,0;M)→ H0,2n(C2+ ∧ S0,0;M) are for each Mackey functor. For M = R, we
get that ϕ is an isomorphism and similarly for the Mackey functor Z, and, so, ϕ : Z→ Z is multiplication
by either 1 or −1. For the case of Z2, we get
0→ Z2ϕ−→ Z2 → · · · ,
and we conclude that the forgetful map is the identify in Z2.
Example 3.3. The argument in example (3.2) allows us to conclude, more generally, that ϕ : H0,n(S0,0;Z2)→
H0,n(C2+ ∧ S0,0;Z2) is an isomorphism for n ≥ 0.
Proposition 3.2. The forgetful maps give ring homomorphisms,
ϕ : H∗,∗(X;M)→ H∗,∗(C2+ ∧X;M).
Proof. The forgetful map in cohomology is induced by a continuous map of G-spaces ∇ : C2+∧X → X.
The result follows from functoriality of the ring structure.
3.1.3 The multiplicative structure
Recall that, if M is a commutative ring, there is a cup product, ^: Hr,s(−;M) × Hr′,s′(−;M) →
Hr+r′,s+s′(−;M), where M is the constant Mackey functor, that endows H∗,∗(X;M) with a graded
commutative ring structure. In fact, consider the map,
µ : (M ⊗X) ∧ (M ⊗ Y )→M ⊗ (X ∧ Y )
(∑i
mixi) ∧ (∑j
njyj) 7−→∑i,j
minj(xi ∧ yj).
35
Let V and W be C2-representations. Given representatives α : Z → M ⊗ SV and β : Z → M ⊗ SW
of HV (Z;M) and HW (Z;M), respectively, the composition,
Z∆−→ Z ∧ Z α∧β−−−→M ⊗ SV ∧M ⊗ SW µ−→M ⊗ SV⊕W ,
where ∆: Z → Z ∧ Z is the diagonal map, classifies the cup product between α ∈ HV (Z;M) and
β ∈ HW (Z;M). We henceforth refer to the cup product, ^, has multiplication.
In this section we compute the ring structure of H∗,∗(pt;M) when M is R, Z or Z2. We begin by
computing the ring structure of H∗,∗(C2;M) and use this computation to analyze the forgetful maps to
singular cohomology studied in examples (3.1), (3.2) and (3.3). These will be critical in understanding
the cohomology ring of a point.
Proposition 3.3. Let M be a ring. There is a ring isomorphism,
H∗,∗(C2;M) ∼= M[u, u−1
],
where u has bi-degree (0, 1).
Proof. For all non-negative integers n, we have isomorphisms,
Hp,q(C2) ∼= Hp,q(C2+) ∼= Hp+n,q+n(Σn,nC2+).
Recall from the proof of corollary (2.16) that there are C2-homeomorphisms, Σn,nC2+∼= Σn,0C2+ which
are multiplication by τ ∈ C2,
Hp+n,q+n(Σn,nC2+) ∼= Hp+n,q+n(Σn,0C2+) ∼= Hp,q+n(C2+) ∼= Hp,q+n(C2).
We conclude that Hp,q(C2) does not depend on the weight of the representation but only on its the
topological dimension and so Hp,q+n(C2) ∼= M for p = 0 and is trivial otherwise.
Let α : Sk,0 ∧ C2+ → M ⊗ Sk,k and β : Sk′,0 ∧ C2+ → M ⊗ Sk′,k′ be generators of H0,k(C2;M) and
H0,k′(C2;M), respectively, where k and k′ are non-negative integers. By corollary (2.16), the equivariant
map α is determined by the inclusion Sk,0 →M ⊗ Sk,k such that (x, e) 7→ 1x and similarly for β.
The product is given by composition µ (α ∧ β) (id ∧∆): Sk+k′,0 ∧ C2+ → M ⊗ Sk+k′,k+k′ which
sends (x ∧ y) ∧ e 7→ 1(x ∧ y). This map represents a generator of H0,k+k′(C2;M).
Using the suspension axioms we can easily generalize to the case where k and k′ are any integers
and the result follows.
The computation of the ring structure for a point is more complicated in general and uses that of G.
Theorem 3.4. There is a ring isomorphism,
H∗,∗(pt;R) ∼= R[v, v−1
],
36
where v ∈ H∗,∗(C2;R) has bidegree (0, 2).
Proof. Recall the exact sequence (3.4),
0→ H−1,q−1(pt;M)→ H0,q(S0,0;M)ϕ−→ H0,q(C2+ ∧ S0,0;M)→ H0,q−1(pt;M)→ H1,q(pt;M)→ 0.
From the analysis of table (3.1) we know that, for all integers n, H−1,2n−1(pt;R) = H0,2n−1(pt;R) = 0.
Therefore, the forgetful map is an isomorphism of abelian groups, ϕ : H0,2n(S0,0;R) → H0,2n(C2+ ∧
S0,0;R).
But ϕ : H∗,∗(S0,0;R) → H∗,∗(C2+ ∧ S0,0;R) is a ring map and so it is a ring isomorphism onto its
image. Therefore,
H0,2∗(pt;R) ∼= R[v, v−1
].
Since all the remaining structure is trivial the result follows.
This result can be obtained in a different way using the forgetful map more explicitly. We include it
here as an example.
Example 3.4. For the case M = R and X = pt, there are forgetful maps ϕ : H0,2q(S0,0)→ H0,2q(C2+ ∧
S0,0) for every integer q. Let v be a generator of H0,2(pt), that is, a non-zero real number. Then v2 is a
generator of H0,4(pt) since ϕ(v2) = ϕ(v) ^ ϕ(v) which is nonzero because ϕ(v) is nonzero by example
(3.2). In the same way, v−2 is a generator of H0,−4(pt) if v−1 generates H0,−2(pt) and so H∗,∗(pt) is a
finitely generated ring, which is commutative because its groups are located in even dimensions. Finally,
v ^ v−1 is also nonzero because ϕ(v ^ v−1) = ϕ(v) ^ ϕ(v−1) 6= 0.
Let X and Y be pointed G-CW-complexes and M a discrete abelian group. Define the maps,
ηX : X → Z⊗X σX,Y : (Z⊗X) ∧ (M ⊗ Y )→M ⊗ (X ∧ Y )
x 7→ 1x, (∑i zixi) ∧ (
∑jmjyj) 7→
∑i,j zimj(xi ∧ yj).
The correspondence V 7→ M ⊗ ΣVX, where V runs over an indexing set of a complete G-universe U ,
defines a G-prespectrum, whose structural maps are given by the compositions,
SW ∧ (M ⊗ ΣVX) (Z⊗ SW ) ∧ (M ⊗ ΣVX) M ⊗ ΣV+WX
x ∧ (∑imixi) 1x ∧ (
∑imixi)
∑imi(xi ∧ x)
ηSW ∧idM⊗ΣV XσSW ,ΣV X
The associated G-spectrum is denoted by M ⊗ Σ∞X except when X = S0, in which case we write
M ⊗ S.
In what follows we will also make use of the natural map, N : X ∧ F (W,Y ) → F (W,X ∧ Y ) defined
such that N(x ∧ ϕ)(w) = x ∧ ϕ(w).
Recall that HM is the Eilenberg-MacLane G-spectrum that represents reduced RO(G)-graded Bre-
don cohomology with coefficients in M . Let k > 0. We denote by Σ0,−kHM the shifted Eilenberg-
MacLane spectrum so that (Σ0,−kHM)p,q := HMp,q−k for any integers p and q such that q − k ≥ 0.
37
Lemma 3.5. Let M be a discrete ring. Let f : Sk,0 → M ⊗ Sk,k be a representative of an element α in
H0,k(pt;M) where k > 0 and and f its adjoint. Then, the map
h : Σ0,−kHM → HM
given for each representation Rp,q by the composition,
M⊗Sp,q−k id∧f−−−→M⊗Sp,q−k∧Ωk,0(M⊗Sk,k)N−→ Ωk,0(M⊗Sp,q−k∧M⊗Sk,k)
Ωk,0µ−−−−→ Ωk,0(M⊗Sp+k,q),
is a map of spectra and it represents multiplication by f in H∗,∗(pt;M).
Proof. Fix a C2-universe U . We have to prove that h commutes with the structure maps of each spectra,
that is, that the diagram,
ΣW−V (Σ0,−kHM)V (Σ0,−kHM)W
ΣW−VHMV HMW
ΣW−V hV
σuV,W
hW
σlV,W
commutes for every V ⊂W ⊂ U .
Let V = Rp′,q′ and W = Rp,q be such that q′ − k ≥ 0, p ≥ p′ and q ≥ q′ and let Σh = idSp−p′,q−q′ ∧ h.
The upper structure map is given by the composition
σuRp′,q′ ,Rp,q = σSp−p′,q−q′ ,Sp′,q′−k (ηSp−p′,q−q′ ∧ idM⊗Sp′,q′−k)
and the lower structure map is
σlRp′,q′ ,Rp,q = σSp−p′,q−q′ ,Sp′,q′ (ηSp−p′,q−q′ ∧ idM⊗Sp′+k,q′ ),
Let x∧∑imixi ∈ Sp−p
′,q−q′∧M⊗Sp′,q′−k. Then, σuRp′,q′ ,Rp,q (x∧∑imixi) =
∑imi(xi∧x) ∈M⊗Sp,q−k.
Let z ∈ Sk,0. Then,
hW
(∑i
mi(xi ∧ x)
)=(Ωk,0µ Ωk,0(id ∧ f)
)(z 7→
∑i
mi(xi ∧ x) ∧ z)
= Ωk,0µ(z 7→∑i
mi(xi ∧ x) ∧ f(z))
= Ωk,0µ(z 7→∑i
mi(xi ∧ x) ∧∑j
njzj) Setting f(z) =∑j
njzj .
= (z 7→∑i,j
minj(xi ∧ x ∧ zj))
38
On the other hand,
ΣW−V hV (x ∧∑i
mixi) = (idSp−p′,q−q′ ∧ hV )(x ∧∑i
mixi)
= x ∧ hV (∑i
mixi)
= x ∧ (z 7→∑i,j
minj(xi ∧ zj)).
Finally, applying σlRp′,q′ ,Rp,q to x ∧ (z 7→∑i,jminj(xi ∧ zj)), we get z 7→
∑i,jminj(xi ∧ x ∧ zj) and
the diagram commutes.
Theorem 3.6. The ring structure of H∗,∗(pt;Z) is completely determined by the following properties:
1. It is commutative.
Let y, ρ and α generate H0,2(pt), H1,1(pt) and H0,−2(pt) respectively. Then,
2. For any generator κ ∈ Hp,q(pt), κ ^ ρ generates Hp+1,q+1(pt) unless Hp,q(pt) is trivial.
3. Multiplication by y, y ^ − : H0,q−2(pt)→ H0,q(pt) is an isomorphism except when q = 0 or q = −1.
4. y ^ α = 2.
Proof. 1. The commutativity of H∗,∗(pt;Z) comes from the fact that the groups consist either of Z2’s
or are in even degrees. In this case the graded commutativity becomes commutativity.
2. The inclusion S1,1 → Z ⊗ S1,1 is a weak C2-equivalence. In fact, by the equivariant Dold-Thom
theorem we know the homotopy groups of Z ⊗ S1,1 and of its C2-fixed points and they agree with
those of S1,1. Let ρ be the inclusion S0,0 → S1,1 '−→ Z ⊗ S1,1 and consider the cofiber sequence
S0,0 → S1,1 → C2+ ∧ S1,0. We get the exact diagram,
H∗,∗(C2+ ∧ S1,0)
H∗,∗(S1,1) H∗,∗(S0,0)ρ∗
δ(3.5)
where δ : Hr,∗(S0,0) → Hr+1,∗(C2+ ∧ S1,0) is the connecting homomorphism in cohomology. It
can, thus, be seen that ρ∗ : Hp,q(S1,1) → Hp,q(pt) is an isomorphism whenever Hp−1,q−1(pt) and
Hp,q(pt) are both nontrivial and p 6= 1 and p 6= 0. We will see that, for the remaining cases,
multiplication by ρ may not be an isomorphism but is, at least, surjective and that is enough to
conclude that it takes generators to generators. In fact, a portion of the exact triangle (3.5) can be
translated into the exact sequence for every integer q,
0→ H0,q(S1,1)ρ∗−→ H0,q(S0,0)→ H1,q(C2+ ∧ S1,0)→ H1,q(S1,1)
ρ∗−→ H1,q(S0,0)→ 0.
Referring to table (3.2), we conclude that the maps ρ∗ : H−1,q−1(pt) → H0,q(pt) are isomor-
phisms for all q of the form q = −2r − 3, where r is a non-negative integer and that the maps
39
ρ∗ : H0,q−1(pt) → H1,q are surjective for all q of the form q = 2r + 1, where r is a non-negative
integer and the result follows. The map ρ∗ is multiplication by ρ in cohomology. In fact, it sends the
identity representative of H∗,∗(S1,1) in ρ and it is a map between free modules over the cohomology
of a point.
3. By example (3.2), ϕ : H0,2q(S0,0)→ H0,2q(C2+ ∧ S0,0) is an isomorphism. By proposition (3.3), yn
generates H0,2n(pt) for every n > 0.
For the negative cone we verify that H0,−n−2(pt)^y−−→ H0,−n(pt) is an isomorphism for n ≥ 2. The
following technique is used in [Dugger, 2005].
Let E be the G-spectrum defined by the cofiber sequence,
Σ0,−2HZ y^−−→ HZ→ E.
The element y can be represented by a class f : S2,0 → Z⊗ S2,2.
By lemma (3.5) we know that multiplication by y is a map of spectra. Thus, for every pointed
G-CW-complex X we have a long exact sequence,
· · · → Hp,q−2(X)→ Hp,q(X)→ Ep,q(X)→ Hp−1,q−2(X)→ · · · . (3.6)
Then, analyzing sequence (3.6) and table (3.2) we get En,0(pt) ∼= Z2 for n = 0 and 0 otherwise.
Also, En,0(C2) = 0 for every n. So we conclude that the Z-graded equivariant cohomology E∗,0
is the Z-graded cohomology theory corresponding to the Mackey functor E0,0(−). It follows that
En,0(X) = Hn(XC2 ;Z2). Therefore, for n > 0, E0,−n(pt) ∼= E0,−n(S0,0) ∼= En,0(Sn,n) ∼= Hn(S0) ∼=
0. In the same way, E1,−n(pt) is trivial and the isomorphism H0,−n−2(pt)^y−−→ H0,−n(pt) follows
for n ≥ 2.
4. From example (3.1), we know that the forgetful map ϕ : H0,−2(S0,0) → H0,−2(C2+ ∧ S0,0) is
multiplication by 2. Consider the composition,
ϕ (^ y) : H0,−2(S0,0)→ H0,0(C2+ ∧ S0,0)
α 7→ ϕ(α ^ y)
We have that ϕ is an isormophism in degree (0, 0) by example (3.2). On the other hand, given that
ϕ is multiplicative, we have that ϕ(α ^ y) = ϕ(α) ^ ϕ(y) is two times a generator by proposition
(3.3).
Theorem 3.7. The ring structure of H∗,∗(pt;Z2) is completely determined by the following properties:
1. It is commutative.
Let τ , ρ and θ be generators of H0,1(pt), H1,1(pt) and H0,−2 respectively. Then,
40
2. For any generator κ ∈ Hp,q(pt), κ ^ ρ generates Hp+1,q+1(pt), except when this group is trivial.
3. Multiplication by τ , τ ^ : H0,q−1(pt) → H0,q(pt) is an isomorphism except when q = 0 or q = −1
for dimensional reasons.
4. θ2 = 0.
Proof. 1. H∗,∗(pt;Z2) is commutative because it consists of Z2’s only.
2. Following the proof of point 2 of the last theorem, let ρ be the inclusion S0,0 → S1,1 → Z2 ⊗ S1,1.
Recall that we have a diagram in cohomology,
H∗,∗(C2+ ∧ S1,0)
H∗,∗(S1,1) H∗,∗(S0,0)ρ∗
δ(3.7)
where δ : Hr,∗(S0,0) → Hr+1,∗(C2+ ∧ S1,0) is the connecting homomorphism in cohomology. The
lower map ρ∗ assigns to S1,1 → Z2 ⊗ S1,1 the restriction S0,0 → S1,1 → Z2 ⊗ S1,1. This is a
map between free modules over the cohomology of a point, H∗,∗(pt), and, hence, it is multiplica-
tion by ρ. From the exactness of diagram (3.7), we conclude that ρ∗ : Hp−1,q−1(pt) → Hp,q(pt)
is an isomorphism whenever p 6= 1 and p 6= 0. The maps ρ∗ : H−1,q−1(pt) → H0,q(pt) and
ρ∗ : H0,q−1(pt) → H1,q(pt) are also isomorphisms in some cases. In fact, for every integer q
there is an exact sequence,
0→ H0,q(S1,1)ρ∗−→ H0,q(S0,0)→ H1,q(C2+ ∧ S1,0)→ H1,q(S1,1)
ρ∗−→ H1,q(S0,0)→ 0,
coming from exactness of diagram (3.7) as in theorem (3.6). Then, referring to table (3.3), we con-
clude that ρ∗ : H−1,q−1(pt)→ H0,q(pt) is an isomorphism whenever q ≤ −2 and ρ∗ : H0,q−1(pt)→
H1,q(pt) is an isomorphism whenever q ≥ 1.
3. The forgetful map gives a ring homomorphism ϕ : H∗,∗(S0,0) → H∗,∗(C2+ ∧ S0,0) ∼= Z2 in each
degree. Therefore, since ϕ(τ) is nonzero by example (3.3), ϕ(τ2) = ϕ(τ) ^ ϕ(τ) is nonzero.
So, by proposition (3.3), τ2 is nonzero and similarly for τn, for n > 2, that is, τn is a generator of
H0,n(pt). The multiplicative structure on the positive cone follows from this and the previous item.
To get the multiplicative structure on the negative cone we use the same approach as used in the
previous theorem. Let E be the C2-spectrum defined by the cofiber sequence,
Σ0,−1HZ2τ^−−→ HZ2 → E.
By lemma (3.5), the map τ ^ is a map of C2-spectra. Thus, for every pointed C2-CW-complex X
we get a long exact sequence,
· · · → Hp,q−1(X)→ Hp,q(X)→ Ep,q(X)→ Hp−1,q−1(X)→ · · · .
41
It is easy to see that En,0(pt) ∼= Z2 if n = 0 and is 0 otherwise, as well as En,0(C2) = 0 for every n.
So, just as in the last theorem we get that En,0(X) = Hn(XC2 ;Z2). Since E1,−n(pt) = E0,−n(pt) =
0 we have that multiplication by τ is an isomorphism H0,−n−1(pt)→ H0,−n(pt) whenever n ≥ 2.
4. Suppose θ2 is the generator of H0,−4(pt). We know that multiplication by τ2 takes generators to
generators and, hence, we would have θ2 ^ τ2 = θ. However, we arrive at a contradiction since
θ ^ τ = 0 by degree reasons.
42
Bibliography
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G. E. Bredon. Equivariant Cohomology Theories. Springer, 1967.
P. F. dos Santos. A note on the equivariant Dold-Thom theorem. J. Pure Appl. Algebra, 183(1-3):
299–312, 2003. ISSN 0022-4049.
D. Dugger. An Atiyah-Hirzebruch spectral sequence for KR-theory. K-theory, 2005.
H. B. Lawson, P. L.-F. Jr., and M.-L. Michelsohn. Algebraic cycles and the classical groups part I, real
cycles. Topology, 2003.
P. Lima-Filho. On The Equivariant Homotopy Of Free Abelian Groups On G-Spaces And G-Spectra.
Mathematische Zeitschrift, 1997.
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J. P. May. Equivariant Homotopy and Cohomology Theory, volume 91. CBMS, 2nd edition, 1996.
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43
44
Index
G-
CW-complex, 12
homeomorphism, 4
homotopy, 10
based, 11
map, 4
space, 4
based, 4
coinduced, 8
induced, 8
universe, 21
complete, 21
trivial, 21
action, 4
free, 5
trivial, 5
averaged cycle, 18
cellular chain, 15
coefficient system, 14
cohomology
RO(G)-graded, 25
Z-graded, 15
reduced
RO(G)-graded, 25
Z-graded, 18
cup product, 26
equivariant
n-cell, 12
Dold-Thom Theorem, 27
Eilenberg-MacLane space, 27
homotopy group, 11
Spanier-Whitehead Duality, 28
fixed point set, 7
functor
adjoint, 9
fixed point, 7
forgetful, 9
Mackey, 22
Burnside ring, 24
constant, 23
homology
RO(G)-graded, 25
Z-graded, 16
reduced
RO(G)-graded, 25
Z-graded, 18
invariant subspace, 6
map
folding, 34
forgetful, 34
restriction, 22
transfer, 22
orbit
category, 13
space, 5
prespectrum, 20
G-, 21
category, 20
Eilenberg-MacLane, 21
map, 20, 22
sphere, 21
suspension, 21, 22
45
representation, 5
completely reducible, 6
dimension, 6
faithful, 6
irreducible, 6
isomorphic, 6
orthogonal, 6
sphere, 7
smash product, 4
spectrum, 20
G-, 22
category, 20
genuine, 22
map, 22
naive, 22
subrepresentation, 6
proper, 6
topological dimension, 7
weak G-equivalence, 11
G-spectra, 22
weight, 7
Weyl group, 7
46