equivariant homotopy and cohomology theory

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Jean Lannes
Jean Lannes

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J.P. May
The author acknowledges the support of the NSF
1980 Mathematics Subject Classication (1985 Revision ). Primary 19L47, 55M35,
55N10, 55N15, 55N20, 55N22, 55N91, 55P42, 55P60, 55P62, 55P91, 55Q91, 55R12,
55R91, 55T25, 57R85; Secondary 18E30, 18G15, 18G40, 19A22, 55P20, 55P25,
55P47, 55Q10, 55Q55, 55R40, 55R50, 55S10, 55S35, 55S45, 55U20, 55U25, 57R75,
57R77, 57S17.
Author addresses:
University of Chicago, Chicago, Il 60637
E-mail address : [email protected]
Chapter I. Equivariant Cellular and Homology Theory
1. Some basic denitions and adjunctions
2. Analogs for based G-spaces
3. G-CW complexes
4. Ordinary homology and cohomology theories
5. Obstruction theory
6. Universal coecient spectral sequences
Chapter II. Postnikov Systems, Localization, and Completion
1. Eilenberg-MacLane G-spaces and Postnikov systems
2. Summary: localizations of spaces
3. Localizations of G-spaces
4. Summary: completions of spaces
5. Completions of G-spaces
Chapter III. Equivariant Rational Homotopy Theory
1. Summary: the theory of minimal models
2. Equivariant minimal models
3. Rational equivariant Hopf spaces
Chapter IV. Smith Theory
1. Smith theory via Bredon cohomology
2. Borel cohomology, localization, and Smith theory
Chapter V. Categorical Constructions; Equivariant Applications 47
1. Coends and geometric realization
2. Homotopy colimits and limits
3. Elmendorf's theorem on diagrams of xed point spaces
4. Eilenberg-MacLane G-spaces and universal F -spaces
Chapter VI. The Homotopy Theory of Diagrams
1. Elementary homotopy theory of diagrams
2. Homotopy Groups
3. Cellular Theory
4. The homology and cohomology theory of diagrams
5. The closed model structure on U J
6. Another proof of Elmendorf's theorem
Chapter VII. Equivariant Bundle theory and Classifying Spaces 71
1. The denition of equivariant bundles
2. The classication of equivariant bundles
3. Some examples of classifying spaces
Chapter VIII. The Sullivan Conjecture
1. Statements of versions of the Sullivan conjecture
2. Algebraic preliminaries: Lannes' functors T and Fix
3. Lannes' generalization of the Sullivan conjecture
4. Sketch proof of Lannes' theorem
5. Maps between classifying spaces
Chapter IX. An introduction to equivariant stable homotopy
1. G-spheres in homotopy theory
2. G-Universes and stable G-maps
3. Euler characteristic and transfer G-maps
4. Mackey functors and coMackey functors
5. RO(G)-graded homology and cohomology
6. The Conner conjecture
Chapter X. G-CW(V ) complexes and RO(G)-graded cohomology
1. Motivation for cellular theories based on representations
2. G-CW(V ) complexes
3. Homotopy theory of G-CW(V ) complexes
4. Ordinary RO(G)-graded homology and cohomology
Chapter XI. The equivariant Hurewicz and Suspension Theorems
1. Background on the classical theorems
2. Formulation of the problem and counterexamples
3. An oversimplied description of the results
4. The statements of the theorems
5. Sketch proofs of the theorems
Chapter XII. The Equivariant Stable Homotopy Category
1. An introductory overview
2. Prespectra and spectra
3. Smash products
4. Function spectra
5. The equivariant case
6. Spheres and homotopy groups
7. G-CW spectra
8. Stability of the stable category
9. Getting into the stable category
Chapter XIII. RO(G)-graded homology and cohomology theories
1. Axioms for RO(G)-graded cohomology theories
2. Representing RO(G)-graded theories by G-spectra
3. Brown's theorem and RO(G)-graded cohomology
4. Equivariant Eilenberg-MacLane spectra
5. Ring G-spectra and products
Chapter XIV. An introduction to equivariant K -theory
1. The denition and basic properties of KG -theory
2. Bundles over a point: the representation ring
3. Equivariant Bott periodicity
4. Equivariant K -theory spectra
5. The Atiyah-Segal completion theorem
6. The generalization to families
Chapter XV. An introduction to equivariant cobordism
1. A review of nonequivariant cobordism
2. Equivariant cobordism and Thom spectra
3. Computations: the use of families
4. Special cases: odd order groups and Z=2
Chapter XVI. Spectra and G-spectra; change of groups; duality
1. Fixed point spectra and orbit spectra
2. Split G-spectra and free G-spectra
3. Geometric xed point spectra
4. Change of groups and the Wirthmuller isomorphism
5. Quotient groups and the Adams isomorphism
6. The construction of G=N -spectra from G-spectra
7. Spanier-Whitehead duality
8. V -duality of G-spaces and Atiyah duality
9. Poincare duality
Chapter XVII. The Burnside ring
1. Generalized Euler characteristics and transfer maps
2. The Burnside ring A(G) and the zero stem 0G(S )
3. Prime ideals of the Burnside ring
4. Idempotent elements of the Burnside ring
5. Localizations of the Burnside ring
6. Localization of equivariant homology and cohomology
Chapter XVIII. Transfer maps in equivariant bundle theory
1. The transfer and a dimension-shifting variant
2. Basic properties of transfer maps
3. Smash products and Euler characteristics
4. The double coset formula and its applications
5. Transitivity of the transfer
Chapter XIX. Stable homotopy and Mackey functors
1. The splitting of equivariant stable homotopy groups
2. Generalizations of the splitting theorems
3. Equivalent denitions of Mackey functors
4. Induction theorems
5. Splittings of rational G-spectra for nite groups G
Chapter XX. The Segal conjecture
1. The statement in terms of completions of G-spectra
2. A calculational reformulation
3. A generalization and the reduction to p-groups
4. The proof of the Segal conjecture for nite p-groups
5. Approximations of singular subspaces of G-spaces
6. An inverse limit of Adams spectral sequences
7. Further generalizations; maps between classifying spaces
Chapter XXI. Generalized Tate cohomology
1. Denitions and basic properties
2. Ordinary theories; Atiyah-Hirzebruch spectral sequences
3. Cohomotopy, periodicity, and root invariants
4. The generalization to families
5. Equivariant K -theory
6. Further calculations and applications
Chapter XXII. Brave new algebra
1. The category of S -modules
2. Categories of R-modules
3. The algebraic theory of R-modules
4. The homotopical theory of R-modules
5. Categories of R-algebras
6. Bouseld localizations of R-modules and algebras
7. Topological Hochschild homology and cohomology
Chapter XXIII. Brave new equivariant foundations
1. Twisted half-smash products
2. The category of L-spectra
3. A1 and E1 ring spectra and S -algebras
4. Alternative perspectives on equivariance
5. The construction of equivariant algebras and modules
6. Comparisons of categories of L-G-spectra
Chapter XXIV. Brave New Equivariant Algebra
1. Introduction
2. Local and C ech cohomology in algebra
3. Brave new versions of local and C ech cohomology
4. Localization theorems in equivariant homology
5. Completions, completion theorems, and local homology
6. A proof and generalization of the localization theorem
7. The application to K -theory
8. Local Tate cohomology
Chapter XXV. Localization and completion in complex bordism
1. The localization theorem for stable complex bordism
2. An outline of the proof
3. The norm map and its properties
4. The idea behind the construction of norm maps
5. Global I-functors with smash product
6. The denition of the norm map
7. The splitting of MUG as an algebra
8. Loer's completion conjecture
Chapter XXVI. Some calculations in complex equivariant bordism387
1. Notations and terminology
2. Stably almost complex structures and bordism
3. Tangential structures
4. Calculational tools
5. Statements of the main results
6. Preliminary lemmas and families in G S 1
7. On the families Fi in G S 1
8. Passing from G to G S 1 and G Zk
This volume began with Bob Piacenza's suggestion that I be the principal lecturer
at an NSF/CBMS Regional Conference in Fairbanks, Alaska. That event took
place in August of 1993, and the interim has seen very substantial progress in this
general area of mathematics. The scope of this volume has grown accordingly.
The original focus was an introduction to equivariant algebraic topology, to stable homotopy theory, and to equivariant stable homotopy theory that was geared
towards graduate students with a reasonably good understanding of nonequivariant algebraic topology. More recent material is changing the direction of the last
two subjects by allowing the introduction of point-set topological algebra into stable homotopy theory, both equivariant and non-equivariant, and the last portion
of the book focuses on an introduction to these new developments. There is a
progression, with the later portions of the book on the whole being more dicult
than the earlier portions.
Equivariant algebraic topology concerns the study of algebraic invariants of
spaces with group actions. The rst two chapters introduce the basic structural
foundations of the subject: cellular theory, ordinary homology and cohomology
theory, Eilenberg-Mac Lane G-spaces, Postnikov systems, localizations of G-spaces
and completions of G-spaces. In most of this work, G can be any topological group,
but we restrict attention to compact Lie groups in the rest of the book.
Chapter III, on equivariant rational homotopy theory, was written by Georgia
Triantallou. In it, she shows how to generalize Sullivan's theory of minimal
models to obtain an algebraization of the homotopy category of (nilpotent) Gspaces for a nite group G. This chapter contains a rst surprise: rational Hopf
G-spaces need not split as products of Eilenberg-Mac Lane G-spaces. This is a hint
that the calculational behavior of equivariant algebraic topology is more intricate
and dicult to determine than that of the classical nonequivariant theory.
Chapter IV gives two proofs of the rst main theorem of equivariant algebraic
topology, which goes under the name of \Smith theory": any xed point space
of an action of a nite p-group on a mod p homology sphere is again a mod p
homology sphere. One proof uses ordinary (or Bredon) equivariant cohomology
and the other uses a general localization theorem in classical (or Borel) equivariant
Parts of equivariant theory require a good deal of categorical bookkeeping, for
example to keep track of xed point data and to construct new G-spaces from
diagrams of potential xed point spaces. Some of the relevant background, such
as geometric realization of simplicial spaces and the construction of homotopy
colimits, is central to all of algebraic topology. These matters are dealt with
in Chapter V, where Eilenberg-Mac Lane G-spaces and universal F -spaces for
families F of subgroups of a given group G are constructed. Special cases of
such universal F -spaces are used in Chapter VII to study the classication of
equivariant bundles.
A dierent perspective on these matters is given in Chapter VI, which was written by Bob Piacenza. It deals with the general theory of diagrams of topological
spaces, showing how to mimic classical homotopy and homology theory in categories of diagrams of topological spaces. In particular, Piacenza constructs a
Quillen (closed) model category structure on any such category of diagrams and
shows how these ideas lead to another way of passing from diagrams of xed point
spaces to their homotopical realization by G-spaces.
Chapter VIII combines equivariant ideas with the use of new tools in nonequivariant algebraic topology, notably Lannes' functor T in the context of unstable
modules and algebras over the Steenrod algebra, to describe one of the most beautiful recent developments in algebraic topology, namely the Sullivan conjecture
and its applications. While many mathematicians have contributed to this area,
the main theorems are due to Haynes Miller, Gunnar Carlsson, and Jean Lannes.
Although the set [X; Y ] of homotopy classes of based maps from a space X to a
space Y is trivial to dene, it is usually enormously dicult to compute. The Sullivan conjecture, in its simplest form, asserts that [BG; X ] = 0 if G is a nite group
and X is a nite CW complex. It admits substantial generalizations which lead
to much more interesting calculations, for example of the set of maps [BG, BH]
for suitable compact Lie groups G and H . We shall see that an understanding
of equivariant classifying spaces sheds light on what these calculations are really
saying. There is already a large literature in this area, and we can only give an introduction. One theme is that the Sullivan conjecture can be viewed conceptually
as a calculational elaboration of Smith theory. A starting point of this approach
lies in work of Bill Dwyer and Clarence Wilkerson, which rst exploited the study
of modules over the Steenrod algebra in the context of the localization theorem in
Smith theory.
We begin the study of equivariant stable homotopy theory in Chapter IX, which
gives a brief introduction of some of the main ideas. The chapter culminates with
a quick conceptual proof of a conjecture of Conner: if G is a compact Lie group
and X is a nite dimensional G-CW complex with nitely many orbit types such
that H~ (X ; Z) = 0, then H~ (X=G; Z) = 0. This concrete statement is a direct consequence of the seemingly esoteric assertion that ordinary equivariant cohomology
with coecients in a Mackey functor extends to a cohomology theory graded on
the real representation ring RO(G); this means that there are suspension isomorphisms with respect to the based spheres associated to all representations, not just
trivial ones. In fact, the interplay between homotopy theory and representation
theory pervades equivariant stable homotopy theory.
One manifestation of this appears in Chapter X, which was written by Stefan
Waner. It explains a variant theory of G-CW complexes dened in terms of representations and uses the theory to construct the required ordinary RO(G)-graded
cohomology theories with coecients in Mackey functors by means of appropriate
cellular cochain complexes.
Another manifestation appears in Chapter XI, which was written by Gaunce
Lewis and which explains equivariant versions of the Hurewicz and Freudenthal
suspension theorems. The algebraic transition from unstable to stable phenomena is gradual rather than all at once. Nonequivariantly, the homotopy groups
of rst loop spaces are already Abelian groups, as are stable homotopy groups.
Equivariantly, stable homotopy groups are modules over the Burnside ring, but
the homotopy groups of V th loop spaces for a representation V are only modules over a partial Burnside ring determined by V . The precise form of Lewis's
equivariant suspension theorem reects this algebraic fact.
Serious work in both equivariant and nonequivariant stable homotopy theory
requires a good category of \stable spaces", called spectra, in which to work.
There is a great deal of literature on this subject. The original construction of the
nonequivariant stable homotopy category was due to Mike Boardman. One must
make a sharp distinction between the stable homotopy category, which is xed
and unique up to equivalence, and any particular point-set level construction of
it. In fact, there are quite a few constructions in the literature. However, only one
of them is known to generalize to the equivariant context, and that is also the one
that is the basis for the new development of point-set topological algebra in stable
homotopy theory. We give an intuitive introduction to this category in Chapter
XII, beginning nonequivariantly and focusing on the construction of smash products and function spectra since that is the main technical issue. We switch to
the equivariant case to explain homotopy groups, the suspension isomorphism for
representation spheres, and the theory of G-CW spectra. We also explain how to
transform the spectra that occur \in nature" to the idealized spectra that are the
objects of the stable homotopy category.
In Chapters XIII, XIV, and XV, we introduce the most important RO(G)graded cohomology theories and describe the G-spectra that represent them. We
begin with an axiomatic account of exactly what RO(G)-graded homology and
cohomology theories are and a proof that all such theories are representable by Gspectra. We also discuss ring G-spectra and products in homology and cohomology
theories. We show how to construct Eilenberg-Mac Lane G-spectra by representing
the zeroth term of a Z-graded cohomology theory dened by means of G-spectrum
level cochains. This implies an alternative construction of ordinary RO(G)-graded
cohomology theories with coecients in Mackey functors.
Chapter XIV, which was written by John Greenlees, gives an introduction to
equivariant K -theory. The focus is on equivariant Bott periodicity and its use to
prove the Atiyah-Segal completion theorem. That theorem states that, for any
compact Lie group G, the nonequivariant K -theory of the classifying space BG is
isomorphic to the completion of the representation ring R(G) at its augmentation
ideal I . The result is of considerable importance in the applications of K -theory,
and it is the prototype for a number of analogous results to be described later.
Chapter XV, which was written by Steve Costenoble, gives an introduction to
equivariant cobordism. The essential new feature is that transversality fails in general, so that geometric equivariant bordism is not same as stable (or homotopical)
bordism; the latter is the theory represented by the most natural equivariant generalization of the nonequivariant Thom spectrum. Costenoble also explains the
use of adjacent families of subgroups to reduce the calculation of equivariant bordism to suitably related nonequivariant calculations. The equivariant results are
considerably more intricate than the nonequivariant ones. While the G-spectra
that represent unoriented geometric bordism and its stable analog split as products of Eilenberg Mac Lane G-spectra for nite groups of odd order, just as in the
nonequivariant case, this is false for the cyclic group of order 2.
Chapters XVI{XIX describe the basic machinery and results on which all work
in equivariant stable homotopy theory depends. Chapter XVI describes xed point
and orbit spectra, shows how to relate equivariant and nonequivariant homology
and cohomology theories, and, more generally, shows how to relate homology and
cohomology theories dened for a group G to homology and cohomology theories
dened for subgroups and quotient groups of G. These results about change of
groups are closely related to duality theory, and we give basic information about
equivariant Spanier-Whitehead, Atiyah, and Poincare duality.
In Chapter XVII, we discuss the Burnside ring A(G). When G is nite, A(G) is
the Grothendieck ring associated to the semi-ring of nite G-sets. For any compact
Lie group G, A(G) is isomorphic to the zeroth equivariant stable homotopy group
of spheres. It therefore acts on the equivariant homotopy groups nG(X ) = n(X G )
of any G-spectrum X , and this implies that it acts on all homology and cohomology
groups of any G-spectrum. Information about the algebraic structure of A(G)
leads to information about the entire stable homotopy category of G-spectra. It
turns out that A(G) has Krull dimension one and an easily analyzed prime ideal
spectrum, making it quite a tractable ring. Algebraic analysis of localizations of
A(G) leads to analysis of localizations of equivariant homology and cohomology
theories. For example, for a nite group G, the localization of any theory at a
prime p can be calculated in terms of subquotient p-groups of G.
In Chapter XVIII, we construct transfer maps, which are basic calculational
tools in equivariant and nonequivariant bundle theory, and describe their basic
properties. Special cases were vital to the earlier discussion of change of groups.
The deepest property is the double coset formula, and we say a little about its
applications to the study of the cohomology of classifying spaces.
In Chapter XIX, we discuss several fundamental splitting theorems in equivariant stable homotopy theory. These describe the equivariant stable homotopy
groups of G-spaces in terms of nonequivariant homotopy groups of xed point
spaces. These theorems lead to an analysis of the structure of the subcategory
of the stable category whose objects are the suspension spectra of orbit spaces.
A Mackey functor is an additive contravariant functor from this subcategory to
Abelian groups, and, when G is nite, the analysis leads to a proof that this
topological denition of Mackey functors is equivalent to an earlier and simpler algebraic denition. Mackey functors describe the algebraic structure that is present
on the system of homotopy groups nH (X ) = n(X H ) of a G-spectrum X , where
H runs over the subgroups of G. The action of the Burnside ring on nG(X ) is
part of this structure. It is often more natural to study such systems than to focus
on the individual groups. In particular, we describe algebraic induction theorems
that often allow one to calculate the value of a Mackey functor on the orbit G=G
from its values on the orbits G=H for certain subgroups H . Such theorems have
applications in various branches of mathematics in which nite group actions appear. Again, algebraic analysis of rational Mackey functors shows that, when G is
nite, rational G-spectra split as products of Eilenberg-Mac Lane G-spectra. This
is false for general compact Lie groups G.
In Chapter XX, we turn to another of the most beautiful recent developments
in algebraic topology: the Segal conjecture and its applications. The Segal conjecture can be viewed either as a stable analogue of the Sullivan conjecture or
as the analogue in equivariant stable cohomotopy of the Atiyah-Segal completion
theorem in equivariant K -theory. The original conjecture, which is just a fragment
of the full result, asserts that, for a nite group G, the zeroth stable cohomotopy
group of the classifying space BG is isomorphic to the completion of A(G) at its
augmentation ideal I . The key step in the proof of the Segal conjecture is due
to Gunnar Carlsson. We explain the proof and also explain a number of generalizations of the result. One of these leads to a complete algebraic determination
of the group of homotopy classes of stable maps between the classifying spaces
of any two nite groups. This is analogous to the role of the Sullivan conjecture
in the study of ordinary homotopy classes of maps between classifying spaces.
Use of equivariant classifying spaces is much more essential here. In fact, the Segal conjecture is intrinsically a result about the I -adic completion of the sphere
G-spectrum, and the application to maps between classifying spaces depends on
a generalization in which the sphere G-spectrum is replaced by the suspension
G-spectra of equivariant classifying spaces.
Chapter XXI is an exposition of joint work of John Greenlees and myself in which
we generalize the classical Tate cohomology of nite groups and the periodic cyclic
cohomology of the circle group to obtain a Tate cohomology theory associated to
any given cohomology theory on G-spectra, for any compact Lie group G. This
work has had a variety of applications, most strikingly to the computation of the
topological cyclic homology and thus to the algebraic K -theory of number rings.
While we shall not get into that application here, we shall describe the general
Atiyah-Hirzebruch-Tate spectral sequences that are used in that work and we shall
give a number of other applications and calculations. For example, we shall explain
a complete calculation of the Tate theory associated to the equivariant K -theory
of any nite group. This is an active area of research, and some of what we say
at the end of this chapter is rather speculative. The Tate theory provides some of
the most striking examples of equivariant phenomena illuminating nonequivariant
phenomena, and it leads to interrelationships between the stable homotopy groups
of spheres and the Tate cohomology of nite groups that have only begun to be
Chapters XXII through XXV concern \brave new algebra", the study of pointset level topological algebra in stable homotopy theory. The desirability of such
a theory was advertised by Waldhausen under the rubric of \brave new rings",
hence the term \brave new algebra" for the new subject. Its starting point is the
construction of a new category of spectra, the category of \S -modules", that has a
smash product that is symmetric monoidal (associative, commutative, and unital
up to coherent natural isomorphisms) on the point-set level. The construction is
joint work of Tony Elmendorf, Igor Kriz, Mike Mandell, and myself, and it changes
the nature of stable homotopy theory. Ever since its beginnings with Adams' use
of stable homotopy theory to solve the Hopf invariant one problem some thirtyve years ago, most work in the eld has been carried out working only \up to
homotopy"; formally, this means that one is working in the stable homotopy category. For example, classically, the product on a ring spectrum is dened only up
to homotopy and can be expected to be associative and commutative only up to
homotopy. In the new theory, we have rings with well-dened point-set level products, and they can be expected to be strictly associative and commutative. In the
associative case, we call these \S -algebras". The new theory permits constructions
that have long been desired, but that have seemed to be out of reach technically:
simple constructions of many of the most basic spectra in current use in algebraic
topology; simple constructions of generalized universal coecient, Kunneth, and
other spectral sequences; a conceptual and structured approach to Bouseld local-
izations of spectra, a generalized construction of topological Hochschild homology
and of spectral sequences for its computation; a simultaneous generalization of the
algebraic K -theory of rings and of spaces; etc. Working nonequivariantly, we shall
describe the properties of the category of S -modules and shall sketch all but the
last of the cited applications in Chapter XXII.
We return to the equivariant world in Chapter XXIII, which was written jointly
with Elmendorf and Lewis, and sketch how the construction of the category of
SG -modules works. Here SG denotes the sphere G-spectrum. The starting point
of the construction is the \twisted half smash product", which is a spectrum level
generalization of the half-smash product X n Y = X+ ^ Y of an unbased Gspace X and a based G-space Y and is perhaps the most basic construction in
equivariant stable homotopy theory. Taking X to be a certain G-space L (j ) of
linear isometries, one obtains a fattened version L (j ) n E1 ^ ^ Ej of the j fold smash product of G-spectra. Taking j = 2, insisting that the Ei have extra
structure given by maps L (1)nEi ?! Ei, and quotienting out some of the fat, one
obtains a commutative and associative smash product of G-spectra with actions
by the monoid L (1); a little adjustment adds in the unit condition and gives the
category of SG-modules. The theory had its origins in the notion of an E1 ring
spectrum introduced by Quinn, Ray, and myself over twenty years ago. Such rings
were dened in terms of \operad actions" given by maps L (j ) n E j ?! E , where
E j is the j -fold smash power of E , and it turns out that such rings are virtually
the same as our new commutative SG -algebras. The new theory makes the earlier
notion much more algebraically tractable, while the older theory gives the basic
examples to which the new theory can be applied.
In Chapter XXIV, which was written jointly with Greenlees, we give a series of
algebraic denitions, together with their brave new algebra counterparts, and we
show how these notions lead to a general approach to localization and completion
theorems in equivariant stable homotopy theory. We shall see that Grothendieck's
local cohomology groups are relevant to the study of localization theorems in
equivariant homology and that analogs called local homology groups are relevant
to the study of completion theorems in equivariant cohomology. We use these
constructions to prove a general localization theorem for suitable commutative
SG -algebras RG . Taking R to be the underlying S -algebra of RG and taking
M to be the underlying R-module of an RG -module MG , the theorem implies
both a localization theorem for the computation of M(BG) in terms of MG (pt)
and a completion theorem for the computation of M (BG) in terms of MG (pt).
Of course, this is reminiscent of the Atiyah-Segal completion for equivariant K theory and the Segal conjecture for equivariant cobordism. The general theorem
does apply to K -theory, giving a very clean description of K(BG), but it does
not apply to cohomotopy: there the completion theorem for cohomology is true
but the localization theorem for homology is false.
We are particularly interested in stable equivariant complex bordism, represented by MUG , and modules over it. We explain in Chapter XXII how simple it
is to construct all of the usual examples of MU -module spectra in the homotopical
sense, such as Morava K -theory and Brown-Peterson spectra, as brave new pointset level MU -modules. We show in Chapter XXIII how to construct equivariant
versions MG as brave new MUG -modules of all such MU -modules M , where G
is any compact Lie group. We would like to apply the localization theorem of
Chapter XXIV to MUG and its module spectra, but its algebraic hypotheses are
not satised. Nevertheless, as Greenlees and I explain in chapter XXV, the localization theorem is in fact true for MUG when G is nite or a nite extension
of a torus. The proof involves the construction of a multiplicative norm map in
MUG , together with a double coset formula for its computation. This depends on
the fact that MUG can be constructed in a particularly nice way, codied in the
notion of a \global I-functor with smash product", as a functor of G.
These results refocus attention on stable equivariant complex bordism, whose
study lapsed in the early 1970's. In fact, some of the most signicant calculational
results obtained then were never fully documented in the literature. In Chapter
XXVI, which was written by Gustavo Comezana, new and complete proofs of these
results are presented, along with results on the relationship between geometric
and stable equivariant complex cobordism. In particular, when G is a compact
Abelian Lie group, Comezana proves that MUG is a free MU -module on even
degree generators.
In Chapter XXVI, and in a few places earlier on, complete proofs are given
either because we feel that the material is inadequately treated in the published
literature or because we have added new material. However, most of the material
in the book is known and has been treated in full detail elsewhere. Our goal
has been to present what is known in a form that is more readily accessible and
assimilable, with emphasis on the main ideas and the structure of the theory and
with pointers to where full details and further developments can be found.
Most sections have their own brief bibliographies at the end; thus, if an author's
work is referred to in a section, the appropriate reference is given at the end of that
section. There is also a general bibliography but, since it has over 200 items, I felt
that easily found local references would be more helpful. With a few exceptions,
the general bibliography is restricted to items actually referred to in the text,
and it makes no claim to completeness. A full list of relevant and interesting
papers would easily double the number of entries. I oer my apologies to authors
not cited who should have been. Inevitably, the choice of topics and of material
within topics has had to be very selective and idiosyncratic.
There are some general references that should be cited here (reminders of their
abbreviated names will be given where they are rst used). Starting with Chapter
XII, references to [LMS] (= [133]) are to
L.G. Lewis, J.P. May, and M. Steinberger (with contributions by J.E. McClure). Equivariant
stable homotopy theory. Springer Lecture Notes in Mathematics Vol. 1213. 1986.
Most of the material in Chapter XII and in the ve chapters XV{XIX is based
on joint work of Gaunce Lewis and myself that is presented in perhaps excruciating
detail in that rather encyclopedic volume. There are also abbreviated references
in force in particular chapters: [L1]{[L3] = [128, 129, 130] in Chapter XI and [tD]
= [55] in Chapter XVII.
The basic reference for the proofs of the claims in Chapters XXII and XXIII is
[EKMM] A. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, modules, and algebras in
stable homotopy theory. Preprint, 1995.
We shall also refer to the connected sequence of expository papers [73, 88, 89]
[EKMM ] A. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Modern foundations for stable
homotopy theory.
[GM1] J. P. C. Greenlees and J. P. May. Completions in algebra and topology.
[GM2] J. P. C. Greenlees and J. P. May. Equivariant stable homotopy theory.
These are all in the \Handbook of Algebraic Topology", edited by Ioan James,
that came out in 1995. While these have considerable overlap with Chapters XXII
through XXIV, we have varied the perspective and emphasis, and each exposition
includes a good deal of material that is not discussed in the other. In particular,
we point to the application of brave new algebra to chromatic periodicity in [GM1]:
the ideas there have yet to be fully exploited and are not discussed here.
In view of the broad and disparate range of topics, we have tried very hard to
make the chapters, and often even the sections, independent of one another. We
have also broken the material into short and hopefully manageable chunks; only
a few sections are as long as ve pages, and all chapters are less than twenty-ve
pages long. Very few readers are likely to wish to read straight through, and the
reader should be unafraid to jump directly to what he or she nds of interest.
The reader should also be unintimidated by nding that he or she has insucient
background to feel comfortable with particular sections or chapters. Unfortunately,
the subject of algebraic topology is particularly badly served by its textbooks. For
example, none of them even mentions localizations and completions of spaces,
although those have been standard tools since the early 1970's. We have tried to
include enough background to give the basic ideas. Modern algebraic topology is
a thriving subject, and perhaps jumping right in and having a look at some of its
more recent directions may give a better perspective than trying to start at the
beginning and work one's way up.
As the reader will have gathered, this book is a cooperative enterprise. Perhaps
this is the right place to try to express just how enormously grateful I am to all
of my friends, collaborators, and students. This book owes everything to our joint
eorts over many years. When planning the Alaska conference, I invited some of
my friends and collaborators to give talks that would mesh with mine and help
give a reasonably coherent overview of the subject. Most of the speakers wrote up
their talks and gave me license to edit them to t into the framework of the book.
Since TeX is refractory about listing authors inside a Table of Contents, I will here
list those chapters that are written either solely by other authors or jointly with me.
Chapter III. Equivariant rational homotopy theory
by Georgia Triantallou
Chapter VI. The homotopy theory of diagrams
by Robert Piacenza
Chapter X. G-CW(V ) complexes and RO(G)-graded cohomology
by Stefan Waner
Chapter XI. The equivariant Hurewicz and suspension theorems
by L. G. Lewis Jr.
Chapter XIV. An introduction to equivariant K -theory
by J. P. C. Greenlees
Chapter XV. An introduction to equivariant cobordism
by Steven Costenoble
Chapter XXI. Generalized Tate cohomology
by J. P. C. Greenlees and J. P. May
Chapter XXIII. Brave new equivariant foundations
by A. D. Elmendorf, L. G. Lewis Jr., and J. P. May
Chapter XXIV. Brave new equivariant algebra
by J. P. C. Greenlees and J. P. May
Chapter XXV. Localization and completion in complex cobordism
by J. P. C. Greenlees and J. P. May
Chapter XXVI. Some calculations in complex equivariant bordism
by Gustavo Costenoble
My deepest thanks to these people and to Stefan Jackowski and Chun-Nip Lee,
who also gave talks; their topics were the subjects of their recent excellent survey
papers [107] and [125] and were therefore not written up for inclusion here. I would
also like to thank Jim McClure, whose many insights in this area are reected
throughout the book, and Igor Kriz, whose collaboration over the last six years has
greatly inuenced the more recent material. I would also like to thank my current
students at Chicago | Maria Basterra, Mike Cole, Dan Isaksen, Mike Mandell,
Adam Przezdziecki, Laura Scull, and Jerome Wolbert | who have helped catch
many soft spots of exposition and have already made signicant contributions to
this general area of mathematics.
It is an especial pleasure to thank Bob Piacenza and his wife Lyric Ozburn for
organizing the Alaska conference and making it a memorably pleasant occasion for
all concerned. Thanks to their thoughtful arrangements, the intense all day mathematical activity took place in a wonderfully convivial and congenial atmosphere.
Finally, my thanks to all of those who attended the conference and helped make
the week such a pleasant mathematical occasion: thanks for bearing with me.
J. Peter May
December 31, 1995
Equivariant Cellular and Homology Theory
1. Some basic denitions and adjunctions
The objects of study in equivariant algebraic topology are spaces equipped with
an action by a topological group G. That is, the subject concerns spaces X together with continuous actions G X ?! X such that ex = x and g(g0x) = (gg0)x.
Maps f : X ?! Y are equivariant if f (gx) = gf (x). We then say that f is a Gmap. The usual constructions on spaces apply equally well in the category GU of
G-spaces and G-maps. In particular G acts diagonally on Cartesian products of
spaces and acts by conjugation on the space Map(X; Y ) of maps from X to Y .
That is, we dene g f by (g f )(x) = gf (g?1x).
As usual, we take all spaces to be compactly generated (which means that
a subspace is closed if its intersection with each compact Hausdor subspace is
closed) and weak Hausdor (which means that the diagonal X X X is a closed
subset, where the product is given the compactly generated topology). Among
other things, this ensures that we have a G-homeomorphism
Map(X Y; Z ) (1.1)
= Map(X; Map(Y; Z ))
for any G-spaces X , Y , and Z .
For us, subgroups of G are assumed to be closed. For H G, we write X H =
fxjhx = x for h 2 H g. For x 2 X , Gx = fhjhx = xg is called the isotropy group
of x. Thus x 2 X H if H is contained in Gx . A good deal of the formal homotopy
theory of G-spaces reduces to the ordinary homotopy theory of xed point spaces.
We let NH be the normalizer of H in G and let WH = NH=H . (We sometimes
write NGH and WGH .) These \Weyl groups" appear ubiquitously in the theory.
Note that X H is a WH -space. In equivariant theory, orbits G=H play the role of
points, and the set of G-maps G=H ?! G=H can be identied with the group
WH . We also have the orbit spaces X=H obtained by identifying points of X
in the same orbit, and these too are WH -spaces. For a space K regarded as a
G-space with trivial G-action, we have
GU (K; X ) = U (K; X G )
GU (X; K ) (1.3)
= U (X=G; K ):
If Y is an H -space, there is an induced G-space G H Y . It is obtained from
G Y by identifying (gh; y) with (g; hy) for g 2 G, h 2 H , and y 2 Y . A bit
less obviously, we also have the \coinduced" G-space MapH (G; Y ), which is the
space of H -maps G ?! Y with left action by G induced by the right action of
G on itself, (g f )(g0) = f (g0 g). For G-spaces X and H -spaces Y , we have the
GU (G H Y; X ) = H U (Y; X )
H U (X; Y ) (1.5)
= GU (X; MapH (G; Y )):
Moreover, for G-spaces X , we have G-homeomorphisms
G H X = (G=H ) X
MapH (G; X ) = Map(G=H; X ):
For the rst, the unique G-map G H X ?! (G=H ) X that sends x 2 X to
(eH; x) has inverse that sends (gH; x) to the equivalence class of (g; g?1 x).
A homotopy between G-maps X ?! Y is a homotopy h : X I ?! Y that is
a G-map, where G acts trivially on I . There results a homotopy category hGU .
Recall that a map of spaces is a weak equivalence if it induces an isomorphism
of all homotopy groups. A G-map f : X ?! Y is said to be a weak equivalence
if f H : X H ?! Y H is a weak equivalence for all H G. We let h GU denote
the category constructed from hGU by adjoining formal inverses to the weak
equivalences. We shall be more precise shortly. The algebraic invariants of Gspaces that we shall be interested in will be dened on the category h GU .
General References
G. E. Bredon. Introduction to compact transformation groups. Academic Press. 1972.
T. tom Dieck. Transformation groups. Walter de Gruyter. 1987.
(This reference contains an extensive Bibliography.)
2. Analogs for based G-spaces
It will often be more convenient to work with based G-spaces. Basepoints are
G-xed and are generically denoted by . We write X+ for the union of a G-space
X and a disjoint basepoint. The wedge, or 1-point union, of based G-spaces is
denoted by X _ Y . The smash product is dened by X ^ Y = X Y=X _ Y . We
write F (X; Y ) for the based G-space of based maps X ?! Y . Then
F (X ^ Y; Z ) = F (X; F (Y; Z )):
We write GT for the category of based G-spaces, and we have
GT (K; X ) = T (K; X G )
GT (X; K ) = T (X=G; K )
for a based space K and a based G-space X . Similarly, for a based G-space X
and a based H -space Y , we have
GT (G+ ^H Y; X ) = H T (Y; X )
H T (X; Y ) = GT (X; FH (G+ ; Y ));
where FH (G+ ; Y ) = MapH (G; X ) with the trivial map as basepoint, and we have
G+ ^H X (2.6)
= (G=H )+ ^ X
FH (G+ ; X ) (2.7)
= F (G=H+ ; X ):
A based homotopy between based G-maps X ?! Y is given by a based Gmap X ^ I+ ?! Y . Here the based cylinder X ^ I+ is obtained from X I by
collapsing the line through the basepoint of X to the basepoint. There results a
homotopy category hGT , and we construct hGT by formally inverting the weak
equivalences. Of course, we have analogous categories hGU , and hGU in the
unbased context.
In both the based and unbased context, cobrations and brations are dened
exactly as in the nonequivariant context, except that all maps in sight are G-maps.
Their theory goes through unchanged. A based G-space X is nondegenerately
based if the inclusion fg ?! X is a cobration.
3. G-CW complexes
A G-CW complex X is the union of sub G-spaces X n such that X 0 is a disjoint union of orbits G=H and X n+1 is obtained from X n by attaching G-cells
G=H Dn+1 along attaching G-maps G=H S n ?! X n . Such an attaching
map is determined by its restriction S n ?! (X n)H , and this allows the inductive
analysis of G-CW complexes by reduction to nonequivariant homotopy theory.
Subcomplexes and relative G-CW complexes are dened in the obvious way. I will
review my preferred way of developing the theory of G-CW complexes since this
will serve as a model for other versions of cellular theory that we shall encounter.
We begin with the Homotopy Extension and Lifting Property. Recall that a map
f : X ?! Y is an n-equivalence if q (f ) is a bijection for q < n and a surjection
for q = n (for any choice of basepoint). Let be a function from conjugacy classes
of subgroups of G to the integers ?1. We say that a map e : Y ?! Z is
a -equivalence if eH : Y H ?! Z H is a (H )-equivalence for all H . (We allow
(H ) = ?1 to allow for empty xed point spaces.) We say that a G-CW complex
X has dimension if its cells of orbit type G=H all have dimension (H ).
Theorem 3.1 (HELP). Let A be a subcomplex of a G-CW complex X of
dimension and let e : Y ?! Z be a -equivalence. Suppose given maps g :
A ?! Y , h : A I ?! Z , and f : X ?! Z such that eg = hi1 and fi = hi0 in
the following diagram:
g }}}
h xxx
f ~~
Ax I
F h~
}} e
A g~
Then there exist maps g~ and ~h that make the diagram commute.
Proof. We construct g~ and ~h on A [ X n by induction on n. When we pass
from the n-skeleton to the (n +1)-skeleton, we may work one cell at a time, dealing
with the cells of X not in A. By considering attaching maps, we quickly reduce
the proof to the case when (X; A) = (G=H Dn+1 ; G=H S n). But this case
reduces directly to the nonequivariant case of (Dn+1 ; S n).
?! Z be a -equivalence and X be a
G-CW complex. Then e : hGU (X; Y ) ?! hGU (X; Z ) is a bijection if X has
Theorem 3.2 (Whitehead). Let e : Y
dimension less than and a surjection if X has dimension .
Proof. Apply HELP to the pair (X; ;) for the surjectivity. Apply HELP to
the pair (X I; X @I ) for the injectivity.
Corollary 3.3. If e : Y
?! Z is a -equivalence between G-CW complexes
of dimension less than , then e is a G-homotopy equivalence.
Proof. A map f : Z ?! Y such that e[f ] = id is a homotopy inverse to e.
The cellular approximation theorem works equally simply. A map f : X ?! Y
between G-CW complexes is said to be cellular if f (X n ) Y n for all n, and
similarly in the relative case.
Theorem 3.4 (Cellular Approximation). Let (X; A) and (Y; B ) be rela-
tive G-CW complexes, (X 0; A0) be a subcomplex of (X; A), and f : (X; A) ?!
(Y; B ) be a G-map whose restriction to (X 0; A0) is cellular. Then f is homotopic
rel X 0 [ A to a cellular map g : (X; A) ?! (Y; B ).
Proof. This again reduces to the case of a single nonequivariant cell.
Corollary 3.5. Let X and Y be G-CW complexes. Then any G-map f :
X ?! Y is homotopic to a cellular map, and any two homotopic cellular maps
are cellularly homotopic.
Proof. Apply the theorem in the cases (X; ;) and (X I; X @I ).
Theorem 3.6. For any G-space X , there is a G-CW complex ?X and a weak
equivalence : ?X ?! X .
Proof. We construct an expanding sequence of G-CW complexes fYi ji 0g
together with maps i : Yi ?! X such that i+1jYi = i . Choose a representative
map f : S q ?! X H for each element of q (X H ; x). Here q runs over the nonnegative integers, H runs over the conjugacy classes of subgroups of G, and x runs
over the components of X H . Let Y0 be the disjoint union of spaces G=H S q ,
one for each chosen map f , and let 0 be the G-map induced by the maps f . Inductively, assume that i : Yi ?! X has been constructed. Choose representative
maps (f; g) for each pair of elements of q ((Yi)H ; y) that are equalized by q (i);
here again, q runs over the non-negative integers, H runs over the conjugacy classes
of subgroups of G, and y runs over the components of (Yi)H . We may arrange that
f and g have image in the q-skeleton of Yi. Let Yi+1 be the homotopy coequalizer
of the disjoint union of these pairs of maps; that is Yi+1 is obtained by attaching a
tube (G=H+ ^ S q I+ via each chosen pair (f; g). Dene i+1 by use of homotopies
h : i f ' ig based at i (y). It is easy to triangulate Yi+1 as a G-CW complex
that contains Yi as a subcomplex. Taking ?X to be the union of the Yi and to
be the map induced by the i , we obtain the desired weak equivalence.
The Whitehead theorem implies that the G-CW approximation ?X is unique
up to G-homotopy equivalence. If f : X ?! X 0 is a G-map, there is a unique
homotopy class of G-maps ?f : ?X ?! ?X 0 such that 0 ?f ' f . That
is, ? becomes a functor hGU ?! hGU such that is natural. A construction
of ? that is functorial even before passage to homotopy is possible (Seymour). It
follows that the morphisms of the category h GU can be specied by
h GU (X; X 0 ) = hGU (?X; ?X 0 ) = hGC (?X; ?X 0 );
where GC is the category of G-CW complexes and cellular maps. From now on,
we shall write [X; X 0]G for this set, or for its based variant, depending on the
Almost all of this works just as well in the based context, giving a theory of
\G-CW based complexes", which are required to have based attaching maps. This
notion is to be distinquished from that of a based G-CW complex, which is just
a G-CW complex with a G-xed base vertex. In detail, a G-CW based complex
X is the union of based sub G-spaces X n such that X 0 is a point and X n+1 is obtained from X n by attaching G-cells G=H+ ^ Dn+1 along based attaching G-maps
G=H+ ^ S n ?! X n . Observe that such G-CW based complexes are G-connected
in the sense that all of their xed point spaces are non-empty and connected.
Nonequivariantly, one often starts proofs with the simple remark that it suces
to consider connected spaces. Equivariantly, this won't do; many important foundational parts of homotopy theory have only been worked out for G-connected
I should emphasize that G has been an arbitrary topological group in this discussion. When G is a compact Lie group | and we shall later restrict attention to
such groups | there are important results saying that reasonable spaces are triangulable as G-CW complexes or have the homotopy types of G-CW complexes. It
is fundamental for our later work that smooth compact G-manifolds are triangulable as nite G-CW complexes (Verona, Illman). In contrast to the nonequivariant
situation, this is false for topological G-manifolds, which have the homotopy types
of G-CW complexes but not necessarily nite ones. Metric G-ANR's have the
homotopy types of G-CW complexes (Kwasik). Milnor's results on spaces of the
homotopy type of CW complexes generalize to G-spaces (Waner). In particular,
Map(X; Y ) has the homotopy type of a G-CW complex if X is a compact G-space
and Y has the homotopy type of a G-CW complex, and similarly for based function
S. Illman. The equivariant triangulation theorem for actions of compact Lie groups. Math. Ann.
262(1983), 487-501.
S. Kwasik. On the equivariant homotopy type of G-ANR's. Proc. Amer. Math. Soc. 83(1981),
T. Matumoto. On G-CW complexes and a theorem of J.H.C. Whitehead. J. Fac. Sci. Univ. of
Tokyo 18(1971), 363-374.
R. M. Seymour. Some functorial constructions on G-spaces. Bull. London Math. Soc. 15(1983),
A. Verona. Triangulation of stratied bre bundles. Manuscripta Math. 30(1980), 425-445.
S. Waner. Equivariant homotopy theory and Milnor's theorem. Trans. Amer. Math. Soc.
258(1980), 351-368.
4. Ordinary homology and cohomology theories
Let G denote the category of orbit G-spaces G=H ; the standard notation is OG.
Observe that there is a G-map f : G=H ?! G=K if and only if H is subconjugate
to K since, if f (eH ) = gK , then g?1Hg K . Let hG be the homotopy category of
G . Both G and hG play important roles and it is essential to keep the distinction
in mind.
Dene a coecient system to be a contravariant functor hG ?! A b. One
example to keep in mind is the system n(X ) of homotopy groups of a based
G-space X : n(X )(G=H ) = n(X H ). Formally, we have an evident xed point
functor X : G ?! T . The map X K ?! X H induced by a G-map f : G=H ?!
G=K such that f (eH ) = gK sends x to gx. Any covariant functor hT ?! A b,
such as n, can be composed with this functor to give a coecient system. It
should be intuitively clear that obstruction theory must be developed in terms of
ordinary cohomology theories with coecients in such coecient systems. The
appropriate theories were introduced by Bredon.
Since the category of coecient systems is Abelian, with kernels and cokernels
dened termwise, we can do homological algebra in it. Let X be a G-CW complex.
We have a coecient system
C n (X ) = H n(X n ; X n?1 ; Z):
That is, the value on G=H is Hn((X n )H ; (X n?1 )H ). The connecting homomorphisms of the triples ((X n )H ; (X n?1)H ; (X n?2 )H ) specify a map
d : C n(X ) ?! C n?1 (X )
of coecient systems, and d2 = 0. That is, we have a chain complex of coecient
systems C (X ). For based G-CW complexes, we dene C~ (X ) similarly. Write
HomG (M; M 0) for the Abelian group of maps of coecient systems M ?! M 0
and dene
CGn (X ; M ) = HomG (C n(X ); M ); with = HomG (d; id):
Then CG (X ; M ) is a cochain complex of Abelian groups. Its homology is the
Bredon cohomology of X , denoted HG (X ; M ).
To dene Bredon homology, we must use covariant functors N : hG ?! A b
as coecient systems. If M : hG ?! A b is contravariant, we dene an Abelian
M G N = M (G=H ) N (G=H )=();
where the equivalence relation is specied by (mf ; n) (m; fn) for a map
f : G=H ?! G=K and elements m 2 M (G=K ) and n 2 N (G=H ). Here we
write contravariant actions from the right to emphasize the analogy with tensor
products. Such \coends", or categorical tensor products of functors, occur very
often in equivariant theory and will be formalized later. We dene cellular chains
CnG(X ; N ) = C n(X ) G N; with @ = d 1:
Then CG(X ; N ) is a chain complex of Abelian groups. Its homology is the Bredon
homology of X , denoted HG (X ; N ).
Clearly Bredon homology and cohomology are functors on the category GC of
G-CW complexes and cellular maps. A cellular homotopy is easily seen to induce
a chain homotopy of cellular chain complexes in our Abelian category of coecient
systems, so homotopic maps induce the same homomorphism on homology and
cohomology with any coecients.
The development of the properties of these theories is little dierent from the
nonequivariant case. A key point is that C (X ) is a projective object in the
category of coecient systems. To see this, observe that C (X ) is a direct sum of
coecient systems of the form
H~ n(G=K+ ^ S n ) (4.4)
= H~ 0(G=K+ ) = H 0(G=K ):
If F denotes the free Abelian group functor on sets, then
(4.5) H 0(G=K )(G=H ) = H0((G=K )H ) = F0((G=K )H ) = F [G=H; G=K ]G :
Therefore HomG (H 0(G=K ); M ) = M (G=K ) via ?! (1G=K ) 2 M (G=K ). In
detail, for a G-map f : G=H ?! G=K , we have f = f (1G=K ),
f : F [G=K; G=K ]G ?! F [G=H; G=K ]G ;
so that (1G=K ) determines via (f ) = f (1G=K ). This calculation implies the
claimed projectivity. It also implies the dimension axiom:
HG (G=K ; M ) = HG0 (G=K ; M ) = M (G=K )
HG (G=K ; N ) = H0G (G=K ; N ) = N (G=K );
these giving isomorphisms of coecient systems, of the appropriate variance, as
K varies.
If A is a subcomplex of X , we obtain the relative chain complex C (X; A) =
~C (X=A). The projectivity just proven implies the expected long exact sequences
of pairs. For additivity, just note that the disjoint union of G-CW complexes is a
G-CW complex. For excision, if X is the union of subcomplexes A and B , then
B=A \ B = X=A as G-CW complexes. We take the \weak equivalence axiom" as
a denition. That is, for general G-spaces X , we dene
HG (X ; M ) HG (?X ; M ) and HG (X ; N ) HG (?X ; N ):

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