RATIONAL S1-EQUIVARIANT STABLE HOMOTOPY
THEORY.
J.P.C.Greenlees
The author is grateful to the Nuffield Foundation for its support.
Author addresses:
School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.
E-mail address: j.greenlees@sheffield.ac.uk
Abstract.We make a systematic study of rational S1-equivariant cohomology t*
*heories,
or rather of their representing objects, rational S1-spectra.
In Part I we construct a complete algebraic model for the homotopy catego*
*ry of S1-
spectra, reminiscent of the localization theorem. The model is of homologic*
*al dimension
one, and simple enough to allow practical calculations; in particular we ob*
*tain a classifi-
cation of rational S1-equivariant cohomology theories.
In Part II we identify the algebraic counterparts of all the usual S1-spe*
*ctra and con-
structions on S1-spectra. This enables us in Part III to give a rational a*
*nalysis of a
number of interesting phenomena, such as the Atiyah-Hirzebruch spectral seq*
*uence, the
Segal conjecture, K-theory and topological cyclic cohomology.
ii
Contents
Chapter 0. General Introduction. 1
0.1. Motivation 1
0.2. Overview 2
Part I
The algebraic model of rational T-spectra.
Chapter 1. Introduction to Part I. 13
1.1. Outline of the algebraic models. 13
1.2. Reading Guide. 17
1.3. Haeberly's example. 18
1.4. McClure's Chern character isomorphism for F-spaces. 18
Chapter 2. First steps. 23
2.1. Natural cells and basic cells 23
2.2. Separating isotropy types. 26
2.3. The single strand spectra E. 29
2.4. Operations. 32
Chapter 3. The Adams spectral sequence. 35
3.1. The Adams short exact sequence. 35
3.2. Construction of the Adams resolution. 36
3.3. Convergence of the Adams resolution. 38
3.4. Torsion Q[cH ]-modules and geometric implications. 39
Chapter 4. Categorical reprocessing. 43
4.1. Recollections about derived categories. 43
4.2. Split linear triangulated categories. 46
iii
iv CONTENTS
4.3. The algebraicization of the category of T-spectra over H. 51
Chapter 5. Assembly and the standard model. 55
5.1. Assembly. 55
5.2. Global assembly. 57
5.3. The standard model category. 59
5.4. The algebraicization of rational T-spectra. 63
5.5. Notation. 66
Chapter 6. The torsion model. 71
6.1. Practical calculations. 71
6.2. The torsion model 72
6.3. Equivalence of derived categories of standard and torsion models. 75
Part II
Algebraic counterparts to standard constructions.
Chapter 7. Introduction to Part II 81
7.1. General outline. 81
7.2. Modelling of the smash product and the function spectrum. 84
7.3. Modelling functors changing equivariance. 86
7.4. Modelling Eilenberg-MacLane spectra and related objects. 88
7.5. Functors between split triangulated categories. 88
Chapter 8. Basic algebra for models and their derived categories. 91
8.1. Euler classes and F-finite torsion OF-modules. 91
8.2. Products in the torsion model. 94
8.3. The tensor hom adjunction. 97
8.4. Hom, tensor and torsion functors in standard models. 101
8.5. Hom, tensor and torsion functors on derived categories. 105
8.6. Products in the standard model. 107
Chapter 9. Products, smash products and function T-spectra. 109
9.1. Maps between injective spectra. 109
9.2. Models of smash products. 110
9.3. Models of function spectra. 112
Chapter 10. Induction, coinduction and geometric fixed points. 117
10.1. Forgetful, induction and coinduction functors. 117
CONTENTS v
10.2. Geometric fixed points. 119
Chapter 11. Algebraic inflation and deflation. 125
11.1. Algebraic inflation and Hausdorff OF-modules. 125
11.2. Algebraic inflation and deflation of OF-modules. 128
11.3. Inflation and its right adjoint on the torsion model category. 130
Chapter 12. Inflation, Lewis-May fixed points and quotients. 135
12.1. The topological inflation and Lewis-May fixed point functors. 135
12.2. Correspondence of Algebraic and geometric inflation maps. 139
12.3. A direct approach to the Lewis-May fixed point functor. 141
12.4. Lewis-May fixed points on objects in the standard model. 143
12.5. Quotient functors. 146
Chapter 13. Homotopy Mackey functors and related constructions. 149
13.1. The homotopy Mackey functor on A. 149
13.2. Eilenberg-MacLane spectra. 152
13.3. coMackey functors and spectra representing ordinary homology. 155
13.4. Brown-Comenetz spectra. 157
Part III
Applications.
Chapter 14. Introduction to Part III. 163
14.1. General Outline 163
14.2. Prospects and problems. 165
Chapter 15. Classical miscellany. 167
15.1. The collapse of the Atiyah-Hirzebruch spectral sequence. 167
15.2. Orbit category resolutions. 169
15.3. Suspension spectra. 171
15.4. K-theory revisited. 172
15.5. The geometric equivariant rational Segal conjecture for T. 175
Chapter 16. Cyclic and Tate cohomology. 179
16.1. Cyclic cohomology. 179
16.2. Rational Tate spectra. 180
16.3. The integral T-equivariant Tate spectrum for complex K-theory. 181
Chapter 17. Cyclotomic spectra and topological cyclic cohomology. 185
vi CONTENTS
17.1. Cyclotomic spectra. 185
17.2. Free loop spaces. 188
17.3. Topological cyclic cohomology of cyclotomic spectra. 190
Appendix A. Mackey functors. 195
Appendix B. Closed model categories. 201
Appendix C. Conventions. 207
C.1. Conventions for spaces and spectra. 207
C.2. Standing conventions. 209
Appendix D. Indices. 211
D.1. Index of definitions and terminology. 211
D.2. Index of notation. 215
Bibliography 225
CHAPTER 0
General Introduction.
0.1.Motivation
Spaces with actions of the circle group T are of particular interest. Loops o*
*ccur in many
constructions, and it is often appropriate to take into account the action of t*
*he circle by
rotation; in particular the free loop space has been the object of much study. *
*This in turn
leads towards the use of the circle group in cyclic cohomology; the refinements*
* of topological
Hochschild homology and topological cyclic constructions are also important in *
*algebraic
K-theory. More prosaically the circle is simply the first infinite compact Lie *
*group, and
it plays a fundamental role in the understanding of all positive dimensional gr*
*oups. For
any one of these reasons it is important to understand equivariant cohomology t*
*heories for
spaces with circle action.
To obtain a reasonably broad and simple picture, we consider the case of rati*
*onal co-
homology theories; these have been considered before for special classes of spa*
*ces (see
for example [5]), but this appears to be the first attempt to obtain a complete*
* algebraic
picture. In any case, the understanding of the rational case is a necessary fi*
*rst step to-
wards a general understanding of T-equivariant cohomology theories. It is well *
*known [14]
that, for finite groups, all cohomology theories are products of ordinary cohom*
*ology the-
ories, but this is false for the circle group. A test case of particular inter*
*est is rational
topological K-theory. The example of J.-P.Haeberly [16] shows that, by contrast*
* with the
case of finite groups of equivariance, there is no Chern character isomorphism.*
* It follows
that T-equivariantly some topological structure remains, even after rationaliza*
*tion. The
author began the present work to understand the T-equivariant Chern character, *
*the T-
equivariant Segal conjecture, the Tate construction on T-equivariant K-theory a*
*nd several
other T-equivariant rational objects that had come to light. The list of conten*
*ts contains
a list of examples treated here.
From now on we let T denote the circle group. We only consider closed subgrou*
*ps, and
the letters H; K and L will denote finite subgroups. The family of all finite s*
*ubgroups will
be denoted by F. We work rationally throughout, without displaying this in the *
*notation;
for example Sn denotes the rationalized n-sphere.
1
2 0. GENERAL INTRODUCTION.
0.2. Overview
Equivariant cohomology theories are represented by equivariant spectra, and w*
*e shall
conduct most of the investigation at the represented level. This gives more pre*
*cise infor-
mation both about individual theories and about natural transformations between*
* them;
indeed, the only loss is any geometric interpretation of the cohomology theory *
*concerned,
which is inevitable in any general study. It is important to be explicit that w*
*e only consider
cohomology theories which admit suspension isomorphisms for arbitrary represent*
*ations;
these are sometimes known as `genuine' or `RO(G)-graded' cohomology theories. *
*The
corresponding representing objects are G-spectra. For these too there are adje*
*ctives to
emphasize the type of spectra concerned: they are `genuine' G-spectra or G-spec*
*tra `in-
dexed on a complete G-universe'. Since these cohomology theories and these G-s*
*pectra
form the most natural classes to consider, we shall not use these adjectives un*
*less required
for emphasis. As made clear by the title, we consider the circle group G = T.
Before summarizing our results we begin by putting the circle group into cont*
*ext. In
fact the circle stands at a watershed: for finite groups of equivariance ration*
*al cohomology
theories may be analysed completely, and any group more complicated than the ci*
*rcle is
substantially harder to understand.
The main problem in analyzing spectra is to choose basic objects which are ea*
*sy to work
with and which give theorems of practical use. It is natural to be guided by on*
*e's favourite
algebraic invariant, and this suggests analysis in terms of Moore spectra or Ei*
*lenberg-
MacLane spectra. For finite groups of equivariance both approaches work well, a*
*nd one
may analyse rational spectra completely. There are two reasons for this: firstl*
*y the group
has no topology, and secondly the classifying space has no rational cohomology.*
* The first
fact means the category of Mackey functors is very simple, and the second means*
* that the
classes of Eilenberg-MacLane spectra, of Moore spectra and of Brown-Comenetz sp*
*ectra
coincide, so that all their characteristic properties can be used at once. Both*
* simplifying
factors fail for infinite groups, and the three basic classes are distinct. Th*
*is means that
different methods must be used: in essence we base our analysis for the circle *
*group on
a slightly embellished version of equivariant homotopy with its primary operati*
*ons. The
reason such a simple invariant suffices is that the rank of the circle group is*
* one. In general
the injective dimension of the category of rational Mackey functors and the Kru*
*ll dimension
of the cohomology of its classifying space are both equal to the rank of the gr*
*oup. When
the rank is one there is no room for extension problems, and some hope of a sim*
*ple answer.
However, even for the group O(2), it is necessary to take into account a topolo*
*gy on the
space of subgroups, and to work with sheaves: it is no longer possible to treat*
* different
conjugacy classes of subgroups entirely separately. This explains why it is wor*
*thwhile to
treat the single case of the circle in such detail.
The work is broken into three parts: Part I in which we construct the algebra*
*ic models
for various classes of T-spectra, Part II in which we identify the algebraic co*
*unterparts of
various general constructions, and Part III in which we consider several classe*
*s of examples
of particular interest. Each part has a detailed introduction of its own, but *
*we give a
0.2. OVERVIEW 3
general outline here.
Part I begins by discussing K-theory. On the one hand, we give Haeberly's ex*
*ample
showing that K-theory cannot be described simply using ordinary cohomology. On *
*the other
hand, we give a generalization of McClure's result that the K-theory Atiyah-Hir*
*zebruch
spectral sequence collapses for F-free spaces. This suggests the necessity of *
*the present
work and that it is practical. We then turn to the main business of constructin*
*g a model:
in this introduction we describe the model in an aesthetically satisfying way, *
*but do not
attempt to explain the proof that it is a model. The introduction to Part I giv*
*es a different
approach to the model which does suggest the proof. We would prefer to achieve *
*these two
ideals simultaneously.
To motivate the form of the model, one should recall the classical localizati*
*on theorem
for semifree T-spaces. This states that if X is a finite space which only has i*
*sotropy groups
T and 1, then the inclusion of the fixed point space XT -! X induces an isomorp*
*hism in
Borel cohomology once the Euler classes E = {1; c1; c21; : :}:are inverted:
~= -1* T -1 * * T
E-1H*(ET+ ^T X) -! E H (ET+ ^T X ) = E H (BT+) H (X ):
We conclude that N = H*(ET+ ^T X), regarded as a module over Q[c1] = H*(BT+) is
very nearly enough to identify the homology of the fixed point space XT, but we*
* need
to pick out a vector subspace V = H*(XT) of E-1N which is a basis in the sense*
* that
E-1N ~=E-1H*(BT+) V . In particular, if X is free then N is E-torsion.
Now T-equivariant cohomology theories are represented by T-spectra, and the l*
*ocaliza-
tion theorem suggests a model which turns out to be a complete invariant. To de*
*scribe it,
we first note that there is a natural homotopy-level analogue of the set of iso*
*tropy groups
which occur. This uses the geometric K-fixed point functor X 7-! K X, which is *
*the func-
tor extending the K-fixed point functor on spaces, in the sense that K (1 Y ) =*
* 1 (Y K);
it also enjoys similar formal properties to the space-level functor. We then de*
*fine the set
of isotropy groups of a spectrum X to be the set of subgroups K for which the g*
*eometric
fixed point spectra K X are non-equivariantly essential. This gives the notion *
*of a free
T-spectrum (alternatively characterized as a T-spectrum X for which ET+ ^ X -! *
*X is
an equivalence). We therefore suppose given a collection H of finite subgroups *
*of T, and
we may consider the class of H-free spectra (i.e. those with isotropy in H), an*
*d the class
of H-semifree spectra (i.e. those with isotropy in H [ {T}). The reader should *
*concentrate
on the case H = {1}, which gives the usual classes of free and semifree spectra*
*, and on
the case H = F: the class of F-semifree spectra is the class of all T-spectra. *
*However the
additional generality makes the picture clearer, and the two special cases are *
*representative
of the two classes of examples: those with H finite, and those with H infinite.*
* Analagous
to the ring H*(BT+) we have the ring of operations
OH = C(H; Q)[c];
where C(H; Q) denotes the Q-valued functions on the discrete set H, and c is of*
* degree
-2. The notation is chosen to suggest that OH is a ring of functions on H. This*
* ring is
Noetherian if H is finite and not otherwise. We let eH 2 C(H; Q) = (OH)0 denot*
*e the
idempotent with support H 2 H, and we let cH = eH c. Next we need the set E = *
*EH
of Euler classes. If H = {1} this is simply the multiplicative subset {1; c1; c*
*21; : :}:of OH
4 0. GENERAL INTRODUCTION.
used for the localization theorem above, but in general it needs a little more *
*explanation.
For any finite subset OE H we have an associated idempotent eOE2 OH, and we ha*
*ve an
Euler class cOE= eOEc + (1 - eOE), which is not a homogeneous element of OH. Th*
*e effect
of cOEon an OH-module M = eOEM (1 - eOE)M is to multiply by c on the first fac*
*tor and
do nothing to the second: thus the result of inverting cOEon M is again a grade*
*d module:
eOEM[c-1] (1 - eOE)M. Thus our second ingredient is the set
EH = {ckOE| OE H finite,k 0}
of Euler classes. The category modelling semifree H-spectra is then the categor*
*y AH of
OH-modules N with a specified graded vector space V to act as a basis of E-1N. *
* It is
convenient to package this as saying that we are given a basing map
fi : N -! (E-1OH) V
which becomes an isomorphism when E is inverted. This makes clear that a morphi*
*sm in
AH is a diagram
M - ! N
ff # # fi
1OE -1
(E-1OH) U - ! (E OH) V:
We refer to N as the nub and V as the vertex. We also refer to an OH-module wit*
*h specified
basing map as a based OH-module, and to a morphism : M -! N for which there is
a compatible map OE as a based map. Note that if H is a singleton the existenc*
*e of a
basing isomorphism E-1N ~=E-1OH V for some V is automatic, but in general it p*
*uts a
restriction on the modules N.
The connection with topology arises since OF = [EF+; EF+]T*, and hence this a*
*cts on
both ssT*(EF+ ^ X) and ssT*(DEF+ ^ "EF ^ X) for any X; if X is H-semifree this *
*action
factors through the projection OF -! OH. Furthermore, since c is of negative d*
*egree
and any element of ssT*(EF+ ^ X) is supported on a finite subspectrum, one sees*
* that
E -1ssT*(EF+ ^ X) = 0. Next, we have a map
DEF+ ^ X -! DEF+ ^ "EF ^ X ' DEF+ ^ "EF ^ T X
with cofibre DEF+ ^ EF+ ^ X ' EF+ ^ X. Since the homotopy of the cofibre is
Euler-torsion, its homotopy
i *
* j
ssA*(X) := ssT*(DEF+ ^ X) -! ssT*(DEF+ ^ "EF ^ T X) = ssT*(DEF+ ^ "EF) ss*(T *
* X)
is therefore an object of AH .
Now we may state our main classification theorem.
Classification Theorem: For any collection H of finite subgroups of the circle *
*T, the
above invariant induces bijections
(i)
{H-free rational spectra}= ' ! {Euler-torsion OH-modules}= ~=
0.2. OVERVIEW 5
where ' denotes homotopy equivalence, and ~=denotes isomorphism, and
(ii)
{H-semifree rational spectra}= ' ! {based OH-modules}= ~=
where ' denotes homotopy equivalence, and ~=denotes isomorphism. In particular*
*, ra-
tional T-equivariant cohomology theories are in bijective correspondence to iso*
*morphism
classes of based OF-modules.
In practice this is derived as a corollary of a theorem identifying the categ*
*ories of spectra
in algebraic terms. More precisely, recall that the derived category of a grade*
*d abelian cat-
egory is the category of differential graded objects with homology isomorphisms*
* inverted,
although for practical purposes a more concrete construction is essential. The*
* theorem
identifies the categories of spectra as the derived category of the associated *
*algebraic cate-
gory:
H-free T-spectra' D(Euler torsion OH-modules)
and
H-semifree T-spectra' D(based OH-modules):
Furthermore, cofibre sequences of spectra correspond to triangles under these e*
*quivalences.
The point here is that both algebraic categories turn out to be abelian and one*
* dimensional,
so that morphisms in the derived category can be calculated from a short exact *
*sequence
involving Hom and Ext in the abelian category.
It is sometimes more practical to identify the place of a spectrum X in the c*
*lassification
by a different route. This amounts to identifying first EF+ ^ X and T X, and th*
*en the
map
qX : "EF ^ T X = "EF ^ X -! EF+ ^ X
of which X is the fibre. It is not enough to identify the effect of qX in homo*
*topy: one
must also take into account the twisting given by representations, and in gener*
*al this
requires both primary and secondary information. Nonetheless, there is a secon*
*d model
for semifree H-spectra based on this approach, which we call the torsion model.*
* We show
it is equivalent to the standard model described above, and it is often the eas*
*iest route to
placing a spectrum in the classification.
There are really three stages to the proof of these theorems. Firstly one sho*
*ws, using
idempotents in the Burnside rings of finite subgroups, that for F-free spectra *
*it is essentially
enough to deal with the case of free spectra. Next, one constructs an Adams sp*
*ectral
sequence for free spectra, which collapses to a short exact sequence and gives *
*a means of
calculation. Because of the particularly simple algebraic behaviour of O1 = Q[c*
*1] this is
enough to identify the entire triangulated category. The final stage is to take*
* this work and
process it: this stage is essentially formal.
Once we have algebraic models for various categories of spectra we naturally *
*want to
understand familiar topological constructions in algebraic terms. This is the b*
*usiness of
Part II. We have followed the order suggested by logic, and therefore begin by *
*studying the
smash product and function spectrum constructions, and then go on to functors c*
*hanging
equivariance. Unfortunately the smash product and function spectrum are by far*
* the
6 0. GENERAL INTRODUCTION.
most complicated examples, and require more algebraic machinery than any of the*
* other
examples we consider. Furthermore, their complexity means that we are not able *
*to show
that our description is functorial, and our approach is necessarily indirect. T*
*his highlights
a shortcoming of our method: the correct proof of our results would follow tha*
*t used
by Quillen in modelling rational homotopy of simply connected spaces. The func*
*torial
identification of smash products and function spectra would then be automatic. *
*At present,
such a proof is not accessible, but the present results strongly suggest that s*
*uch a proof
exists. In any case, the model of the smash product is essentially the left der*
*ived tensor
product, and the model of function spectra is its right adjoint. There are two *
*warnings here:
in the categories of H-free spectra, there are not enough flat objects, so the *
*left derived
tensor product must be calculated in a larger category; it results in an Euler-*
*torsion object
since it coincides with the suspension of the right derived torsion product. Wi*
*th this caveat,
if the spectra X and Y are modelled by M and N respectively then
X ^ Y is modelled byM L N:
There is also a caveat for function objects, which we now explain. It is conv*
*enient in
both cases to consider the larger algebraic category in which no condition is p*
*laced on the
behaviour of Euler classes. For H-free spectra this is the category of all OH-m*
*odules, and
for H-semifree spectra it is the category of all maps N -! E-1OH V . It turns *
*out that
the internal Hom functor in the abelian category is the composite functor Hom *
*(M; N),
where Hom (M; N) is an object in the category with no condition on behaviour u*
*nder
inversion of Euler classes, and where is the right adjoint to the inclusion of*
* the smaller
category. For example, in the case of H-free spectra Hom (M; N) is simply the O*
*H-module
of OH-morphisms, and for an arbitrary OH-module M0, the Euler-torsion module M *
*is
defined to be the kernel of M -! E-1M. In the semifree case both functors are h*
*arder
to describe, and we refer the reader to Chapter 8. It turns out that the right *
*adjoint of
M 7-! M L N is not the right derived functor of P 7-! Hom (N; P ), but rather *
*it is
P 7-! R RHom (N; P ). Thus if the spectra Y and Z are modelled by N and P , th*
*en
The internal function spectrum of maps from Y to Z is modelledRby RHom (N; P *
*):
An essential step in identifying the function spectrum on objects is to give a *
*functorial
identification of the product. In these terms we may say that if Xiis modelled *
*by Mithen
Y
The internal product of the spectra Xiis modelledRby Mi;
i
and this model is functorial.
The other topological functors we consider can be modelled functorially, and *
*we shall
discuss only the full category of T-spectra. The forgetful functor and its lef*
*t and right
adjoints, induction and coinduction, are straightforward. Similarly the geomet*
*ric fixed
point functor X 7-! T X is the passage-to-vertex functor given as part of the s*
*tructure.
The first interesting functor is the geometric fixed point functor K : T - sp*
*ectra -!
T=K - spectra for a finite subgroup K. This turns out to be easy to describe: w*
*e simply
let e 2 C(F; Q) denote the idempotent supported on the set [ K] of subgroups co*
*ntaining
K. The algebraic model of_K_ is multiplication by_e; this make sense since eOF *
*is naturally
identified with the ring O __Fof operations for T = T=K. As usual, the Lewis-M*
*ay fixed
0.2. OVERVIEW 7
point functor K : T - spectra -! T=K - spectra (the spectrum K X is written XK *
*in
[18]) is much harder to understand, and we only describe its behaviour here for*
* F-free and
F-contractible spectra, referring the reader to Chapters 11 and 12 for details_*
*of how these
are spliced. On F-contractible spectra X ' "EF ^ T X, we have K (X) = "EF^ T X,
so this is easy. We have seen that an F-spectrum X is modelled by an Euler-tors*
*ion OF-
module N; from the form of Euler classes it follows that this is equivalent to *
*specifying the
function
[N] : F-! torsionQ[c] - modules
H 7-! eH N:
The Lewis-May fixed point functor groups these modules together according to th*
*e be-
haviour of the subgroup on passage_to quotient. More precisely,_we observe that*
* passage
to quotient q : T -! T=K = T defines a map q* : F -! F on finite subgroups. If *
*the
function [N] models the F-free spectrum X then the function [K N] modelling K X*
* is
the map __
F_ -! torsionQ[c]L- modules
H 7-! q*(H)=__H[N](H):
A little thought shows that it is not a trivial matter to see how the F-free an*
*d F-
contractible parts should be spliced together. Because the Lewis-May fixed poin*
*t functor
is so complicated, we actually approach it via its left adjoint, the inflation *
*map infTT=K:
T=K - spectra -! T - spectra. This is the functor given by regarding a T=K spec*
*trum
as a T-spectrum by pullback along the quotient, and then building in representa*
*tions (it is
written q# in [18], but more commonly i* by abuse of notation; we shall stick t*
*o the more
descriptive notation)._From our description_of Lewis-May_fixed points it is eas*
*y_to_deduce
inflation on F-contractible_and F-free specta. On F-contractible_spectra Y ' "E*
*F^ TY
we have infTT=KY = "EF ^ TY . If [P ] is the model of the F-spectrum Y then the*
* model
[infTT=KP ] of infTT=KY is the composite
q* __[P]
F -! F -! torsionQ[c] - modules:
In cases where N is Euler-torsion, the right adjoint of the inflation map is al*
*so its left
adjoint; it therefore also gives a model for the topological quotient when X is*
* K-free.
The final chapter of Part II turns to ordinary cohomology and its variants. A*
*fter Eilen-
berg and Steenrod we define a cohomology theory to be ordinary if its coefficie*
*nts are non-
zero only in degree 0, and similarly in homology. For each integer q, an equiva*
*riant coho-
mology theory FG*(.) specifies a contravariant additive functor G=H+ 7-! FGq(G=*
*H+) = FHq
on the stable category of orbits; such a functor is called a Mackey functor. As*
* in the clas-
sical case, ordinary cohomology theories are classified by their non-zero Macke*
*y functor M
in degree 0, and we write H*G(.; M) for this theory and HM for its representing*
* spectrum.
Similarly, for each integer q a homology theory F*G(.) defines a covariant addi*
*tive functor
G=H+ 7-! FqG(G=H+) on the stable category of orbits; such a functor is called a*
* coMackey
functor. Ordinary homology theories are classified by their associated coMackey*
* functors
N, and we write HG*(.; N) for this functor and JN for the representing spectrum*
*. For
finite groups G the stable orbit category is self-dual, so that a coMackey func*
*tor can also
be viewed as a Mackey functor; in this case the ordinary homology theory classi*
*fied by a
8 0. GENERAL INTRODUCTION.
Mackey functor M is also represented by HM. However, for positive dimensional g*
*roups
such as the circle, the functor given by a homology theory cannot usually be vi*
*ewed as a
Mackey functor.
Our first task is to identify objects of the form HM and JN in our model; we *
*find that
they are well behaved but by no means trivial. Finally, whenever one has an in*
*jective
Mackey functor I one may consider the cohomology theory defined by Brown-Comene*
*ntz
I-duality
hIqG(X) = Hom(ss_Gq(X); I);
and its representing spectrum hI. Again, in the case of a finite group all rati*
*onal Mackey
functors are injective, and HM = JM = hM. Indeed, this is the basis of a simple*
* proof
that all rational cohomology theories are ordinary for finite groups. However, *
*for the circle
group the spectrum hI is rather complicated, and in particular it is unbounded;*
* we identify
it exactly in our model.
In Part III we apply the general theory of Parts I and II to several examples*
* of particular
interest. First we answer a number of obvious general questions. To begin with,*
* we relate
the model we have used to the use of Postnikov towers and the use of cells. In *
*fact, we
can understand the Atiyah-Hirzebruch spectral sequence H*T(X; K_*T) =) K*T(X) f*
*or F-free
spectra X completely, in terms of our model. It collapses at the E2 page if an*
*d only if
KT*(EF+) is injective over OF. The latter condition holds for complex K-theory*
*, so we
recover McClure's theorem that the Atiyah-Hirzebruch spectral sequence for the *
*rational
K-theory of an F-space collapses at E2. However, in general there are arbitrar*
*ily long
differentials. The contrast with the simplicity of the one dimensional nature o*
*f the category
of Euler-torsion OF-modules suggests that the Postnikov tower is a poor way to *
*study T-
spectra. On the other hand, because of the simplicity of the graded maps betwee*
*n cells,
we can contemplate homological algebra over it, and it is easy to construct a c*
*onvergent
spectral sequence based on cellular resolutions with a calculable E2 term. Unfo*
*rtunately
the spectral sequence does not appear to be useful in general.
We do not have the means to detect purely unstable phenomena, but the splitti*
*ng the-
orem of Segal and tom Dieck shows that suspension spectra of T-spaces are very *
*special,
and we briefly comment on the implications of this for their algebraic model.
Finally we return to complex K-theory and identify its algebraic model. It i*
*s simple
to describe in terms of representation theory, and is well behaved algebraicall*
*y (`formal'
in the torsion model). However there remain many interesting questions that we*
* have
not treated. Firstly, a qualitative comparison of the F-spectrum Euler classes *
*and the K-
theory Euler classes is sufficient for our purpose, but an exact comparison usi*
*ng the Chern
character, along the lines of Crabb's work [5], would be illuminating. Secondly*
*, it would be
interesting to compare our model with that of Brylinski [3]. Presumably these q*
*uestions
would be useful preparation for the more substantial project of modelling T-equ*
*ivariant
elliptic cohomology as constructed by Grojnowski [8] and Ginzburg-Kapranov-Vase*
*rrot [6].
The other motivating problem was that of understanding the T-equivariant anal*
*ogue of
the Segal conjecture. We had the ironic situation that we understood the harder*
* profinite
part by virtue of work on the Segal conjecture for finite groups, whilst we cou*
*ld not un-
derstand the rational part. Using the model described here, it is now an easy e*
*xercise to
0.2. OVERVIEW 9
identify DET+ in the torsion model as the composite
E-1OF Q E-1OF -! E-1OF -! E-1OF=OF -! Q[c1; c-11]=Q[c1]
where the first map is the product. It is quite instructive to view this as a s*
*pecial case of
the identification of the function spectrum.
Turning to more specialised examples, we reach Tate cohomology theories in th*
*e sense of
[14]. This construction on T-spectra corresponds precisely to Tate cohomology i*
*n commu-
tative algebra in the sense of [10]. Perhaps more interesting is our study of t*
*he integral Tate
spectrum of complex equivariant K-theory. We are able to identify the exact ho*
*motopy
types of both t(KZ) ^ EF+ and t(KZ) ^ "EF and the map q of which t(KZ) is the f*
*ibre:
the first is rational, and identified using our general theory, and the second *
*is formed from
K-theory with suitable coefficients by inflating and smashing with "EF.
Finally we turn to examples gaining their importance from algebraic K-theory.*
* The mo-
tivation for the notion of a cyclotomic spectrum comes from the free loop space*
* X =
map(T; X) on a T-fixed space X. This has the property that if we take K-fixed p*
*oints we
obtain the T=K-space map(T=K; X), and if we identify the circle T with the circ*
*le T=K
by the |K|th root isomorphism we recover X. For spectra one also needs to worry*
* about
the indexing universe, but a cyclotomic spectrum is basically one whose geometr*
*ic fixed
point spectrum K X, regarded as a T-spectrum, is the original T-spectrum X. Aft*
*er the
suspension spectrum of a free loop space, the principal example comes from the *
*topological
Hochschild homology of T HH(F ) of a functor F with smash products. Given such *
*a cyclo-
tomic spectrum X one may construct the topological cyclic spectrum T C(X) of B"*
*okstedt-
Hsiang-Madsen [2], which is a non-equivariant spectrum. An intermediate constru*
*ction of
some interest is the T-spectrum T R(X). Although these constructions are princi*
*pally of
interest profinitely, it is instructive to identify the cyclotomic spectra in o*
*ur model and fol-
low the constructions through. In fact we show that cyclotomic spectra, are tho*
*se spectra
X so that the function [N] : F -! torsionQ[c] - modules modelling EF+ ^ X is co*
*nstant,
and so that the structure map E-1OF V -! N commutes with any translation of the
finite subgroups. It therefore factors through E-1OF V -! (E-1OF)=OF V , and *
*the
map (E-1OF)=OF V -! N is a direct sum of copies of Q[c; c-1]=Q[c] V -! [N](1).
Furthermore, we may recover Goodwillie's theorem that for any cyclotomic spectr*
*um X
we have T C(X) = XhT: topological cyclic cohomology coincides with cyclic cohom*
*ology in
the rational setting.
This summarises the contents of the body. There are also a number of appendic*
*es. Ap-
pendix A gives the structure of rational Mackey functors, and is of independent*
* interest:
in particular the category is of projective and injective dimension 1. Appendi*
*x B gives
Quillen closed model category structure on the algebraic categories. Finally we*
* suggest the
reader glance at Appendix C summarising our conventions. There are also a numb*
*er of
indices.
It is appropriate to comment briefly on reading this document. Formally, Part*
* I is the
basis of all that follows, and is cumulative. Part II consists of an introduct*
*ory chapter,
followed by the treatment of four classes of examples. Since it gives algebraic*
* models of
10 0. GENERAL INTRODUCTION.
topological constructions it must therefore develop the relevant algebra before*
* comparing
it to topology. Thus Chapters 8 and 11 are purely algebraic, and are prerequis*
*ites for
Chapters 9 and 12 respectively. Otherwise the chapters are independent of each*
* other,
but the geometric results depend on Part I. Finally, the chapters of Part III *
*are again
independent, and depend only on Part I and the appropriate results from Part II*
*. We have
made some effort to ensure it is possible for the trusting reader to read a par*
*t without
previously reading its predecessors.
We expect there will be those only interested in Chapters 1 to 3. There may *
*also be
those wanting to gain a feel for the behaviour of certain functors, who may fin*
*d Part II
worthwhile, even without reading Part I. Finally, there may be those who want t*
*o begin
with Part III and read earlier chapters as necessary.
The author is grateful to the Nuffield Foundation for its support, to the Uni*
*versities
of Georgia (Athens) and Chicago for their hospitality, and to the towns of Karl*
*sruhe and
Worms. The author also thanks L.Hesselholt, J.P.May and N.P.Strickland for usef*
*ul com-
ments and conversations.
Part I
The algebraic model of rational T-spectra.
12
CHAPTER 1
Introduction to Part I.
This chapter motivates Part I and provides a map for it. In Section 1.1 we exp*
*lain the
strategy used in Part I to analyse the category of rational T-spectra, and in S*
*ection 1.2
give a brief guide to help readers with particular interests. This is followed *
*in Sections 1.3
and 1.4 by accounts of Haeberly's example and a generalization of McClure's the*
*orem: this
is designed to show there is a need for analysis and some hope of achieving it.
1.1.Outline of the algebraic models.
The main business of Part I is to construct a complete algebraic model of the*
* category
of rational T-spectra. Since spectra represent cohomology theories, this gives *
*a complete
algebraic classification of rational T-equivariant cohomology theories. Having*
* given the
overview in the General Introduction, we concentrate here on the practical appr*
*oach. In
fact, we lead the reader through the investigative process to the algebraic mod*
*el of T-
spectra. This should help explain the how geometric information is packaged in *
*the model,
and how the algebraic model can be used.
The main problem in analyzing T-spectra is to choose basic objects which are *
*easy to
work with and which give theorems of practical use. We explained in the introdu*
*ction that
the building blocks familiar from finite groups of equivariance are not suitabl*
*e: Eilenberg-
MacLane spectra, Moore spectra and Brown-Comenetz spectra form distinct classes*
*. This
means that different methods must be used.
The redeeming feature is that there is no complication at all from representa*
*tion theory
since the Weyl groups are all connected. This means we can return to geometric *
*intuition
and concentrate on isotropy groups. It is appropriate for our present purpose t*
*o think of
T-spectra as generalized stable spaces. It is standard practice in transformat*
*ion groups
to consider various fixed point spaces XH of a space X. In particular, spaces w*
*ith a free
action are especially approachable. One reason for this is that only one subgro*
*up occurs as
an isotropy group. In the rational case the behaviour at each finite subgroup i*
*s reasonably
similar and reasonably simple. Therefore it is common to consider spaces X all *
*of whose
isotropy groups are finite. These are variously called F-spaces, F-free spaces,*
* almost free
spaces, or spaces without fixed points. We shall call them F-spaces, and concen*
*trate on
the fact that they are equivalent to spaces constructed from cells G=H x En wit*
*h H finite.
13
14 1. INTRODUCTION TO PART I.
In any case, our analysis follows this time-honoured pattern, by breaking any*
* object X
into into F-free and F-contractible parts by the isotropy separation cofibering
qX
X -! X ^ "EF -! X ^ EF+:
We thus consider X in two parts: the F-contractible object X(T) = X ^ "EF and t*
*he F-
free object X(F) = X ^ EF+. The object X(T) is determined by its T-homotopy gro*
*ups
as rational vector spaces. The main content of the analysis is therefore in und*
*erstanding
F-objects such as X(F), and how they may be stuck to F-contractible objects X(T*
*). By
use of idempotents in Burnside rings it is easy to see that X(F) splits as a we*
*dge of objects
X(H), one for each finite subgroup H, where only the isotropy group H is releva*
*nt to
X(H). The category of these will be called the category of T-spectra over H and*
* denoted
T-Spec=H ; the mathematical core of the whole enterprise is the analysis of thi*
*s category of
objects X(H). It turns out that ssT*(X(H)) is a torsion module over the ring OH*
* = Q[cH,]
in which cH is an Euler class, and of degree -2, and that the category T-Spec=H*
* of objects
X(H) is equivalent to the derived category of differential graded torsion Q[cHQ*
*]-modules.
The object X(F) is thus determined by the torsion module ssT*(X(F)) over OF = *
*H Q[cH.]
Because we are working rationally it is not difficult to calculate homotopy gro*
*ups of any
precisely described spectrum, so this description is of practical use.
Finally we must determine the assembly map qX : X(T) -! X(F). Note first that
ssT*(X(T)) is not naturally a module over OF, and also that ssT*(qX ) may be ze*
*ro without qX
being zero. The answer is to take into account the twisting available from repr*
*esentations
of T. This twisting is measured by Euler classes, and since there are Thom isom*
*orphisms
for arbitrary F-spectra we may consider the ring E-1OF formed from OF by invert*
*ing all
Euler classes. We denote this ring tF*, sinceLit is in fact the F-Tate cohomolo*
*gyQof S0 in
the sense of [14]. It turns out that tF*is HQ in positive even degrees and H*
* Q in even
degrees 0. By construction, tF*is a OF-module, and qX determines a map
^qX: tF* ssT*(X(T)) -! ssT*(X(F))
in the derived category of differential graded OF-modules. It transpires that ^*
*qXis a com-
plete invariant of qX , so that X is determined by the rational vector space ss*
*T*(X(T)), the
torsion Q[cH-]modules ssT*(X(H)), and the derived OF-map ^qX. Continuing from t*
*his stage,
it is not hard to identify which triples (ssT*(X(T)); ssT*(X(F)); ^qX) occur, a*
*nd to identify the
relevant algebraic triangulated category.
In fact we may consider the torsionLmodel category Atwhose objects are maps t*
*F*V -!
T of OF-modules, T being a sum HT (H) with T (H) a torsion Q[cH ]-module. It *
*turns
out that this category is abelian and of injective dimension 2. One may therefo*
*re consider
differential graded objects in At, and invert homology isomorphisms to form the*
* derived
category DAt . This category is equivalent to the category of rational T-spec*
*tra, and
provides the complete algebraic model we seek. However we prefer not to emphasi*
*ze this
model: the analysis is only possible by introducing a second model, which we c*
*all the
standard model. This proves to be more convenient for most purposes. The real d*
*ifficulty
is that, since At is of dimension 2, it is rather hard to get a precise hold on*
* morphisms
in the derived category. On the other hand the standard model is of dimension *
*1. The
identification of the standard model is the most important result of the analys*
*is.
1.1. OUTLINE OF THE ALGEBRAIC MODELS. 15
It will help to explain the construction of algebraic models for four triangu*
*lated categories
of T-spectra in increasing order of complexity. They are (i) the category of fr*
*ee T-spectra,
or more generally the category T-Spec=H of T-spectra in which only the isotrop*
*y group
H is important, (ii) the category of T-Spec=F of F-spectra, (iii) the category*
* T-Specsf
of semifree T-spectra and (iv) the category of all rational T-spectra. For eac*
*h of these
categories C, we find an abelian category A = A C of dimension 1, and a linear*
*ization
functor ssA*: C -! AC. Because the abelian category AC is so simple in each cas*
*e, it is
possible to reconstruct the original triangulated category C from it. Recall th*
*at the derived
category of an abelian category A is the category formed from the category of d*
*ifferential
graded objects by inverting homology isomorphisms; if A is finite dimensional, *
*the derived
category may be constructed explicitly.
Theorem 1.1.1. If C is one of the above four categories of rational T-spectra*
*, there is
a category A = AC which is abelian and one dimensional so that there is an equi*
*valence of
triangulated categories
C ' DA ;
where DA is the derived category of A. Hence in particular, for any objects X *
*and Y of
C, there is a natural short exact sequence
0 -! ExtA(ssA*(X); ssA*(Y )) -! [X; Y ]T*-! Hom A(ssA*(X); ssA*(Y )) -! 0;
which splits unnaturally.
Before making the theorem explicit for the four categories we make some gener*
*al remarks
about the levels at which the theorem is useful. Firstly, every geometric obje*
*ct X of C
has an algebraic model ssA*(X) and there is a bijection between isomorphism cla*
*sses in
C and isomorphism classes in A . Next, if we know the algebraic models of two *
*objects
X and Y , the short exact sequence allows us to use the algebra of the abelian *
*category
to calculate the group [X; Y ]T*of maps between them. Finally, we may model all*
* primary
constructions (such as formation of cofibres, smash products, function spectra,*
* composition
of functions and calculation of Toda brackets) in the algebraic category. This*
* much is
internal to the category, but in addition, all homotopy functors of T-spectra h*
*ave their
algebraic counterparts. It is very illuminating to identify the algebraic behav*
*iour of various
well known functors.
We now make Theorem 1.1.1 explicit in the four cases.
Theorem 1.1.2. If C = T-Spec=H is the category of T-spectra over H, then A i*
*s the
category of torsion Q[cH ]-modules. The functor ssA*is simply T-equivariant hom*
*otopy ssT*.
This category A is abelian and one dimensional. Accordingly, for two T-spectra *
*X and Y
over H there is a split short exact sequence
0 -! ExtQ[cH](ssT*(X); ssT*(Y )) -! [X; Y ]T*-! Hom Q[cH](ssT*(X); ssT*(Y ))*
* -! 0: __|_|
The proof of this will be completed in Section 4.3. The short exact sequence *
*is Theorem
3.1.1, and it is the central result of the analysis of Part I.
16 1. INTRODUCTION TO PART I.
Theorem 1.1.3. If C = T-Spec=F is the categoryLof F-spectra, then A is the *
*full
subcategory of OF-modules M of the form M = H M(H) for torsion Q[cH ]-modules
M(H). We refer to these as F-finite torsion modules, and they may also be desc*
*ribed
as the OF-modules annihilated by inverting all Euler classes. The functor ssA**
*is simply
T-equivariant homotopy ssT*. The category of F-finite torsion modules is abelia*
*n and one
dimensional. Accordingly, for two F-spectra X and Y there is a split short exac*
*t sequence
0 -! ExtOF(ssT*(X); ssT*(Y )) -! [X; Y ]T*-! Hom OF(ssT*(X); ssT*(Y )) -! 0*
*: __|_|
The proof of this will also be completed in Section 4.3.
Theorem 1.1.4. If C = T-Specsf is the category of semi-free spectra, then A i*
*s the
category whose objects are morphisms M - ! Q[c; c-1] V of Q[c]-modules (for s*
*ome
graded vector space V ) which become isomorphisms when c is inverted. This cate*
*gory A
is abelian and one dimensional. The functor ssA*is defined by
i j
ssA*(X) := ssT*(X ^ DET+) -! ssT*(X ^ DET+ ^ "EF):
Accordingly, for two semifree T-spectra there is a split short exact sequence
0 -! ExtA(ssA*(X); ssA*(Y )) -! [X; Y ]T*-! Hom A(ssA*(X); ssA*(Y )) -! 0:*
* __|_|
Finally the model of all rational T-spectra is as follows.
Theorem 1.1.5. If C = T-Spec then A is the category whose objects are morphis*
*ms
M -! tF* V of OF-modules (for some graded vector space V ) which become isomor-
phisms when all Euler classes are inverted (i.e. the kernel and cokernel are F-*
*finite torsion
modules). This category A is abelian and one dimensional. The functor ssA*is de*
*fined by
i j
ssA*(X) := ssT*(X ^ DEF+) -! ssT*(X ^ DEF+ ^ "EF):
Accordingly, for two T-spectra there is a split short exact sequence
0 -! ExtA(ssA*(X); ssA*(Y )) -! [X; Y ]T*-! Hom A(ssA*(X); ssA*(Y )) -! 0:*
* __|_|
The proof of this is given in Section 5.4. It should be emphasized that Hom A*
*(M; N) and
ExtA (M; N) are routinely computable, and that, because we are working rational*
*ly, there
is usually no serious trouble in calculating ssA*(X).
Part I begins with the concrete and moves towards the abstract in two steps. *
*Thus we
begin with the cohomology theories, move on to homotopy theory, pass to algebra*
* by an
Adams spectral sequence, and finally package this in categorical terms. Here i*
*s a more
detailed outline of contents.
We begin with two sections which can be expressed in classical terms. These g*
*ive evidence
that there is some complexity in rational T-equivariant cohomology theories, bu*
*t not too
much. In particular they give some evidence for the simplicity of F-objects.
1.2. READING GUIDE. 17
After this, the discussion is conducted in the Lewis-May [18] stable category*
* of T-spectra.
The first step is to introduce the basic building blocks and the methods for br*
*eaking general
objects up. This gives us the setting to construct an Adams spectral sequence,*
* which
provides the connection between topology and algebra. Once the Adams spectral s*
*equence
for T-Spec=H has been constructed we need only do some algebra and certain for*
*mal
manipulations to obtain and exploit all the algebraic models. We have taken the*
* view that
an abstract machine should only be introduced when there is a particular case o*
*n which
its operation can be illustrated. Accordingly we have not described the transit*
*ion from an
Adams spectral sequence to an algebraic model (in Section 4.2) until we have co*
*nstructed
the simplest instance to which it applies. On the other hand Section 4.2 may be*
* relevant in
quite different settings, and it is written axiomatically so that it can be rea*
*d and applied
independently of the preceding sections.
Once the general analysis is completed we consider standard T-spectra and con*
*structions
on T-spectra in Part II. In Part III we consider in more detail certain example*
*s of estab-
lished interest. More detailed accounts of the contents of Parts II and III may*
* be found in
their introductions.
1.2. Reading Guide.
Some readers may not wish to read all of the material in Part I, so we provid*
*e further
guidance here.
Those only interested in the Atiyah-Hirzebruch spectral sequence for the K-th*
*eory of
an F-space will only need to read Sections 1.3, 1.4, 2.1, referring to Appendix*
* A for the
necessary facts about Mackey functors. Sections 1.3 and 1.4 are not used elsewh*
*ere in Part
I. We shall return to the Atiyah-Hirzebruch spectral sequence in Section 15.1 o*
*f Part III,
where we give more complete results.
Those interested in Mackey functors should read Section 2.1 and then refer to*
* Appendix
A. Mackey functors are not used until we consider ordinary cohomology theories *
*in Chapter
13 from Part II.
The central material constructing the main Adams spectral sequence for the ca*
*tegories
of F-spectra and T-spectra over H is to be found in Chapters 2 and 3. Maps fro*
*m F-
contractible spectra to F-free spectra are deduced in Sections 5.1 and 5.2. Thi*
*s is sufficient
to answer most direct questions about particular T-spectra, and may satisfy som*
*e readers.
On the other hand readers wishing to understand the shape of the algebraic mode*
*ls without
reading these chapters.
In Chapter 4, we explain the abstract process of reaching an algebraic model *
*from an
Adams spectral sequence and we illustrate it for T-spectra over H. However the *
*goal of a
full algebraic model is fulfilled in Chapter 5. We deduce the remaining topolog*
*ical input
from the Adams spectral sequence in Sections 5.1 and 5.2, and construct the alg*
*ebraic
model in Section 5.3. It is then a simple matter to show in Section 5.4 that th*
*e algebra
does indeed model the topology. Chapter 6 completes the circle by introducing t*
*he torsion
model, closely following geometric intuition, and by showing that it gives a mo*
*del equivalent
to the standard model.
18 1. INTRODUCTION TO PART I.
1.3. Haeberly's example.
We give Haeberly's example [16] showing there is no Chern character isomorphi*
*sm, for T-
equivariant K-theory. This simply involves constructing a T-space X whose equiv*
*ariant K-
theory is concentrated in even degrees, but whose ordinary cohomology with coef*
*ficients in
the rationalized representation ring functor is nonzero in odd degrees. Since t*
*he homotopy
functors of the K-theory spectrum are in even degrees the K-theory cannot be a *
*product
of copies of ordinary cohomology. In the next section we give a proof of a gene*
*ralization of
McClure's result that there is a Chern isomorphism for T-spaces X with XT trivi*
*al.
To explain Haeberly's example it is convenient to consider the group = T x T*
*0where
both T and T0are copies of the circle group. The group has a 3-dimensional com*
*plex
representation V = (1 t t2) t0, where t is the natural representation of T o*
*n C, and
similarly for T0. We may consider the unit sphere S(V ) as a -space, give it a*
* disjoint
basepoint and then form the T-space X = S(V )+=T0. We could equally well descri*
*be X
as a copy of CP+2on which T acts via s(z0 : z1 : z2) = (z0 : sz1 : s2z2). From*
* the first
description it is easy to calculate the K-theory since we have K*T(X) = K*(S(V *
*)+), because
S(V ) is free as a T0-space. Indeed, the cofibre sequence S(V )+ -! S0 -! SV of*
* -spaces
gives an exact sequence
(V )i 0 i i+1 V
. .-.! Ki(SV ) -! K (S ) -! K (S(V )+) -! K (S ) -! . .:.
Now by Bott periodicity Ki(SV ) is R() if i is even and 0 if i is odd, and beca*
*use the degree
0 Euler class (V ) = (1-t0)(1-tt0)(1-t2t0) is not a zero divisor in R() = Z[t; *
*t-1; t0; (t0)-1]
we find
K0T(X) = R()=(V ) andK1T(X) = 0:
In particular the K-theory of X is entirely in even degrees.
On the other hand from the second description2it is not hard to see that X ha*
*s isotropy
groups T; C2 and 1. Furthermore XC2 = (St _S0)+ and X may be given a T-CW struc*
*ture
with two free 1-cells, one free 2-cell and one free 3-cell. Hence for any Macke*
*y functor M
we see that H*T(X; M) is the cohomology of a complex
0 d1 d2
3M(T) -d!M(C2) 2M(1) -! M(1) -! M(1);
and it is easy to see that d1 is surjective. Thus H3(X; M) = M(1), and in parti*
*cular if M
is the rationalized representation ring Mackey functor this is the non-zero gro*
*up Q.
1.4. McClure's Chern character isomorphism for F-spaces.
McClure has observed that the if X is an F-space then the Atiyah-Hirzebruch s*
*pectral
sequence for the K-cohomology of X does collapse at E2. His proof involves appe*
*aling to
unstable results and the work of Slominska. We shall give a proof of the corre*
*sponding
statement for any cohomology theory whose homotopy functors are concentrated en*
*tirely
in even degrees, and of the corresponding statement for homology theories. Of c*
*ourse this
applies in particular to K theory, by the Bott periodicity theorem. In Section *
*15.1 of Part 3
we shall give a necessary and sufficient condition for the collapse of the Atiy*
*ah-Hirzebruch
spectral sequence for F-spaces, which will give an alternative to the proof of *
*this section.
1.4. MCCLURE'S CHERN CHARACTER ISOMORPHISM FOR F-SPACES. 19
Before stating the theorem, we recall that for each integer k it is appropria*
*te to consider
the entire system of homotopy groups ssHk(X) = [G=H+ ^ Sk; X]T as H runs through
all subgroups of T. It is appropriate to regard this as a functor ss_Tk(X) : G*
*=H+ 7-!
[G=H+ ^Sk; X]T, on the category of stable orbits. An additive functor of this f*
*orm is called
a Mackey functor; we examine the algebraic structure of the category of rationa*
*l Mackey
functors in Appendix A, but for the present we only need the basic terminology.*
* In line
with the usual abbreviation we write the coefficient functor ss_Tk(K) as K_Tk.
Since the orbits are the equivariant analogues of points, an ordinary cohomol*
*ogy theory
is one for which the cohomology of each orbit is concentrated in degree zero. T*
*hus ordi-
nary cohomology theories correspend to Mackey functors M, and they are represen*
*ted by
Eilenberg-MacLane spectra HM.
Theorem 1.4.1. If K is any rational T-spectrum with homotopy functors K_Tm= 0*
* for
all odd integers m then for any F-space X there are isomorphisms
(a)
Y T
K*T(X) ~= H*T(2nX; K_-2n)
n2Z
and
(b)
M T
KT*(X) ~= HT*(2nX; K_2n):
n2Z
This follows from a geometric statement.
Theorem 1.4.2. If K is any rational T-spectrum with homotopy functors K_Tm= 0*
* for
all odd integers m then
(a)
Y T
F (EF+; K) ' F (EF+; 2nH(K_2n))
n2Z
and
(b)
_ T
K ^ EF+ ' EF+ ^ 2nH(K_2n):
n2Z
To see how Theorem 1.4.1 follows from 1.4.2 we use a lemma which is immediate*
* from
the definition of EF+ and its unreduced suspension "EF.
Lemma 1.4.3.For any F-spectrum X,
(a) X ^ "EF ' * and hence X ' EF+ ^ X; also
(b) for any T-spectrum Y we have F (X; Y ^ "EF) ' * and hence F (X; Y ^ EF+) '
F (X; Y ). __|_|
By 1.4.3 (a), Theorem 1.4.1 follows by applying F (X; ) to Part (a) of 1.4.2 *
*and X^ to
Part (b) of 1.4.2 and taking homotopy groups.
20 1. INTRODUCTION TO PART I.
Proof: We turn to the proof of 1.4.2. Note first that it is enough to prove Par*
*t (b); indeed,
by 1.4.3 (b), Part (a) follows by applying F (EF+; .) toWthe equivalence of Par*
*t (b).
It is enough to construct a T-map : K ^ EF+ -! EF+ ^ n2Z2nH(K_2n) which is
an H-equivalence for all finite subgroups H. By the Whitehead theorem it is su*
*fficient
that induces an isomorphism of ssH*for all finite subgroups H. By 1.4.3 (b) ag*
*ain, it is
equivalent to give the composite
_
0: K ^ EF+ -! 2nH(K_2n);
n2Z
and since this wedge is equivalent to the product we may specify 0by giving its*
* components.
These are elements of the cohomology groups [K ^ EF+; HM]*T= H*T(K ^ EF+; M) for
various Mackey functors M. Accordingly we set about calculating the cohomology*
* of
K ^ EF+.
The idea is to filter EF+ so that the subquotients are analogues of cells, bu*
*t with all
elements of finite order as isotropy groups. This extends the idea of [9]. Thus*
* we note that
if H L we have a projection T=H -! T=L, and that the subgroups of finite order*
* form a
directed set. We may therefore let T==F+:= holim!T=H+ where the limit is over a*
*ll finite
H
subgroups H (or over a cofinal sequence if that appears more comfortable). Anal*
*ogously,
if H is a finite subgroup of order n we may let V (H) denote the representation*
* tn with
kernel H, and there are maps mV (H) -! mV (L) (of degree |L=H|m ) for all m. We*
* let
S(mV (F))+ := holim!S(mV (H))+ for 0 m 1. The usefulness of these constructio*
*ns
H
is summarized in a lemma.
Lemma 1.4.4. The infinite sphere S(1V (F))+ is a model for EF+. We thus have*
* a
filtration
* = S(0V (F))+ S(1V (F))+ S(2V (F))+ . . .S(1V (F))+= EF+
and the subquotients are generalized cells
S(mV (F))+=S((m - 1)V (F))+' S2m-2^ T==F+
for 1 m < 1.
Proof: Since (S(mV (H)))L = ; if L 6 H or S(mV (H)) if L H the fact that S(1V *
*(F))+
is a universal space is clear. To identify the quotients we use the fact that *
*the cofibre
sequences
S((m - 1)V (H))+ -! S(mV (H))+ -! S2m-2^ T=H+
fit into a direct system. __|_|
In other words we have
EF+ = T==F+ [ T==F+ ^ e2 [ T==F+ ^ e4 [ T==F+ ^ e6 [ . .:.
Thus, for any spectrum K, we may form the spectral sequence of the filtered s*
*pectrum
K ^ EF+ which will have the form
Es;t1= Hs+tT(K ^ (EF(s)+=EF(s-1)+); M) ) Hs+tT(K ^ EF+; M):
1.4. MCCLURE'S CHERN CHARACTER ISOMORPHISM FOR F-SPACES. 21
Indeed, from the form of the filtration, we find the spectral sequence is conce*
*ntrated in the
first quadrant in terms with even s where we have
E2m;t1= HtT(K ^ T==F+; M):
Of course, using the change of groups isomorphism H*T(K ^ T=H+; M) = H*H(K; M),*
* we
have a Milnor exact sequence
0 -! lim1Ht-1H(K; M) -! HtT(K ^ T==F+; M) -! lim HtH(K; M) -! 0:
H H
It is in the analysis of this exact sequence that it is essential we are workin*
*g rationally.
Indeed, because H is finite, every rational H-spectrum is a product of Eilenber*
*g-MacLane
spectra and these are necessarily also Moore spectra. It now follows that, prov*
*ided K has
its homotopy functors in even degrees, the groups HtH(K; M) are only nonzero fo*
*r even t.
The collapse of the spectral sequence is thus ensured once we show the lim1term*
*s vanish.
In fact the restriction maps
HtL(K; M) -! HtH(K; M)
are surjective. Perhaps the quickest way to see this is to note that H*H(HM0; *
*M) =
[HM0; HM]*H= HomH (M0; M), for any Mackey functors M0and M. We may then use the
corresponding fact for Mackey functors, that
HomL(M0; M) -! HomH (M0; M)
is surjective. This surjectivity is due to the fact that all Weyl groups are co*
*nnected, and it
is easily deduced from Appendix A.
We conclude that if K has all its homotopy functors in even degrees then
Y
H*T(K ^ EF+; M) = lim H*H(2mK; M);
m2Z H
and in particular we can find a map
02m: K ^ EF+ -! 2mH(K_T2m)
inducing the identity in ssH2m(o) for all finite subgroups H. The map
_ T
0: K ^ EF+ -! 2nH(K_2n)
n2Z
is thus an F-equivalence and hence is a homotopy equivalence as required. __|*
*_|
In Section 15.1 of Part III we shall complete the picture of Atitiyah-Hirzebr*
*uch spectral
sequences for F-spaces by giving an analysis without hypothesis on the rational*
* cohomol-
ogy theory. We characterize those theories K*T(.) for which the spectral sequen*
*ce always
collapses at E2, show that arbitrarily high differentials occur, and give a geo*
*metric expla-
nation of them in terms of universal examples. The behaviour of the spectral se*
*quence for
arbitrary spaces X is much more complicated.