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
160
Part III
Applications.
162
CHAPTER 14
Introduction to Part III.
14.1.General Outline
The material in Part III is a collection of applications of the general theor*
*y developed in
Parts I and II. Accordingly, the chapters are completely independent of each ot*
*her. Since
most of the work has already been incorporated in the general framework, the se*
*ctions are
rather short, and uncluttered by technicalities.
We begin with Chapter 15, which consists of five independent sections answeri*
*ng the
main questions motivating this study. Perhaps the most obvious problem of all, *
*in the light
of Haeberly's example, is to understand the behaviour of the Atiyah-Hirzebruch *
*spectral
sequence. We show that it collapses at E2 for all F-spaces if and only if ssT*(*
*K ^ EF+) is
injective. In general, the differentials encode the Adams short exact sequence,*
* but there
are differentials of arbitrary length. For arbitrary spaces, the main message *
*is that the
Atiyah-Hirzebruch spectral sequence is not a natural tool. Alternatively, we ca*
*n always use
a cellular decomposition of the domain to try to understand maps. In our case, *
*the graded
orbit category can be made quite explicit, and we can therefore do homological *
*algebra
over the category of additive functors on this category. We may then construct *
*a spectral
sequence, whose E2 term is calculable and given by homological algebra over the*
* graded
orbit category. It is also obviously convergent, but it does not seem a practi*
*cal tool in
general, because it is usually a half-plane spectral sequence. The moral is tha*
*t we should
not decompose spectra by Eilenberg-MacLane spectra or by cells, but rather by i*
*njectives
in the standard model.
Another basic construction of spectra is by taking the suspension spectrum of*
* a space.
Equivariantly it is known that suspension spectra have certain special properti*
*es, such as
tom Dieck splitting, which are not enjoyed by all spectra. In particular, the i*
*nclusion of
the H-fixed points can be regarded as a map of T-spaces, and this has implicati*
*ons for the
model. However, since our model has no record of purely unstable information, w*
*e cannot
hope for a precise characterization of suspension spectra.
The special case of K-theory is interesting because it has Bott periodicity, *
*and hence
Euler classes of its own. The representing spectrum also turns out to be forma*
*l in the
torsion model, so that the K-theory of any spectrum depends only on its homolog*
*y in the
torsion model. On the other hand, our analysis is really only the beginning of*
* a study
163
164 14. INTRODUCTION TO PART III.
melding the formalisms of the present work with the geometric information of K-*
*theory:
in particular it would be interesting to understand the Chern character in more*
* detail and
to relate our results to the work of Brylinski and collaborators [3].
After our analysis of function spectra, the rational Segal conjecture is littl*
*e more than an
elementary example: in the torsion model DET+ is formal, and represented by the*
* natural
map
tF* tF*-! tF*-! tF*=OF = 2I -! Q[c; c-1]=Q[c]:
We also present a more naive approach by way of comparison.
In Chapter 16 we consider the well known T-equivariant cohomology theory given*
* by
cyclic cohomology. This is very simple rationally; more generally rational Tate*
* spectra are
also rather simple, but we may make certain intriguing algebraic connections. F*
*inally we
are able to identify the integral Tate spectrum t(KZ) of integral complex K-the*
*ory KZ.
This is of interest because t(KZ) is known to be H-equivariantly rational for a*
*ll finite
subgroups H. In fact, writing KZ for integral complex K-theory for emphasis, we*
* identify
the spectra t(KZ) ^ "EF and t(KZ) ^ EF+ together with the map of which t(KZ) is
the fibre. The first is obtained from K-theory with suitable coefficients by i*
*nflating and
smashing with "EF, and the second is rational, and we can identify it as an inj*
*ective Euler-
torsion OF-module.
The final Chapter 17 is more substantial. We turn to examples gaining their im*
*portance
from algebraic K-theory. B"okstedt, Hsiang and Madsen define the topological c*
*yclic co-
homology of a ring or a space [2]. It is obtained by performing various constru*
*ctions on
the topologicial Hochschild homology spectrum and can be used as a close approx*
*imation
to the completed algebraic K-theory of suitable rings. Hesselholt and Madsen [1*
*7] identify
the structure required of a T-spectrum X for one to be able to construct T C(X)*
*. Spec-
tra X with this structure are called cyclotomic spectra. The motivation for th*
*e notion
of a cyclotomic spectrum comes from the free loop space X = map(T; X) on a T-fi*
*xed
space X. This has the property that if we take K-fixed points we obtain the T=K*
*-space
map(T=K; X), and if we identify the circle T with the circle T=K by the |K|th r*
*oot isomor-
phism we recover X. For spectra, one also needs to worry about the indexing uni*
*verse,
but a cyclotomic spectrum is basically one whose geometric fixed point spectrum*
* K X,
regarded as a T-spectrum, is the original T-spectrum X. After the suspension sp*
*ectrum 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 cyclotomic spectru*
*m X,
one may construct the topological cyclic spectrum T C(X) of B"okstedt-Hsiang-Ma*
*dsen [2],
which is a non-equivariant spectrum. An intermediate construction of some inter*
*est is the
T-spectrum T R(X). Although these constructions are principally of interest pro*
*finitely, it
is instructive to identify the cyclotomic spectra in our model and follow the c*
*onstructions
through. We identify the rational cyclotomic spectra in the torsion model: they*
* are the
spectra X so that the function [N] : F -! torsionQ[c] - modules modelling EF+ ^*
* X is
constant, and so that the structure map tF* V - ! N commutes with any translati*
*on
of the finite subgroups. It therefore factors through tF* V -! tF*=OF V and th*
*e map
OF=OF V - ! N is a direct sum of copies of Q[c; c-1]=Q[c] V - ! [N](1). Fur-
thermore, we may recover Goodwillie's theorem [7] that for any cyclotomic spect*
*rum X
14.2. PROSPECTS AND PROBLEMS. 165
we have T C(X) = XhT: topological cyclic cohomology coincides with cyclic cohom*
*ology in
the rational setting.
14.2. Prospects and problems.
The main theoretical problem is to show that the equivalence of the category *
*of T-spectra
and the derived category of the standard model can be obtained from a chain of *
*equivalences
arising from adjoint pairs of functors on underlying Quillen model categories. *
*This would
inevitably be linked with a better understanding of the meaning of the standard*
* model.
One of the most interesting prospective applications is that of understanding*
* rational
T-equivariant elliptic cohomology. Constructions have recently been given by Gr*
*ojnowski
[8] and by Ginzburg-Kapranov-Vasserot [6], and the cohomology of any T-space is*
* a sheaf
over an elliptic curve. One can ask if these theories are represented. This wou*
*ld involve
considering sheaves of T-spectra over an elliptic curve, and it would seem a se*
*nsible first
step to consider sheaves of objects of A.
Both in this case and that of K-theory, there is the task of relating the gen*
*eral model
to the geometry of the cohomology theory: in practice this will involve concent*
*ration on
the Chern character, and comparison with the work of Brylinski [3]. There are a*
* number
of other classes of spectra which we do not understand as well as we should lik*
*e, such as
suspension spectra, free loop spaces, THH, ......
In the present work we have concentrated entirely on the circle group T. Alt*
*hough it
is unlikely to be possible to give so complete a picture as we have done for T-*
*spectra,
we hope to consider other small groups in due course. The continuous quaternio*
*n and
dihedral groups are prime candidates, both by virtue of their simplicity and th*
*e prospects
for applications. However, consideration of the case of Mackey functors [11] sh*
*ows that it
is necessary to replace the underlying algebra of OF-modules by that of sheaves*
* over spaces
of subgroups, since the topology on the space of subgroups can no longer be ign*
*ored. For
groups of rank greater than 1, the injective dimension of the relevant algebrai*
*c categories
will be greater than 1. There is therefore no prospect of obtaining splittings*
* for formal
reasons, and models must be based on a more complete geometric understanding th*
*an we
have used here.
166 14. INTRODUCTION TO PART III.
CHAPTER 15
Classical miscellany.
The sections in this chapter are independent of each other; each answers a natu*
*ral question
about rational T-spectra.
In Section 15.1 we give a complete analysis of the behaviour of the Atiyah-Hi*
*rzebruch
spectral sequence for F-spectra, generalising the study in Section 1.4. Sectio*
*n 15.2 sets
up a calculable spectral sequence for calculating maps between T-spectra from a*
* cellular
decomposition, based on the graded orbit category. Section 15.3 shows how the e*
*xistence of
tom Dieck splitting makes the models of suspension spectra very special. Sectio*
*n 15.4 finally
returns to complex T-equivariant K-theory, and identifies its place in the tors*
*ion model,
showing that it is formal. Finally, Section 15.5 identifies the functional dual*
* DET+, giving
the rational analogue of the geometric equivariant Segal conjecture.
15.1.The collapse of the Atiyah-Hirzebruch spectral sequence.
The purpose of this section is to analyse the Atiyah-Hirzebruch spectral sequ*
*ence for
F-spectra.
We suppose given an arbitrary T-equivariant cohomology theory K*T(.), and con*
*sider the
Atiyah-Hirzebruch spectral sequence
Es;t2= HsT(X; K_tT) =) K*T(X):
This may be constructed either by using the skeletal filtration of X or, in com*
*plete gener-
ality, by using the Postnikov filtration of K. It is conditionally convergent i*
*f X is bounded
below.
First, let us suppose that K = ET(2n)+; we observe that, if X is free, the on*
*ly relevant
part of the Mackey functor K_*Tis the nonequivariant homotopy of K. Thus the sp*
*ectral
sequence is concentrated on the lines q = 1 and q = 2n + 2. There are therefor*
*e no
differentials except
d2n+2: Hp(X=T) = HpT(X; K_2n+2T) -! Hp+2n+2T(X; K_1T) = Hp+2n+2(X=T):
If we take the special case X = ET(2m)+, we see that the differential must be g*
*iven by
multiplication by cn+1 in order to give the correct answer, as calculated by th*
*e Adams
short exact sequence. In particular, the differential is non-zero if and only i*
*f m > n.
167
168 15. CLASSICAL MISCELLANY.
Theorem 15.1.1. If X is an F-spectrum then the Atiyah-Hirzebruch spectral coll*
*apses
at E2 if ssT*(K) is injective over OF. Conversely, if the Atiyah-Hirzebruch Spe*
*ctral sequence
collapses at E2 for all F-spectra X, then ssT*(K) is injective.
More precisely, any differential d2i+1is zero, and the nonzero differentials d*
*2n+2are all
explained by the above example, in a sense to be made precise in the proof.
Proof: The first observation is that if X is an F-spectrum then K*T(X) = [X; K]*
**T=
[X; K ^ EF+]*T. Since the spectral sequence is natural, we may suppose that bot*
*h X and
K ^ EF+ have homotopy in even degrees. We argue that if ssT*(K) is injective th*
*en the E2
term is entirely in even total degrees, and hence the spectral sequence collaps*
*es.
First note that for any F-spectrum T ,
K*T(T ) = Hom(ssT*(T ); ssT*(K ^ EF+)):
Thus, taking T = oe0H, we see that, for any finite subgroup H, the H-equivarian*
*t basic
homotopy groups are purely in odd degrees, since ssT*(oe0H) = Q is odd. Thus t*
*he part
of the graded Mackey functor K_*Tover F is entirely in odd degrees. On the othe*
*r hand,
since X is an F-spectrum, [X; HM]*T= [X; HM ^ EF+]*T, and HM ^ EF+ is a wedge of
copies of E,Lwith one factor for each basis element of V (H) = eH M(H). Thus*
*, if we
let I M = HI(H) V (H) we have
H*T(X; M) = Hom(ssT*(X); I M);
which is entirely in odd degrees. Thus E*;*2= H*T(X; K_*T) is in even total deg*
*rees as claimed.
Now suppose that the Atiyah-Hirzebruch spectral sequence does not collapse, an*
*d that
x 2 Ep;qrsupports a non-zero differential dr(x) = y 6= 0. We shall prove that r*
* = 2n + 2 for
some n, and that the differential is explained by naturality and the differenti*
*als described
above.
We may pick a representative x02 Ep;q1= [X(p)=X(p-1); K]p+qTfor x. This shows*
* that
x0is supported on a map oepH- ! X=X(p-1)-! p+qK. Replacing X by X=X(p-1)and
suspending, we may assume that X is (-1)-connected and p = 0. We may therefore *
*replace
K by its connective cover K10without changing the fate of x in the spectral seq*
*uence. Now,
letting Knmdenote the Postnikov section of K with non-zero homotopy groups in d*
*egree i
with m i n, we consider the Postnikov tower of K:
Kr-1r-1-! Kr-10 -! Krr
#
Kr-2r-2-! Kr-20 -! Kr-1r-1
#
K22 -! K20 -! K33
#
K11 -! K10 -! K22
#
x
X - - -! K00 -! K11
By hypothesis, x : X -! K00lifts to x(r): X -! Kr-20so that the composite x(r):*
* X -!
Kr-20-! Kr-1r-1is essential and represents y. Since X is an F-spectrum, the beh*
*aviour
15.2. ORBIT CATEGORY RESOLUTIONS. 169
is unaltered if the diagram is smashed with EF+. Since HM ^ EF+ is injective, a*
*ll maps
at the E2-term are detected by their d-invariant, and it is thus appropriate to*
* examine the
effect of taking homotopy of the above diagram smashed with EF+. The basic obse*
*rvation
is that,
ssT*(HM ^ EF+) = I M;
which is in odd degrees. Furthermore, x is detected by ssT1, and we need only l*
*ook at the odd
graded part. This immediately shows that all odd differentials are zero, so tha*
*t r = 2n + 2
for some n.
Next, we note that the maps
ssT*(K2s+10^ EF+) -! ssT*(K2s0^ EF+)
are injective in odd degrees, whilst the maps
ssT*(K2s+20^ EF+) -! ssT*(K2s+10^ EF+)
are surjective in odd degrees. Furthermore, the image of the composite consists*
* of elements
divisible by c. We thus find the diagram
ssT*(K2n0^ EF+) -! ssT*(K2n+12n+1^ EF+) = 2n+3I K_2n*
*+1T
% #
ssT*(X)-! ssT*(K00^ EF+) = I K_0T ;
and we know that some element "zof ssT2n+3(X) maps to z 2 ssT2n+3(K2n0^ EF+), a*
*nd cn+1z
detects x, whilst the image of z in ssT2n+3(K2n+12n+1^ EF+) detects y. We there*
*fore find a
map
ssT*(E(2n+2)) = I2n+20-! ssT*(X)
with "zas the image of the top class, and "xas the image of the bottom class. B*
*y the Adams
short exact sequence, this is realised by a map E(2n+2)-! X. __|_|
15.2. Orbit category resolutions.
Integrally one expects the cellular decomposition to be unhelpful in global c*
*alculations
because one does not know the stable homotopy groups of spheres. Rationally, ev*
*erything
is much simpler. For finite groups, cells are Eilenberg-MacLane spectra, and h*
*ence the
cellular decomposition is simply another way of viewing the complete splitting *
*[14]. In the
present T-equivariant context, cells are not all Eilenberg-MacLane spectra, but*
* one may
understand the entire graded category of natural or basic cells. We shall conce*
*ntrate on the
graded category hSB* of basic cells (i.e. the full subcategory of the graded st*
*able category
with the basic cells as objects).
One thus views the entire homotopy functor ss_T*(X) of X as a module over the*
* graded
category hSB*. By the Yoneda lemma, the case when X is a cell plays the role of*
* a free
object, and a resolution is form of cellular approximation. We understand maps *
*of degree 0
from the discussion of Mackey functors presented in Appendix A. Referring to 2.*
*1.4, we see
that composition in hSB* is usually zero for dimensional reasons. In fact, ther*
*e are no maps
of degree 2 between any pair of objects, and the only case with maps of degree *
*more than 1
170 15. CLASSICAL MISCELLANY.
is [oe0T; oe0T]T*= Q(QF[c-1]). For maps of degree 1 we have [oe0T; oe0T]T1= QF,*
* [oe0T; oe0H]T1= Q,
[oe0H; oe0T]T1= 0 and [oe0H; oe0H]T1= Q; for maps of degree 0 we have [oe0T; oe*
*0T]T0= Q, [oe0T; oe0H]T0= 0,
[oe0H; oe0T]T0= Q and [oe0H; oe0H]T0= Q. It therefore remains to deal with the*
* composite of a
degree 1 morphism (which must be of form xTT: oe1T-! oe0T, xHH: oe1H-! oe0Hor a*
* multiple
of the transfer trTH: oe1T-! oe0H) and a degree 0 morphism (which is either a m*
*ultiple of
the relevant identity or a multiple of the projection prHT: oe0H-! oe0T). We no*
*w deal with
these remaining cases.
Lemma 15.2.1. The composites of the x's and y's are as follows.
(i) xTTprHT= 0 and prHTxHH= 0,
(ii) trTHprHT 6= 0 and prHTtrTHcorresponds to the inclusion of the Hth factor i*
*n QF =
[oe0T; oe0T]T1.
Proof: Part (i) is clear since the composites lie in the zero group. The first *
*fact in Part (ii)
follows from the explicit geometric construction of the transfer. The second fa*
*ct in Part
(ii) follows by construction of the isomorphism in tom Dieck splitting [18, V.1*
*1]. __|_|
It is thus natural to take oH = trTHprHTas the basic generator of [oe0H; oe0H]*
*T1and ffiH =
prHTtrTHas a standard basis element of [oe0T; oe0T]T1.
Now, suppose given any T-spectrum X, we consider [.; X]T*as a contravariant fu*
*nctor on
the graded orbit category. As such, we may form a projective resolution, and we*
* may realise
it. In fact we may construct a map P0 -! X, which is surjective in graded equiv*
*ariant
homotopy for all subgroups of T, and in which P0 is a wedge of cells. Now let X*
*1 be the
cofibre of this, and iterate to form the diagram
X = X0 -! X1 -! X2 -! . . .
" " "
P0 P1 P2
By construction, all the maps Xs -! Xs+1 are zero in H-equivariant homotopy for*
* all
subgroups H, and so holim!Xs is contractible by the Whitehead theorem.
s
It is convenient to form the dual diagram with Xp-1 = fibre(X -! Xp), so that *
*X-1 = *
and X ' holim!Xp:
* = X-1 -! X0 -! X1 -! . . .
" " "
-1P0 -1P1 -1P2
Replacing the maps by inclusions, we view this as a filtration of X with subquo*
*tients
Xp=Xp-1 = Pp. We may now construct a spectral sequence by applying [.; Y ]*Tto*
* the
diagram. It has Ep;q1= [Pp; Y ]p+qT, and Dp;q1= [Xp; Y ]p+qT. The spectral sequ*
*ence lies in the
right half-plane, and the differentials are cohomological, so that dr : Ep;qr-!*
* Ep+r;q-r+1r,
and it is evidently conditionally convergent. Finally, we see by construction t*
*hat the se-
quence
. .-.! ss_T*(-2P2) -! ss_T*(-1P1) -! ss_T*(P0) -! ss_T*(X) -! 0
15.3. SUSPENSION SPECTRA. 171
is exact. Hence we may identify the E2 term as an Ext group, and the spectral s*
*equence
takes the form
Ep;q2= Extp;qhSB(ss_T*(X); ss_T*(Y )) =) [X; Y ]p+qT:
*
The fact that the category of Mackey functors is one dimensional gives a vani*
*shing line if
X is bounded below. Indeed, since cells are also (-1)-connected we may ensure P*
*0 is (-1)-
connected, and hence that X1 is 0-connected. This formality proves that if the *
*resolution
is dimensionally minimal, Xp is (p - 1)-connected. However, if we use basic cel*
*ls and the
fact that the category of Mackey functors is of projective dimension 1, we may *
*ensure that
X2 is 2-connected. By iteration we see that X2sand P2sare (3s - 1)-connected; s*
*imilarly
X2s+1and P2s+1are 3s-connected. Thus the map X2s-1-! X is (3s - 2)-connected, a*
*nd
the map X2s -! X is (3s - 1)-connected. Unfortunately this only seems to be use*
*ful if
the homotopy of the spectrum Y is bounded above. Thus if YHq= 0 for q -1 and a*
*ll H,
then the nonzero entries are in the first quadrant and lie on or above the line*
* q = (p - 1)=2.
15.3.Suspension spectra.
In this section we suppose given a based T-space Z, and we identify the place*
* of its
suspension spectrum in our classification. We follow our usual convention of o*
*mitting
notation for the suspension spectrum functor, and using the notation T for Lewi*
*s-May
fixed point functor.
Our basic tool is tom Dieck splitting, which states that the Lewis-May fixed *
*points of
the suspension spectrum of Z is
_
TZ = Z _ ET=K+ ^T=K ZK ;
K
furthermore this is natural, and applies to stable retracts of spaces. The cruc*
*ial simplifica-
tion for spaces is that there is a map ZT -! Z of T-spectra, and hence a diagram
"EF ^ ZT - ! EF+ ^ ZT
'# #
E"F ^ Z - ! EF+ ^ Z:
The structure map "EF ^ Z -! EF+ ^ Z thus factors through the corresponding map
for ZT, which we understand completely, since ZT is rationally a wedge of spher*
*es.
On the other hand, by naturality of tom Dieck splitting, we find
ssT*(Z ^ E) = ss*(ET=H+ ^T=H ZH ) = H*(ET=H+ ^T=H ZH );
which we may certainly regard as computable.
Summary 15.3.1. The algebraic model of the suspension spectrum of a space Z is
M M
tF*H*(ZT) -! 2IH*(ZT) = 2 H*(ET=H+^T=HZT) -! 2 H*(ET=H+^T=HZH ):
H H
172 15. CLASSICAL MISCELLANY.
The first map in the diplayed composite necessarily has zero e-invariant, and *
*is simply
induced by the quotient tF*-! 2I. However the second map is induced by the incl*
*usion
XT -! X, and may have non-zero d and e invariant. Again we may be satisfied th*
*at
the d-invariant is given by a homology calculation. For the e invariant, since*
* the above
discussion applies to retracts of spaces, we may assume that XT and EF+ ^ X are*
* of pure
parity, and then identify the e invariant with the Borel homology of the cofibr*
*es XH =XT
as in [1].
There remains the question of whether this characterizes suspension spectra. M*
*ore pre-
cisely, we cannot distinguish T-fixed suspensions so that we are asking if ever*
*y model of
the above sort is the model of a spectrum n1 X for some T-space X and some inte*
*ger
n. We do not have available the option of identifying the suspension spectrum *
*functor,
since there is no algebraic model of rational T-spaces. One might hope to reali*
*ze finitely
generated examples by explicit construction, but one would expect a certain amo*
*unt of
suspension to be necessary to achieve stability in each case. To obtain a globa*
*l realization
one would need a bound on these suspensions; since the model contains no data r*
*elevant
to the achievment of stability, there is no ready way to do this.
15.4. K-theory revisited.
Let us consider the structure of the T-spectrum K representing rational equiva*
*riant
K-theory. We know that KH* = R(H)[fi; fi-1] for all subgroups H, where R(H) is*
* the
rationalization of the complex representation ring and fi 2 K-2 is the Bott ele*
*ment. Now
R = R(T) = Q[z; z-1], and R(H) = Q[z]=(zn - 1) when H is of order n. The restri*
*ction
maps are implicit in the notation here, and the induction maps to T are zero (h*
*olomorphic
induction maps are not part of the structure). Now, by Bott periodicity we have*
* K-theory
Euler classes c(H) of degree -2 for each finite subgroup H, obtained by applyin*
*g the Thom
isomorphism to the image of e(V (H)) in K-theory. We may apply Bott periodicity*
* to obtain
the usual K-theory Euler class (H) 2 R(T). In other words if H is of order n we*
* have
(H) = 1 - zn and c(H) = fi(H):
Furthermore Y
(H) = d;
d|n
where d is the dth cyclotomic polynomial. Let S be the multiplicative set gener*
*ated by
the Euler classes 1 - zn, and T be generated by the cyclotomic polynomials d; i*
*n practice
the geometry localizes so as to invert S, which is algebraically the same as in*
*verting T , and
the latter is easier to understand. Let F = S-1R = T -1R, so that
ssT*(K ^ "EF) ~=F [fi; fi-1] and ssT*(K ^ EF+) ~=(F=R)[fi; fi-1]:
Since the cyclotomic polynomials are coprime, an element of F=R can be written *
*uniquely
as a sum of terms fd(z)=d(z)n for n 1 where fd(z) 2 R is not divisible by d(z)*
*. Hence
M
F=R = R[1=n]=R:
n
We should relate this to our geometric decompositions.
15.4. K-THEORY REVISITED. 173
Lemma 15.4.1. If K(H) denotes the part of K in T-Spec=H as usual then
ssT*(K(H)) = (R[1=n]=R)[fi; fi-1]
and cH acts as multiplication by fin times an automorphism.
Remark 15.4.2. Our map cH is defined up to a non-zero rational number, whilst*
* the
K-theory Euler class is defined absolutely. The multiple is therefore well defi*
*ned up to a
non-zero scalar, but its exact value is not relevant to us at present. Crabb co*
*nsiders studies
this value in greater detail [5].
Proof: The only part requiring proof is that the action of cH is as claimed. W*
*e shall
show that cH acts as c(V (H)) times an automorphism on K ^ E. To do this, we*
* let
V = V (H), and unravel definitions.
We have a K-theory Thom isomorphism t : K ^ SV -'! K ^ S|V,|and the F-spectrum
Thom isomorphism o : SV ^ E -'! S|V |^ E; we need to know they are compat*
*ible
in the sense that the composites
K ^ S0 ^ E -! K ^ SV ^ E -! K ^ S|V |^ E
are equal. Now, we observe that both are maps of K-module spectra, and hence i*
*t is
sufficient to show both induce the same map S0^ E -! K ^ S|V |^ E. Maps o*
*f this
form are classified by K|VT|(E) ~=lim K|V(|E(2n)). Indeed the maps are cl*
*assified
n
by what they induce in homotopy:
~= 1 1
[E ^ K; S|V |^ E ^ K]K;T0-! HomR(R=n ; R=n ):
We know the K-theory Thom isomorphism gives multiplication by n. We may express*
* the
action of cH in its n-adic expansion as multiplication by x0+ x1n + x22n+ . .,.*
*where
xi2 R. It suffices to prove that the F-spectrum Euler class is multiplication b*
*y n mod
2nfor a non-zero scalar , i.e. that x0 = 0 and x1 = .
First, we know that the map oe0H-! oe0H^ K induces the permutation module map
Q = eH A(H) -! eH R(H) = R=n. Now, we need to understand something of the map
E(2n)-! K ^ E(2n), and we can infer enough by considering the diagram
0 Q Q
k k k
~= T 2k
ssT2k+1(E(2k-2))-! ssT2k+1(E(2k))-! ss2k+1(oeH)
# # #
ssT2k+1(E(2k-2)^ K)-! ssT2k+1(E(2k)^ K)-! ssT2k+1(oe2kH^ K)
k k k
R=kn R=k+1n R=n:
This shows that the image gk of the generator of ssT2k+1(E(2k)) in ssT2k+1(E*
*(2k)) =
R=k+1nis k modulo knfor a non-zero rational number k.
174 15. CLASSICAL MISCELLANY.
Now consider the diagram
Q Q
k k
~= T kV (0)
ssT2k+1(E(2k))-! ss2k+1( E )
~=# #~=
ckH T kV
ssT2k+1(E) -! ss2k+1( E)
# #
ssT2k+1(E ^ K)-! ssT2k+1(kVE ^ K)
k k
R=1n R=1n:
We see that ckHtakes the image of gk 2 R=k+1nto the image of 1 2 R=n. Now gk is
mapped to k=k+1nmodulo elements annihilated by kn. We conclude from the case k *
*= 1
that x1 = 1 as required. The general case shows that k = k1. __|_|
Corollary 15.4.3. The Q[cH ]-module ssT*(K(H)) is injective. Indeed, if Linis *
*the space
of Laurent polynomials in z with poles of order at most i at a primitive nth ro*
*ot of unity,
then multiplication by n gives an isomorphism Li+1n=Lin-! Lin=Li-1n. Thus ssT*(*
*K(H)) ~=
I(H) (L1n=L0n)[fi; fi-1], and hence
K(H) ' E ^ S0[ (L1n=L0n)[fi; fi-1] ]: __|_|
Accordingly K is characterized by tF* ssT*(K ^ "EF), ssT*(K ^ EF+) and the hom*
*omor-
phism between them. To make sense of the following statement, note that (F=R)[f*
*i; fi-1] is
a module over OF, since it is F-finite; more explicitly, if x 2 R[1=n]=R and H *
*is of order
n, then cH x = nxfi as mentioned above. Of course, multiplication by -1nis not *
*defined
on F=R, but it makes sense for F .
Theorem 15.4.4. The T-spectrum K is the unique T-spectrum for which tF* ssT*(K*
* ^
"EF) -! ssT*(K ^ EF+) is the map
^qK: tF* F [fi; fi-1] -! (F=R)[fi; fi-1]
__
described as follows._ For x 2 OF of degree -2k we have ^qK(x ffil) = xffil+k,*
* and
^qK(c-kH ffil) = -knffil-k.
Proof: The value of ^qK(x ffil) is immediate from our method of calculating ss*
*T*(K ^
EF+).
For ^qK(c-kH ffil) we apply 6.1.2, using the compatibility statement in 15.4.1*
* to relate
it to our present naming of elements. Consider the diagram
K -1^e K ^ S-kV (H) -'! K ^ S-2k -'! K
# # # #
K ^ "EF -' K ^ "EF ^ S-kV (H) -'! K ^ "EF ^ S-2k -'! K ^ "EF
# r # # #
K ^ EF+ - K ^ EF+ ^ S-kV (H) -'! K ^ EF+ ^ S-2k -'! K ^ EF+:
15.5. THE GEOMETRIC EQUIVARIANT RATIONAL SEGAL CONJECTURE FOR T. 175
The first horizontal is induced by e : S-kV (H)-! S0, the second is the K-theor*
*y Thom
class, and the third is multiplication by the integer Bott class. By 6.1.2 the *
*relevant map
is induced by r.
Applying ssT*we see by definition that the composite from right to left in th*
*e first row
is multiplication by (kV (H)), and hence this is also true in the second row. *
*Since the
bottom right hand vertical induces projection, we identify the second vertical *
*r in the lower
ladder as stated. __|_|
___
It is tempting to rewrite the description of ^qKabove as ^qK(x ffil) = xffil*
*+k for all
x 2 tF*, but this makes no sense, since F is not an OF-module. This suggests we*
* should
perform some algebraic completion to F . This can be achieved geometrically by *
*replacing
K with F (EF+; K). This is a reasonable thing to do since the fibre of the comp*
*letion map
is the F-contractible spectrum F (E"F; K), which is determined by its homotopy *
*groups.
15.5. The geometric equivariant rational Segal conjecture for T.
In this section we aim to analyse the functional dual DET+ = F (ET+; S0); the*
* title
is something of a misnomer since neither Segal nor anyone else has made a conje*
*cture
about DET+. It completes the description given in [9, 12] of the integral funct*
*ional dual.
It would be possible to give an entirely algebraic treatment since we have a mo*
*del for
function spectra, but we shall first present a more direct treatment so as to a*
*void relying
on the more complicated bits of algebra.
Since ET+ = E<1>, we should really discuss theWmore generalQquestion of ident*
*ifying
DE. Indeed, we should also consider DEF+ ' D( HE) ' H DE. Since the
initial stages of the analysis are easier to understand for EF+, we begin with *
*that.
We already know EF+ ^ DEF+ ' EF+, and that ssT*(DEF+) = tF*; accordingly we h*
*ave
the cofibre sequence
DEF+ -! "EF ^ S0[tF*] -a! EF+;
where a induces projection onto the positive dimensional part. In the algebrai*
*c model
DEF+ is thus described by the element ^a2 Hom OF(tF*tF*; 2I) given by smashing *
*a with
DEF+ and looking in ssT*. For definiteness we emphasize that the second copy of*
* tF*is the
new one.
Lemma 15.5.1. The map ^a: tF* tF*-! 2I is given by ^a(x y) = a*(xy).
Note that ^arealizes Tate duality between negative and positive parts of tF*.
Proof: Given an OF-map : tF*-! M, so that (1) = m the value of on OF tF*
follows, and if the components of m are uniquely divisible by the relevant cH t*
*hen is
determined. The result follows provided x is of positive degree. The real conte*
*nt of the
lemma is that the formula is valid also when a*(x) = 0.
To begin with we note how a nontrivial map "EF -! 2k+1EF+ is detected. As mot*
*iva-
tion, we observe that the obvious example is the quotient of "EF = S1V (F)by Sk*
*V (F). If it
were possible to simply desuspend by smashing with a putative spectrum S(-k-1)V*
* (F), then
the map would be detected in homotopy. Since the spectrum SV (F)is not inverti*
*ble we
must be slightly less direct by concentrating on a single subgroup H at a time,*
* and using
176 15. CLASSICAL MISCELLANY.
oekV (H)instead of SkV (F). This part of the analysis is given in the proof of *
*15.5.2 below. __|_|
The following identifies DE, and the special case H = 1 gives DET+.
Proposition 15.5.2.There is a cofibre sequence
DE -! "EF ^ S0[ Q[cH ; c-1H] ] -b! E
and the extension is determined by the fact that
^b2 Hom OF(tF* Q[cH ; c-1H]; 2I(H)) = Hom Q[cH](Q[cH ; c-1H] Q[cH ; c-1H]; *
*2I(H))
is given by ^b(x y) = b*(xy), which represents a perfect duality of Q[cH ; c-1*
*H]
Proof: We smash the standard cofibre sequence EF+ -! S0 -! "EF with DE; the
terms are identified with those in the statement by the following lemma.
Lemma 15.5.3. (i)There is an equivalence
EF+ ^ DE ' E:
(ii)There is an isomorphism
ssT*(TDE) ~=Q[cH ; c-1H]:
Remark: Part (i) of 15.5.3 with H = 1 corrects statements in the rational analy*
*sis of [9].
More precisely, the space EF+ should be replaced by EG+ in 1.6, Theorem B, 4.8,*
* and on
the right hand side of 4.5 and page 359 line -3. The correction is discussed in*
* more detail
in [12].
Proof of 15.5.3: (i) The equivalence EF+ ^DEW' E follows since EF+ ^DE
is a retract of EF+ ^ DEF+ ' EF+. Indeed EF+ ' H E, and the idempotent for *
*all
subgroups K 6= H annihilates DE. Hence EF+ ^ DE is a retract of E, and*
* by
homotopy groups it is an equivalence.
(ii) The identification of TDE is immediate from 2.4.1. __|_|
It remains to show that ^b(xy) = b*(xy); this follows when x is of positive de*
*gree exactly
as in 15.5.1. Now suppose x = c-kHfor k 0; the verification that ^b(c-kH y) = *
*b*(c-kHy) in
this case will also complete the proof of 15.5.1.
We take the cofibre sequence in the statement and smash it with oenV (H). Sinc*
*e oenV (H)
is formed from S0 and various basic cells with isotropy H, we have oenV (H)^ "E*
*F ' "EF; by
the Thom isomorphism E ^ oenV (H)' E ^ S2n, and also
oenV (H)^ DE ' D(E ^ oe-nV (H)) ' D(E ^ S-2n):
Thus the cofibre sequence becomes
S2n^ DE -! "EF ^ S0[ Q[cH ; c-1H] ] 1^b-!2n+1E:
Because the final term is 2n-connected and the first has ssT*only in degrees 2*
*n it follows
that ssT*(1 ^ b) is surjective for all n. Taking n -k - 1 establishes the requ*
*ired formula
for ^b. __|_|
15.5. THE GEOMETRIC EQUIVARIANT RATIONAL SEGAL CONJECTURE FOR T. 177
Now let us turn to the general question of what can be said about DX = F (X; *
*S0)
if we already understand X in the standard model. Thus we suppose that X has mo*
*del
B = (N -! tF* V ), and we let Q -! tF* H denote the torsion model of DX.
First we recall that S0 has model (OF -! tF*), this has torsion part I, and t*
*he natural
fibrant model is as the fibre of the map e(Q) -! f(2I). We have seen in Section*
* 9.3 that
if S is the torsion part of X, the torsion part of DX is
Rf Hom(S; I):
The vertex is described by the fibre sequence
H -! V *-! E-1Hom(tF* V; 2I):
However the real aim here is to give a description of the complete model. We sh*
*ow that
the obvious cofibre sequence DX -! DX ^ "EF -! DX ^ EF+ is also natural from the
algebraic point of view.
Lemma 15.5.4. If B = (N -! tF* V ), then the functional dual of B is describe*
*d by
i j
R^ RHom (B; S0) = fibre e(V *) -! R^ f(Hom(N; 2I)) :
Proof: We use the injective resolution S0 -! e(Q) -! f(2I), and deduce that
i j
RHom (B; S0) = fibre Hom (B; e(Q)) -! Hom (B; 2f(I)) :
Now Hom (B; e(Q)) is the injective e(V *), whilst Hom (B; 2f(I)) = f(Hom(N; 2*
*I)).
Applying ^ we obtain the result. __|_|
We can say a little more about the term R^ f(Hom(N; 2I)) in the description F*
*or
an arbitrary OF-module M we proved in Example 8.5.8 that there is a fibre seque*
*nce
R^f(M) -! e(E-1M) -! f(E-1M=M).
178 15. CLASSICAL MISCELLANY.
CHAPTER 16
Cyclic and Tate cohomology.
This is a short chapter, but fits naturally between its neighbours. The first s*
*ection identifies
rational cyclic cohomology, the second gives an algebraic description of the Ta*
*te construc-
tion on rational spectra, and the third gives a description of the Tate spectru*
*m of integral
complex equivariant K-theory.
16.1. Cyclic cohomology.
In this section we consider periodic cyclic cohomology. We begin by observing*
* that the
rationalisation of the integral cyclic cohomology is the cyclic cohomology of t*
*he rationals,
so that the two possible interpretations coincide.
It was proved in [14] that the representing spectrum for cyclic cohomology wi*
*th coef-
ficients in an abelian group A is the Tate spectrum t(HA) = F (ET+; HA) ^ "ET. *
* The
following lemma is special to bounded cohomology theories.
Lemma 16.1.1. For any Mackey functor A the rationalization of t(HA) is t(H(A *
* Q)).
Proof: The essential point is that F (ET+; HA) = holim F (ET(n)+; HA), and that*
* the
n
maps induced in [X; .]T by those of the inverse system are ultimately isomorphi*
*sms for
each finite X. This inverse limit therefore commutes with direct limit under de*
*gree zero
selfmaps of HA.
Thus
t(HA) ^ S0Q = holim!( F (ET+; HA) ^ "ET; m! )
m
' F (ET+; holim!(HA; m!)) ^ "ET
m __
' F (ET+; HA Q) ^ "ET= t(H(A Q)): |_|
Henceforth we suppose A is rational.
Lemma 16.1.2. Provided A is rational, t(HA) is F-contractible. We therefore h*
*ave an
equivalence
t(HA) ' "EF ^ S0[ssT*(t(HA))]:
179
180 16. CYCLIC AND TATE COHOMOLOGY.
Proof: Indeed t(HA)|H = t(HA|H ), and the rational Tate cohomology of any finit*
*e group
is 0. __|_|
For any abelian group A we have ssT*(t(HA)) = ^HC* A = A[c; c-1].
16.2.Rational Tate spectra.
In this section we discuss the Tate construction of [14], which generalizes th*
*e periodic
cyclic cohomology discussed in the previous section. Recall that the Tate const*
*ruction on
a T-spectrum is defined by t(X) = F (ET+; X) ^ "ET. This simplifies considerabl*
*y in the
rational case, and it seems worth giving a complete description of the Tate con*
*struction in
the category of rational T-spectra.
We begin with the warning that if X is integral, the map t(X) -! t(X ^ S0Q) ne*
*ed
not be a rational equivalence, so that Lemma 16.1.1 above is special to suitabl*
*y bounded
theories like HA. An example is given by complex K-theory, since t(KZ)|H is non*
*-trivial
and rational for all non-trivial finite subgroups H [10, 14, 15]; the following*
* lemma shows
this is false for t(KQ).
We revert to our global assumption that all spectra are rational.
Lemma 16.2.1. The natural map
t(X) = F (ET+; X) ^ "ET-! F (ET+; X) ^ "EF
is an equivariant equivalence. Thus t(X) is an F-contractible spectrum determin*
*ed by its
homotopy groups.
Proof: We give two proofs. Firstly, the Tate construction commutes with restri*
*ction:
t(X)|H = t(X|H ). The lemma follows from the fact that the Tate construction is*
* trivial on
rational spectra for finite groups. One way of seeing this is to use the fact t*
*hat if H is finite
and e 2 A(H) is the idempotent with support 1 then EH+ = eS0 and "EH = (1 - e)S*
*0.
For the second proof, we compare the cofibre sequence ET+ - ! S0 -! E"T with
EF+ -! S0 -! "EF. We see that the lemma is equivalent to showing that the natur*
*al
map f : F (ET+; X) ^ ET+ -! F (ET+; X) ^ EF+ is an equivalence. However, the co*
*fibre
of f is a wedge of terms F (ET+; X) ^ E with H 6= 1; this is contractible, a*
*s one sees
from the fact that F (ET+; X) ^ oe0H' * by using cofibre sequences and passing *
*to direct
limits. __|_|
Proposition 16.2.2.If X is a rational T-spectrum with associated module M = ss*
*T*(X^
ET+) over Q[c1], then t(X) is the F-contractible spectrum with homotopy groups
^H0(c(M) H^-1(M)
1) (c1)
where ^H*(c1)denotes local Tate cohomology in the sense of [10].
16.3. THE INTEGRAL T-EQUIVARIANT TATE SPECTRUM FOR COMPLEX K-THEORY. 181
Remark 16.2.3. There are two methods for calculating the local Tate cohomolog*
*y; since
(c1) is principal both are very simple. The second description simplifies furth*
*er because M
is torsion. In fact, the local Tate cohomology is only non-zero in codegrees 0 *
*and -1, and
for these cases we have
^H-i(c(M) = (L(c1)M)[1=c ] = limi(M; c );
1) i 1 1
where L(c1)*denotes the left derived functors of completion at (c1). Note that *
*this is only
likely to be equal to (c1)-adic completion when M is finitely generated. Howeve*
*r, since M is
torsion, when it is finitely generated it is already complete; the Tate cohomol*
*ogy therefore
vanishes, as we know it must for geometric reasons.
Proof: Observe F (ET+; X) ' F (ET+; X ^ ET+), so that if M = ssT*(X ^ ET+), the*
*re is
an exact sequence
0 -! Ext(2I; M) -! [ET+; X]T*-! Hom(I; M) -! 0:
This is precisely parallel to the algebraic situation. We may split X into eve*
*n and odd
parts, and thus the exact sequence splits. Therefore F (ET+; X) is modelled by *
*the com-
plex Hom(P K(c1); M) where P K(c1) is a complex of projectives approximating th*
*e stable
Koszul complex Q[c1]-! Q[c1; c-11]; the homology of this complex calculates the*
* left de-
rived functors of c1-completion [13]. In particular, when X is even, [ET+; X]T**
*is L(c1)0M in
even degrees and L(c1)1M in odd degrees.
Now conclude that there is a split exact sequence
0 -! Ext(2I; M)[1=c] -! t(X)T*-! Hom(I; M)[1=c] -! 0:
Therefore, if T T(c1)(M) is the complex of the second avatar in the notation of*
* [10], t(X) is
modelled by the corresponding torsion free model, e(T T(c1)(M)). Thus, if X is *
*even, t(X)T*
is ^H0(c1)(M) in even degrees and H^-1(c1)(M) in odd degrees. __|_|
16.3. The integral T-equivariant Tate spectrum for complex K-theory.
In this section we apply the general theory to identify the Tate spectrum of *
*complex
equivariant K-theory KZ integrally. However, we note that t(KZ) is not rational*
*, and its
rationalization is not t(KQ), so this is not an application of the previous sec*
*tion.
Before we state the theorem, recall that the representation ring R(T) = Z[z; *
*z-1], and
that the Euler class of the representation zn is 1-zn. In particular, we let O *
*= 1-z and find
R(T)^(O)= Z[[O]]; indeed, z = 1 - O is invertible in Z[O]=(On), so that Z[O] -!*
* Z[O; z-1] =
Z[z; z-1] induces an isomorphism of (O)-completions. We write Z((O)) for the lo*
*calization
Z[[O]][O-1], and S for the mulitiplicative set generated by the Euler classes. *
*Note that if
n 2 the Euler class 1 - zn is a multiple of O. However, although the multiplie*
*r is a unit
in Q((O)), it is not a unit in Z((O)) Q.
182 16. CYCLIC AND TATE COHOMOLOGY.
Theorem 16.3.1. The Tate spectrum t(KZ) is F-equivalent to a rational spectrum,
and is thus determined by the homotopy type of T KZ and its rational type. Ther*
*e is an
equivalence of KZ-module spectra
T KZ ' KS-1Z((O)):
The rational spectrum t(KZ) ^ S0Q is classified in the torsion model by
M
tF* S-1Z((O))[fi; fi-1] -! S-1Z((O))=Z((O))[fi; fi-1] = Z((O))=1|H|[fi; *
*fi-1];
H6=1
where n is the nth cyclotomic polynomial, and the structure map is described as*
* in 15.4.4.
Proof: First note that t(KZ) = F (ET+; KZ)^E"T, so that we may calculate its co*
*efficient
ring from the Atiyah-Segal completion theorem. First, equivariant K-theory has *
*coefficients
R(T)[fi; fi-1], with R(T) = Z[z; z-1], and the K-theory Euler class of zn is 1 *
*- zn. By the
Atiyah-Segal completion theorem, ssT*(F (ET+; KZ)) = R(T)^(O)[fi; fi-1].
Consider the cofibre sequence
t(KZ) -! t(KZ) ^ "EF -! t(KZ) ^ EF+:
We shall identify t(KZ) ^ "EF, t(KZ) ^ EF+, and the map between them in turn.
Firstly, since O is an Euler class,
ssT*(t(KZ) ^ "EF) = S-1Z((O))[fi; fi-1];
where S is the multiplicative set generated by 1 - zn for n 1. Now S-1Z((O)) i*
*s flat over
Z, and hence the coefficients of T t(KZ) are the same as those of K-theory with*
* coefficients
in S-1Z((O)).
Lemma 16.3.2. There is an equivalence
T KZ ' KS-1Z((O))
of non-equivariant KZ-module spectra.
Proof: First note that T t(KZ) is a module over KZ. Now let MS-1Z((O)) be a no*
*n-
equivariant Moore spectrum, and construct a map f : MS-1Z((O)) -! t(KZ) inducing
an isomorphism in ssT0. Now form the composite
KS-1Z((O)) = K ^ MS-1Z((O)) -! K ^ T t(KZ) -! T t(KZ)
in which the first map is obtained from f by applying K ^ T (.) to f, and the s*
*econd uses
the module structure. By construction this induces an isomorphism in homotopy, *
*and is
therefore an equivalence. __|_|
Next, we claim that t(KZ) ^ EF+ is rational. This is immediate from the fact *
*that,
t(KZ)|H is rational for all finite subgroups H [10,W14, 15]. Indeed, we know th*
*at t(KZ) ^
T=H+ is induced from t(KZ)|H . Rationally EF+ ' HE, so t(KZ) ^ EF+ has a
corresponding splitting. The summand for E<1> = ET+ is trivial, so we may choos*
*e H 6= 1
and consider t(KZ) ^ E. From the identification of cH , we find it has homo*
*topy
groups Z((O))=1nin each even degree, and in particular it is injective.
16.3. THE INTEGRAL T-EQUIVARIANT TATE SPECTRUM FOR COMPLEX K-THEORY. 183
Finally, the map t(KZ)E"F -! t(KZ) ^ EF+ factors through the rationalization
t(KZ) ^ "EF - ! t(KZ) ^ "EF ^ S0Q, and the resulting map t(KZ) ^ "EF ^ S0Q -!
t(KZ) ^ EF+ is classified by its d-invariant since the codomain is injective. *
*The map
from tF* S-1Z((O))[fi; fi-1] is the analogue of that in Theorem 15.4.4. __|_|
184 16. CYCLIC AND TATE COHOMOLOGY.
CHAPTER 17
Cyclotomic spectra and topological cyclic cohomology.
In this chapter, we study various T-spectra arising from algebraic K-theory. Va*
*rious con-
structions are used to define suitable targets for trace maps from algebraic K-*
*theory, and
the most sophisticated takes B"okstedt's Topological Hochschild homology of a r*
*ing, and
forms the associated topological cyclic spectrum in the sense of B"okstedt-Hsia*
*ng-Madsen
[2]. Madsen has recently given a very helpful general survey [20].
The topological cyclic construction can be applied to any T-spectrum with app*
*ropriate
extra structure, and we begin in Section 17.1 by identifying the extra structur*
*e involved
in specifying such a `cyclotomic' spectrum. In the following section, we illus*
*trate this
by considering the basic examples: free loop spaces on a T-fixed space, and top*
*ological
Hochschild homology of a functor with smash products. Finally, in Section 17.3 *
*we analyse
the topological cyclic construction on rational cyclotomic spectra.
17.1.Cyclotomic spectra.
We must begin by recalling the definition of a cyclotomic spectrum. The basic*
* idea is
that it is a spectrum X with the property analogous to that of the free loop sp*
*ace Z, on a
T-fixed based space Z, namely that for any finite subgroup K the fixed point se*
*t (Z)K is
equivalent to the original space Z. The analogue should be that any fixed point*
* spectrum
K X is equivalent to X again. Of course K X is really a T=K-spectrum, so we m*
*ust begin
by explaining exactly how we interpret it as a T-spectrum indexed on the origin*
*al universe.
In addition, we want to avoid redundant structure, so we simply require that th*
*e resulting
equivalences are transitive. __
We wish to consider the group T and all its quotients T = T=K by finite subgr*
*oups
~= *
*__
K. We want transitive systems of structure, so we first let ae = aeK : T -! *
*T be the
isomorphism given by taking_the |K|th root. If we index our T-spectra on a com*
*plete
universe U, we index our T-spectra on the complete universe UK . However we wan*
*t these
universes to be comparable, so we say that a complete T-universe U is cyclotomi*
*c if it is
~= * K __
provided with isomorphisms U -! aeK U . Identifying T and T via aeK , this als*
*o specifies
isomorphisms UL -! ae*K=L(U L)K=L. We require that these are transitive in the *
*sense that
if L K then the composite
U -! ae*LUL-! ae*L(ae*K=LUL)K=L = ae*KUK
185
186 17. CYCLOTOMIC SPECTRA AND TOPOLOGICAL CYCLIC COHOMOLOGY.
is the isomorphismLfor K. One such cyclotomic universe is the direct sum U = n2*
*ZUn
where Un = i2Nzn with the isomorphisms suggested by the indexing.
Suppose then that X is a T-spectrum indexed_on the cyclotomic universe U . Th*
*us,
for any finite subgroup K, K X is a T-spectrum indexed on UK ; by pullback alo*
*ng the
isomorphism aeK we obtain a T-spectrum ae*KK X indexed on ae*KUK, which may be *
*viewed
as a T-spectrum ae!KK X indexed on U by using the cyclotomic structure of the u*
*niverse.
A cyclotomic structure on X consists of a transitive system of T-equivalences
rK : ae!KK X -'! X:
By transitivity, the essential pieces of the structure come from the cases that*
* K is of prime
order.
Although this structure is really designed to capture profinite information, t*
*here is
enough residue rationally to make it worthwhile identifying the cyclotomic obje*
*cts in the
algebraic model of rational T-spectra. The essential idea is that in a cyclotom*
*ic spectrum
all finite subgroups behave in an analogous way, differing only in the multipli*
*city with which
information occurs. There is no significant constraint on total fixed points T *
*X. The first
step of our analysis was to split F-spectra into the parts over different subgr*
*oups, so it is
easy to describe the cyclotomic structure in these terms. A spectrum X is cyclo*
*tomic if we
have specified equivalences X(C1) ' X(C2) ' X(C3) ' : :.:This uniformity itself*
* imposes
constraints on the assembly map of a T-spectrum.
Let us now describe the algebraic model for cyclotomic spectra more precisely.*
* It is
useful to bear in mind the torsion model rather than the standard model.
Definition 17.1.1.The ring of cyclotomic operations is the polynomial ring Q[c*
*0]on a
single generator c0 of degree -2. The standard injective I0 is defined by the e*
*xact sequence
0 -! Q[c0]-! Q[c0; c-10] -! 2I0 -! 0: The cyclotomic torsion category Cthas obj*
*ects
(2I0 V -! T0) where V is a graded vector space and T0 is a torsion Q[c0]-module*
*. The
morphisms are given by commutative squares as usual. __|_|
Lemma 17.1.2. The category Ctis abelian and 2 dimensional. Hence we may form t*
*he
derived category DCt.
Proof: The proof is precisely analogous to that for the torsion model category *
*At. __|_|
Again, it is convenient to have a 1-dimensional model; the analogue of the sta*
*ndard
model is considerably simplified in the present context.
Definition 17.1.3.The standard cyclotomic category C has objects Q[c0]-maps N0*
* -!
2I0 V with N0 a torsion module. The morphisms are given by commutative squares *
*as
usual.
Lemma 17.1.4. The cyclotomic category C is abelian and 1-dimensional. Hence we*
* may
form the derived category DC . Furthermore passage to fibre dgC -! dgCt and pas*
*sage to
cofibre dgC -! dgCt induce inverse equivalences of derived categories, so that *
*DC ' DCt.
17.1. CYCLOTOMIC SPECTRA. 187
Proof: The proof is similar to the case of the standard model, but with the sim*
*plification
that the cofibre functor arrives in the correct category before passing to homo*
*logy. __|_|
Now define a functor
: Ct-! At
as follows. For an object we define (2I0 V -s0!T0) to be the composite
M LH s0M
(tF* V -! 2I V = 2I0 V0 -! T0):
H H
Here the first map is induced by the quotient tF*-! tF*=OF = 2I, and the second*
* is the
direct sum of countably many copies of s0 made into a OF-module in the obvious *
*way.
The functor is obviously exact and hence induces a functor
: DCt -! DAt :
We may now state a precise theorem.
Theorem 17.1.5. A T-spectrum admits the structure of a cyclotomic spectrum if*
* and
only if it corresponds to an object of DAt equivalent to one in the image of .
Note that the condition in the theorem gives a rather satisfactory characteri*
*zation of
cyclotomic spectra. It essentially says that a cyclotomic spectrum is one that*
* has two
properties. Firstly, the structure map factors through that for its geometric *
*fixed point
spectrum (as happens for suspension spectra) and secondly, that all finite subg*
*roups behave
alike.
If a spectrum admits a cyclotomic structure then a structure is imposed by ch*
*oosing
particular equivalences between the idempotent pieces of the torsion part of th*
*e model.
Note that for a spectrum X with torsion model tF* V -! T admitting a cyclotomic
structure the corresponding cyclotomic spectrum is simply 2I0 V - ! T0 where T*
*0 =
e1T = ssT*(ET+ ^ X) and the map is obtained by factoring s through the projecti*
*on and
applying the idempotent e1.
Proof: We have explained how to put a cyclotomic structure on an object in the *
*image of
. Any imprecision will be eliminated in the course of the proof in the reverse *
*direction.
Suppose then that X is a cyclotomic spectrum with cyclotomic structure maps r*
*K :
ae!KK X -'! X as required. We already know from Section 10.2 the effect of pass*
*age to
geometric fixed points. Indeed, by Theorem 10.2.6, if M is the model of X then *
*eM is the
model_for K X where e is the idempotent supported_on_the_subgroups containing *
*K. Here
O __Fis identified with eOF by letting a subgroup H of T correspond to its inve*
*rse image in
T. The effect_of ae!Ksimply results from identifying subgroups of T with those *
*of the same
order in T. __ __ *
* __
Define nK : F -! F by letting n(H ) be the subgroup of T with the same order *
*as H ,
~= __
and consider the induced ring isomorphism n*K: OF -! O__F.
188 17. CYCLOTOMIC SPECTRA AND TOPOLOGICAL CYCLIC COHOMOLOGY.
__
Lemma 17.1.6. The functor ae!K: T-Spec -! T-Spec corresponds to pullback alo*
*ng
n*Kin the usual sense that the diagram
__ ae!K
T-Spec -! T-Spec
'# #'
__ nK#
DA -! DA
commutes up to natural isomorphism, and similarly for torsion models. __|_|
It is then clear (for example by using the cyclotomic structure for K itself) *
*that the part
eK M of the model over any subgroup K will be the same as the piece e1M over th*
*e trivial
subgroup. This also forces the map to factor as specified.
If M is an object of the torsion model with zero differential, we see that the*
* structure
map must be zero on any element of form 1 v; otherwise it would have nonzero i*
*mage
in eH T for some H, and hence for all finite subgroups H. This contradicts F-fi*
*niteness of
T . Since this argument passes to an injective resolution, it applies to all di*
*fferential graded
objects. __|_|
17.2. Free loop spaces.
For a based space Z, we intend to identify the place of the free loop space Z *
*in the
present scheme. In particular we may consider K*T(Z), which is a conjectural ap*
*proxima-
tion to Ell*(Z).
We restrict attention to the case that Z = Y is a suspension. Here, Carlsson a*
*nd Cohen
[4] prove the splitting
_
ET+ ^T Y = (ECn)+ ^Cn Y ^n:
n
Hence, rationally we have
M
ssT*(ET+ ^ Y ) = {H*(Y )n }Cn
n
with trivial H*(BT) action. This leaves us to describe a map
M
2I0 H*(Y ) -! 2 {H*(Y )}Cn:
n
This necessarily has zero d-invariant. Indeed, this is obvious if H*Y has even *
*parity. In the
general case we see that the map is induced by ET+ ^TY -! ET+ ^TY ; by duality *
*it
is sufficient to consider cohomology, and the domain has torsion free cohomolog*
*y whilst the
codomain has torsion cohomology. Now, exactlyLas in the case of suspension spec*
*tra, the e
invariant is the element of Ext(4I0 H*(Y ); n 2H*(Y )) corresponding to the ex*
*tension
obtained by applying homology to
ET+ ^T Y -! ET+ ^T (Y )=Y -! ET+ ^T 2Y:
17.2. FREE LOOP SPACES. 189
Since Q[c1]acts trivially on H*(Y ), one might hope the extension is always obt*
*ained by
tensoring a universal extension
M
0 -! Q -! E -! 2I0 -! 0
n
with 2H*(Y ).
There is another important example of cyclotomic spectra.
Example 17.2.1. Topological Hochschild Homology:
Suppose that F is a functor with smash products in the sense of B"okstedt. O*
*ne may
define a cyclotomic spectrum T HH(F ), which comes with a spectral sequence
HH*(F (S0)*) =) T HH(F )*
for calculating its homotopy groups. One may then hope to calculate ssT*(ET+ ^T*
*T HH(F ))
using the skeletal filtration of ET+.
It is always the case that T T HH(F ) ' S0, and so the structure map of the c*
*yclotomic
spectrum T HH(F ) takes the form
2I0 -! 2T HH(F )hT*:
By definition, we always have a map from the identity functor to F and hence a *
*cyclotomic
map S0 = T HH(I) -! T HH(F ). Since this is an equivalence of geometric fixed p*
*oints,
and the structure map for S0 has zero d-invariant we deduce that the structure *
*map for
an arbitrary functor F has zero d-invariant. It would be interesting to unders*
*tand its
e-invariant more precisely.
One case of particular interest is when the FSP arises from a ring R. In thi*
*s case
ssT*(ET+ ^ T HH(R)) = HC*(R), which can be calculated by the algebraic Loday-Qu*
*illen
double complex. It remains to identify the torsion model structure map, but we *
*can obtain
information by naturality from the unit Q -! R. In fact we have the diagram
"EF ^ DEF+ - ! EF+
'# #'
"EF ^ DEF+ ^ T HH(Q) - ! EF+ ^ T HH(Q)
'# #
"EF ^ DEF+ ^ T HH(R) - ! EF+ ^ T HH(R)
in which we understand the top row precisely as the structure map of the sphere*
*. Thus
we only need to understand the algebraic map 2I0 = 2HC*(Q) -! 2HC*(R). This is
in fact either zero or injective: this follows from the Tate spectral sequence *
*by naturality.
Indeed t(T HH(R)) is a module over t(T HH(Q)), and hence over the ring spectrum*
* t(HQ),
whose coefficients are Q[c0; c-10]: thus the behaviour is completely determined*
* by the image
of the unit. If R is augmented, then of course the map is injective.
This special case is not too far from the general case, because any rational *
*FSP arises
from a simplicial ring, and ssT*(ET+ ^T HH(Ro)) can be calculated algebraically*
*, since there
is a T-map T HH(Ro) -! HH(Ro) which is a non-equviariant rational equivalence. *
* __|_|
190 17. CYCLOTOMIC SPECTRA AND TOPOLOGICAL CYCLIC COHOMOLOGY.
17.3. Topological cyclic cohomology of cyclotomic spectra.
In the first instance, the topological cyclic cohomology of a ring is designed*
* to be the
target of a refined trace map from the algebraic K-theory. Hesselholt and Mads*
*en have
shown that the cyclotomic trace is very close to being an isomorphism in many c*
*ases
[17]. The construction of the topological cyclic cohomology in this case begin*
*s with the
topological Hoschschild homology, and the definition of a cyclotomic spectrum a*
*bstracts
precisely what is required to make the construction.
Goodwillie has identified the topolical cyclic cohomology of the topological H*
*oschschild
homology of a rational functor with smash products [7], and we generalize this *
*to an
arbitrary cyclotomic spectrum. This is not a deep result, but it demonstrates t*
*he character
of the topological cyclic cohomology and illustrates the adequacy of the presen*
*t theory.
The author is grateful to L. Hesselholt for many helpful discussions.
We must begin by describing the construction. For any T-spectrum X, if L K we*
* have
an inclusion of the Lewis-May fixed points FLK : K X -! LX; the letter F is cho*
*sen
because it corresponds to the Frobenius map in algebraic K-theory. To avoid co*
*nfusion,
the reader should ignore for the duration of the present section the fact that *
*FLK induces
the restriction map from K-equivariant to L-equivariant homotopy groups. The cy*
*clotomic
structure supplies a second set of maps RKL: K X -! LX defined as follows. Firs*
*t we let
L* be the subgroup of K with order |K=L|. Now consider*the inclusion*X -! X ^E"*
*[6 L*];
applying Lewis-May L*-fixed points we obtain a map L X -! L X. Applying ae!L*a*
*nd
the cyclotomic structure we obtain
* ! L* rL*
ae!L*L X -! aeL* X -! X;
finally we apply L-fixed points and obtain the required map
*L* L ! L* L
ae!K=LK X = ae!K=LK=L X = (aeL* X) -! X:
Again, the letter R is chosen because the induced map is the restriction map in*
* algebraic
K-theory.
To simplify notation, we index F and R simply by the order of the quotient K=L*
*. Thus
we find F1 = R1 = 1, FrFs = Frsand RrRs = Rrs. It turns out that the Frobenius *
*and
restriction maps also commute.
The most familiar version of the topological cyclic cohomology construction is*
* simply to
take the the homotopy inverse limit of the system of non-equivariant fixed poin*
*t spectra
under the restriction and Frobenius maps:
T C0(X) = holim(holim(K X; R)); F ) = holim(holim(K X; F ); R):
It may help later motivation to view this as the homotopy fixed point object of*
* an `action
of a category'. It turns out that the intermediate object
T R0(X) = holim (K X; R);
K
has significance of its own, so we prefer the first description T C0(X) = holim*
*(T R0(X); F ).
Furthermore, we note that the above construction shows that the map RKL: ae!KK *
*X -!
17.3. TOPOLOGICAL CYCLIC COHOMOLOGY OF CYCLOTOMIC SPECTRA. 191
ae!LLX is a map of T-spectra so
T R(X) = holim (ae!KK X; R)
K
is a T-spectrum with underlying spectrum T R0(X). However, we warn that the ide*
*ntifica-
tion of all terms with T R(X) means that the Frobenius maps are not maps of T-s*
*pectra.
We shall identify the relevant equivariance below.
For non-profinite work, Goodwillie points out that the diagram given by the r*
*estriction
and Frobenius maps should be augmented by adding in the circle action; we may n*
*ow think
of an action by a topological category. Since R also commutes with the Frobeniu*
*s, passing
to limits under R, we obtain a diagram with a copy of T R(X) for each finite su*
*bgroup,
and Frobenius maps relating them; the quotient category acts on T R(X). For the*
* present
we view all objects as the same and hence we think of having an action of the m*
*onoid M
occurring in a split exact sequence 1 -! T -! M -! Z>0 -! 1; in fact if w; z 2 *
*T then
(wFr)(zFs) = wzrFrs. This leads to the definition
T C(X) = T R(X)hM ' (T R(X)hT)hF:
The following result may simply be regarded as evidence that the definition is *
*a reasonable
one: rationally, the topological cyclic construction is a complicated way of do*
*ing something
familiar.
Theorem 17.3.1. (Goodwillie ) For any rational cyclotomic spectrum X we have *
*an
equivalence of rational spectra
T C(X) ' XhT;
so that the topological cyclic cohomology agrees with the Borel cohomology.
Goodwillie proves this in the case that X = T HH(F ) for a rational functor F*
* with
smash products [7, 14.2].
Proof: The first step is to note that homotopy fixed points commute with homoto*
*py inverse
limits, and that the homotopy fixed point spectrum of a non-equivariantly contr*
*actible
spectrum is contractible. Thus
T R(X)hT = (holim ae!KK X)hT
K
= holim ((ae!KK X)hT)
K
= holim ((ET+ ^ ae!KK X)hT)
K
This shows that it is really only necessary to understand X(1) = ET+ ^ X. Of co*
*urse the
end result is simply a non-equivariant rational spectrum, so it is only necessa*
*ry to calculate
homotopy groups.
The main result of Part I is that the T-free spectrum ET+ ^ ae!K X is determi*
*ned by
its homotopy groups as modules over Q[c1].
Lemma 17.3.2. If X is a cyclotomic spectrum with ssT*(ET+ ^ X) = T0 then
M
ssT*(ET+ ^ ae!KK X) = T0
LK
192 17. CYCLOTOMIC SPECTRA AND TOPOLOGICAL CYCLIC COHOMOLOGY.
and hence _
ET+ ^ ae!KK X = X(1):
LK
Proof: This is immediate from 17.1.6 together with our exact identification of *
*Lewis-May
fixed points in Theorem 12.2.2, (or more directly from 12.3.2). __|_|
The relevant inverse system thus has Kth term given as a wedge of copies of th*
*e spectrum
X(1) indexed by the subgroups of K. It will perhaps be clearest if we think of *
*this as the
set of functions from the finite set [ K] of subgroups of K to X(1). The advan*
*tage is
that it permits a helpful notation for maps: any function f : A -! B of finite *
*sets induces
f* : X(1)B -! X(1)A. If f is an inclusion the map f* is simply projection.
Lemma 17.3.3. When L K, the restriction map RKL induces the projection corre-
sponding to the function KL: [ L] -! [ K] defined by requiring KL(H) to have or*
*der
|H| . |K=L|, in the sense that the diagram
1^RKL !L
ET+ ^ ae!KK X -! ET+ ^ aeL X
'# #'
W (KL)* W
HK X(1) -! HL X(1)
commutes.
Proof:*Recall*that L* denotes*the subgroup of K with order |K=L|. The effect of*
* the map
L X -! L (X ^ "E[6 L*]) = L X follows from our account of the Lewis-May fixed
points. Now we just need to rename subgroups using ae!L*17.1.6, and apply L-fix*
*ed points
as described in 12.3.2. __|_|
Corollary 17.3.4. We have an equivalence
Y __
T R(X)hT ' X(1)hT: |_|
H
Notice that the above argument could also be used to identify the T-spectrum T*
* R(X)
exactly in the algebraic model. Indeed the maps ae!KK X -! ae!LLX are all iden*
*tified
exactly, and we can form the homotopy inverse limit in the algebraic model. Ho*
*wever,
since inverse limits do not preserve F-free objects, the answer is not very att*
*ractive. Our
present purpose requires much less; indeed, since T R(X)hT is just a rational s*
*pectrum, and
it remains only to understand the action of F on homotopy groups.
Lemma 17.3.5. When L K the Frobenius map FLKinduces the projection correspond-
ing to the inclusion iKL: [ L] -! [ K] in the sense that the diagram
1^FKL L
ET+ ^ K X -! ET+ ^ X
'# #'
W (iKL)*W
HK X(1) -! HL X(1)
17.3. TOPOLOGICAL CYCLIC COHOMOLOGY OF CYCLOTOMIC SPECTRA. 193
commutes.
Proof: The first necessity is to understand the statement. We begin with a map *
*K X -!
LX, which we may view as a map of T spectra indexed on UK . Once we have smashed
with ET+ the universe may be replaced by a complete one, and we obtain a map in*
* the
category for which we have a model.
To understand the map we factor K X -! LX as K X -! infK X -! LX. If
we view these as maps of T=L spectra, the second map is the counit of the K=L f*
*ixed point
adjunction, completely understood by 12.2.2 and the contents of Section 11.3. *
*The first
map has the property that it is a nonequivariant equivalence. The result now fo*
*llows from
our description of the adjunction. __|_|
It remains only to index the terms so that the relevant structure is visible,*
* and to verify
that the circle action does not get in the way.
We begin by replacing subgroups by their orders, and defining a category as f*
*ollows.
The object set Z>0 x Z>0 consists of integer points in the strictly positive or*
*thant. There
are morphisms (OEs; aet) : (m; n) -! (ms; nt) for s; t 2 Z>0. Next, consider t*
*he diagram
D of divisors defined by D(m; n) = {d | d dividesmn} and OEs : D(m; n) -! D(ms;*
* n) is
inclusion, aet: D(m; n) -! D(m; nt) is the multiplication by t. Finally, for an*
* object Y we
consider the contravariant functor Y D defined by taking functions from D into *
*Y ; thus on
objects, Y D(m; n) = Y D(m;n). The connection with the restriction and Frobeniu*
*s diagram
is immediate from 17.3.5 and 17.3.3. The maps will be clearest if we replace D(*
*m; n) by the
set of rational numbers i=jwhere i divides m and j divides n. It is easy to che*
*ck that these
fractions are in bijective correspondence to divisors of mn: if d divides mn th*
*e relevant
fraction is d=n. With this indexing, both R and F simply drop irrelevant coordi*
*nates. Now,
passing to limits under restriction maps we obtain Y D(m) := lim Y D(m; n), whi*
*ch simply
n
consists of sequences (yi=j) with i dividing m. The map Fs : Y D(ms) -! Y D(m) *
*again
simply drops coordinates with numerator dividing ms but not m. In other words, *
*if we now
identify Y D(m) with Y D(1) by dividing the coordinate indexes by m we find Fs *
*is the shift
map specified by mutiplying indices by s and ignoring fractions with an integer*
* numerator
bigger than 1. The system consists of surjections, so lim1(Y D(1); F ) = 0, and*
* evidently
m
the only compatible families are those with all coordinates equal: lim (Y D(1);*
* F ) = Y .
m
This description suggests that we should have a means for discussing subgroups *
*of T with
fractional orders, which suggests we should be considering the solenoid S := li*
*m(T; s)
which is the inverse limit of copies of the circle under the power maps s .
We must now check that the fact we have taken homotopy T-fixed points between*
* the
R and F stages does not invalidate the above procedure. The time has come to be*
* precise
about the equivariance of the Frobenius maps. First, note that although we hav*
*e the
behaviour Fsz = zsFs in the monoid M, so that Fs is identified with the map s *
*: ET+ -!
*sET+, we expect the reverse type of behaviour for the objects acted upon.
Lemma 17.3.6. The Frobenius map induces a map of T-spectra along s, in the s*
*ense
that Fs : *sT R(X) -! T R(X) is a map of naive T-spectra.
194 17. CYCLOTOMIC SPECTRA AND TOPOLOGICAL CYCLIC COHOMOLOGY.
Proof: We must remember that the map Fs arose from the inclusions K X -! LX,
which is a map of T-spectra. However, when we have applied ae!in the appropriat*
*e way, we
must insert the power map s to retrieve the equivariance. __|_|
The relevant map T R(X)hT -! T R(X)hT is then obtained by passage to fixed poi*
*nts
from
F ( s; Fs) : *sF (ET+; T R(X)) = F ( *sET+; *sT R(X)) -! F (ET+; T R(X)):
The relevant untwisting result is as follows.
Lemma 17.3.7. The sth power map s : ET+ -! *sET+ is a stable rational equiva-
lence. __|_|
Q *
Let Y = n X(1), and consider the map Fs : sY -! Y of T-spectra. the commutat*
*ive
diagram
F(1;Fs)
F (ET+; *sY ) -! F (ET+; Y )
F ( s; 1) "' "=
F( s;Fs)
F ( *sET+; *sY )-! F (ET+; Y )
*
* Q hT
has an equivalence in its left hand vertical. Hence we can untwist the action o*
*n n X(1) .
Corollary 17.3.8. Rationally,Qwe may identify the system of copies of T R(X)hT*
* under
the Frobenius map with n>0X(1)hT and with the Frobenius Fs acting via multipli*
*cation
by s shifts. __|_|
The theorem now follows. __|_|