AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR
KRTHEORY
DANIEL DUGGER
Contents
1. Introduction 1
2. Background 6
3. Equivariant Postnikovsection functors 10
4. The Postnikov tower for Z x BU 15
5. Properties of the spectral sequence 17
6. Connective KRtheory 20
7. 'Etale analogs 23
8. Postnikov sections of spheres 26
Appendix A. Symmetric products and their group completions 31
Appendix B. Computations of coefficient groups 33
References 36
1.Introduction
In recent years much attention has been given to a certain spectral sequence
relating motivic cohomology to algebraic Ktheory [Be, BL, FS, V3]. This spectr*
*al
sequence takes on the form
Hp(X, Z(q_2)) ) Kp+q(X),
where the Hs(X; Z(t)) are the bigraded motivic cohomology groups, and Kn(X)
denotes the algebraic Ktheory of X. It is useful in our context to use topolog*
*ists'
notation and write Kn(X) for what Ktheorists call Kn (X). The above spectral
sequence is the analog of the classical AtiyahHirzebruch spectral sequence rel*
*ating
ordinary singular cohomology to complex Ktheory, in a way that is explained
further below.
It is well known that there are close similarities between motivic homotopy t*
*heory
and the equivariant homotopy theory of Z=2spaces (cf. [HK1 , HK2 ], for exampl*
*e).
In fact there is even a forgetful map of the form
(motivic homotopy theory over)R! (Z=2equivariant homotopy theory),
discussed in [MV , Section 3.3] and [DI, Section 5]. Our aim in this paper is *
*to
construct the analog of the above motivic spectral sequence in the Z=2equivari*
*ant
context. The spectral sequence takes on the form
q_ p+q
Hp,2(X, Z_) ) KR (X),
____________
Date: April 6, 2003.
1
2 DANIEL DUGGER
where the analog of algebraic Ktheory is Atiyah's KRtheory [At]. The analog of
motivic cohomology is RO(G)graded EilenbergMacLane cohomology, with coeffi
cients being the constant Mackey functor Z_. The indexing conventions have been
chosen for their analogy with the motivic situation, and will be elucidated fur*
*ther
in just a moment.
The fact is that constructing the above spectral sequence is not at all diffi*
*cult,
and there are many ways it could be done. In equivariant topology one has so ma*
*ny
tools to work with that the arguments end up being very simple. Unfortunately
most of these tools are not yet available in the motivic context. This paper tr*
*ies
to develop the spectral sequence in a way that might eventually work motivicall*
*y,
and which accentuates the basic properties of the spectral sequence. We introdu*
*ce
certain `twisted' Postnikov section functors, and use these to construct a tower
for the equivariant space Z x BU in which the layers are equivariant Eilenberg
MacLane spaces. The homotopy spectral sequence for the tower is essentially what
we're looking for_although technically speaking this only produces half the spe*
*ctral
sequence, and to get the other half we must stabilize. The approach here is sim*
*ilar
to the one advocated in [V2 ], but was worked out independently (in fact there *
*are
several differences, one being that [V2 ] takes place in the stable category).
We'll now explain the methods of the paper in more detail, starting with our
basic notation. Recall that every real vector space V with an involution gives *
*rise
to a Z=2sphere SV by taking its onepoint compactification. If R and R denote
the onedimensional vector spaces with trivial and sign involutions, respective*
*ly,
then any V will decompose as Rp (R )q for some p and q. So the spheres SV
form a bigraded family, and when V is as above we'll use the notation
SV = Sp+q,q.
Here the first index is the topological dimension of the sphere, and the second*
* index
is called the weight. Note that when V = Cn (regarded as a real vector space wi*
*th
the conjugation action) then SV = S2n,n; in particular, CP 1~=SC = S2,1. The
reader should be warned that this differs from the bigraded indexing introduced
in [At] and later used in [AM , Ar].
Recall from [LMS ] that to give an equivariant spectrum E is to give an as
signment V 7! EV together with suspension maps W EV ! EV W which are
compatible in a certain sense. One then has cohomology groups EV (X) for any re*
*p
resentation V , and when V is as above we will likewise write EV (X) = Ep+q,q(X)
to correspond with our bigraded indexing of the spheres. This `motivic indexin*
*g'
is quite suggestive, and ends up being a useful convention.
For the group Z=2, every real representation is contained in a Cn for a large
enough value of n. The definition of equivariant spectra can then be streamline*
*d a
bit by only giving the assignment Cn 7! E(2n,n)= En together with structure maps
S2,1^ En ! En+1. This is the approach first taken in [AM ]_albeit with different
indexing conventions, as mentioned above_and was later used in [V1 , J]. We will
treat spectra this way throughout the paper.
Our first example of such an object is the KR spectrum. The space ZxBU has an
obvious Z=2action coming from complex conjugation on the unitary group U. From
another perspective, one could model ZxBU by the infinite complex Grassmannian,
again with the action of complex conjugation. The reduced canonical line bundle
over CP 1is classified by an equivariant map S2,1= CP 1! Z x BU, and so one
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 3
gets S2,1^ (Z x BU) ! Z x BU by using the multiplication in Z x BU. So we
have a Z=2spectrum in which every term is Z x BU, and this is called the KR
spectrum. In fact, it is an Omegaspectrum: equivariant Bott periodicity shows
that the maps Z x BU ! 2,1(Z x BU) are equivariant weak equivalences, or that
KRs,t(X) ~=KRs2,t1(X). The reference for this fact is [At].
The second spectrum we will need is the equivariant EilenbergMacLane spec
trum HZ_. The easiest way to construct this, by analogy with the nonequivariant
case, is to consider the spectrum Cn 7! AG (S2n,n). Here AG (X) denotes the free
abelian group on the space X, given a suitable topology. The structure maps are*
* the
obvious ones, induced in the end by the isomorphisms S2,1^ S2n,n~=S2n+2,n+1. It
is proven in [dS] that this spectrum represents EilenbergMacLane cohomology wi*
*th
coefficients in the constant Mackey functor Z_, and that it is an Omegaspectru*
*m.
The nth space AG (S2n,n) is therefore an equivariant EilenbergMacLane space,
and will be denoted K(Z(n), 2n). This is the last of the basic notation needed *
*to
describe our results.
Our goal in this paper will be to construct certain functors P2n on the categ*
*ory of
Z=2spaces, which are analogs of the classical Postnikov section functors. Roug*
*hly
speaking, P2nX will be built from X by attaching cones on all maps from spheres
`bigger than' S2n,n. There are different possible choices for what is meant by *
*this,
for which we refer the reader to Section 3.
As was pointed out above there is a Bott map fi :S2,1! ZxBU which classifies
the reduced canonical line bundle over CP 1; let fin denote its nth power S2n,n!
Z x BU. Applying Postnikov section functors gives the induced map P2n(S2n,n) !
P2n(Z x BU). The main goal of this paper is the following:
Theorem 1.1. There are Postnikov functors P2n on the category of Z=2spaces
with the properties that
(a)P2n(S2n,n) isnweakly equivalent to K(Z(n), 2n), and
(b)P2n(S2n,n) fi!P2n(Z x BU) ! P2n2(Z x BU) is a homotopy fiber sequence.
Corollary 1.2. The tower
. ._.__//P4(Z x BU)____//P2(Z x BU)____//P0(Z x BU)
has the following properties:
(i)The homotopy fiber Fn of the map P2n(Z x BU) ! P2n2(Z x BU) is an
equivariant EilenbergMacLane space K(Z(n), 2n).
(ii)The Adams operation _k: Z x BU ! Z x BU induces a selfmap of the
tower, whose action on Fn coincides with the multiplicationbykn map on the
EilenbergMacLane space K(Z(n), 2n).
Looking at the homotopy spectral sequence of the above tower then gives the
following:
Corollary 1.3.qThere is a fringed spectral sequence Ep,q2) KRp+q,0(X) where
Ep,q2= Hp,_2(X; Z_) when p + q 0 and q is even, and Ep,q2= 0 otherwise. The
spectral sequence converges conditionally for p + q < 0, is multiplicative, and*
* has
an actionqof the Adams operations _k in which _k acts on Ep,q2as multiplication
by k_2.
4 DANIEL DUGGER
One can also stabilize the spectral sequence to avoid the awkward truncation,*
* but
then one loses the action of the Adams operations. To this end, we let Wn denote
the homotopy fiber of Z x BU ! P2n2(Z x BU). In nonequivariant topology the
Wn's are the connective covers of ZxBU, and are also the spaces in the spectr*
*um
for connective Ktheory bu. The following result shows the same for the Z=2cas*
*e:
Proposition 1.4. There are weak equivalences Wn ! 2,1Wn+1, unique up to
homotopy, making the diagrams
Wn ________//_ 2,1Wn+1
 
 
fflffl fflffl
Z x BU ____//_ 2,1(Z x BU)
commute (where the bottom map is the Bott periodicity map).
The corresponding Z=2spectrum whose nth object is Wn will be denoted kr
and called the connective KRspectrum.
Theorem 1.5. There is a `Bott map' fi : 2,1kr ! kr with the following propertie*
*s:
(a)The cofiber of fi is HZ_;
(b)The telescope of the tower
. .!. 2,1kr ! kr ! 2,1kr ! 4,2kr ! . . .
is weakly equivalent to the spectrum KR (where each map in the tower is the
obvious suspension or desuspension of fi);
(c)The homotopy inverse limit of the above tower is contractible.
The above tower of course yields a spectral sequence for computing KR*(X) for
any Z=2space X, which could be considered the Bockstein spectral sequence for
the map fi:
Theorem 1.6. For any Z=2space X, there is aqconditionally convergent, multi
plicative spectral sequence of the form Hp,_2(X, Z_) ) KRp+q,0(X).
This spectral sequence is interesting even when X is a point, in which case it
converges to the groups KO*; it is drawn in detail in section 6.4. Also, note t*
*hat
there is really a whole family of spectral sequences of the form
q_ p+q,r
Hp,r2(X, Z_) ) KR (X),
but these can all be shifted back to the case r = 0 by using Bott periodicity
KRs,t(X) = KRs+2,t+1(X).
1.7. Acknowledgments. Most of the results in this paper were taken from the
author's MIT doctoral dissertation [D1 ]. The author would like to thank his th*
*esis
advisor Mike Hopkins, and would also like to acknowledge very helpful conversat*
*ions
with Gustavo Granja. The final year of this research was generously supported by
a Sloan Dissertation Fellowship.
Since there has been a long delay between [D1 ] and the appearance of this pa*
*per,
a brief history of related work might be in order. Very shortly after [D1 ] was*
* written,
Friedlander and Suslin released [FS ] which constructed the more interesting mo*
*tivic
spectral sequence, using very different methods. In early 2000 the paper [V2 ] *
*was
released, outlining via conjectures a homotopytheoretic approach similar to the
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 5
one given here (but working in the stable category, and using a different defin*
*ition
of the Postnikov sections). These ideas were developed a little further in [V3*
* ].
Sometime in 20002001 Hopkins and Morel also announced proofs of results along
these lines, although the details have yet to appear. At the end of 2002, the p*
*aper
[V4 ] proved a stable result similar to Theorem 1.1(a) in the motivic context, *
*over
fields of characteristic zero. An analog of Theorem 1.1 for the unstable motiv*
*ic
category has never been claimed or proven, as far as I know.
1.8. Organization of the paper. The paper has been written with a good deal
of exposition, partly because the literature on these subjects is not always so*
* clear.
Sections 2 and 3 set down the necessary background, in particular giving the co*
*n
structions of equivariant Postnikov functors. In these sections we often work o*
*ver
an arbitrary finite group, because it is easier to understand the ideas in this*
* gen
erality. In this context everything is graded by orthogonal Grepresentations, *
*as is
standard from [LMS ]. When specializing to the Z=2 case we always translate into
the motivic (p, q)indexing. Section 2 also recalls the basic facts we will nee*
*d about
the theory H*,*(; Z_).
The real work takes place in section 4, where we analyze the Postnikov tower *
*for
ZxBU. Section 5 discusses the basic properties of the associated spectral seque*
*nce,
most of which follow immediately from the way the tower was constructed. Section
6 is concerned with passing to the stable case. Section 7, which goes back to b*
*eing
very expository, deals with the `'etale' version of the spectral sequence and t*
*he
analog of the QuillenLichtenbaum conjecture. Finally, in section 8 we give the
proof of Theorem 1.1(a).
6 DANIEL DUGGER
2. Background
2.1. Basic setup. Throughout this paper we will be working in the world of equi
variant homotopy theory over a finite group G (usually with G = Z=2). Unless
otherwise indicated, `space' means `equivariant space' and `map' means `equivar*
*iant
map'. If X and Y are spaces, then [X, Y ] denotes the set of equivariant homoto*
*py
classes of maps. When H is a subgroup of G, [X, Y ]H is the set of Hequivariant
homotopy classes of Hequivariant maps; in particular, [X, Y ]e is the set of n*
*on
equivariant homotopy classes. The phrase `weak equivalence' means `equivariant
weak equivalence': this refers to a map X ! Y such that XH ! Y H is an ordinary
weak equivalence for every subgroup H G.
2.2. Connectivity. Let V be an orthogonal Grepresentation. Waner [W , Section
2] introduced the notion of an equivariant space being V connective, generaliz*
*ing
the nonequivariant notion of nconnectivity. The key observation is that one c*
*an
make sense of the set [SV +k^ G=H+ , X]* not just for k 0, but for k V H
(here, and elsewhere, W  denotes the real dimension of the vector space W ). *
* If
VH denotes V regarded as an Hrepresentation, then there is a decomposition
VH = V (H) V H, where V (H) is the orthogonal complement of the fixed space
V H. One then considers the chain of equalities
H+k H
[SV +k^ G=H+ , X]* ~=[SVH+k , X]H*~=[SV (H)+V , X]*
and observes that the righthand set makes sense for k V H. We can therefo*
*re
take this as a definition for the lefthand set when k is negative.
A pointed Gspace X is called V connected if [SV +k^ G=H+ , X]* = 0 for all
subgroups H and all 0 k V H. Waner proved that this is equivalent to
requiring that XH is V Hconnected for all subgroups H. This result eventual*
*ly
appeared, in expanded form, in [Lw2 , Lemma 1.2].
The following result of Lewis [Lw3 , Lemma 3.7] will be used often:
Lemma 2.3. Suppose V 1 (the trivial representation), and let X and Y be
pointed Gspaces which are both (V  1)connected. Then a map X ! Y is a weak
equivalence if and only if for every k 0 and every subgroup H it induces an
isomorphism [SV +k^ G=H+ , X]* ~=[SV +k^ G=H+ , Y ]*.
2.4. EilenbergMacLane spectra.
When G is a finite group, let Or(G) denote the orbit category of G_the full sub*
*cat
egory of Gspaces whose objects are the orbits G=H. Recall that a Mackey functor
for G is a pair of functors (M*, M*) from Or(G) to Abelian groups having the
properties that
(a)M* is contravariant and M* is covariant;
(b)M*(G=H) = M*(G=H) for all H;
(c)For every t : G=H ! G=H one has t* O t* = id;
(d)The double coset formula holds.
We will not write down what the last condition means in general, but see [M ,
XIX.3].
The importance of Mackey functors is that if E is an equivariant spectrum and
X is any pointed space, then the assignment G=H 7! [ 1 (G=H+ ^ X), E] has a
natural structure of a Mackey functor. In the case X = SV , this Mackey functor*
* is
denoted ß_V(E) or E_V.
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 7
In the case G = Z=2 the orbit category is quite simple, having the form
t_________________
_________________________________________________*
*___
__________________________________________i
Z=2 ____//_e
where it = i and t2 = id. It follows that a Mackey functor for Z=2 consists of
Abelian groups M(Z=2) and M(e) together with restriction and transfer maps
__i*//_ _t*_//_
M(e) oo___M(Z=2), M(Z=2) oo___M(Z=2)
i* t*
satisfying the following conditions:
(i)(Contravariant functoriality) (t*)2 = id and t*i* = i*;
(ii)(Covariant functoriality) (t*)2 = id and i*t* = i*;
(iii)t* O t* = id;
(iv)(Double Coset formula) i* O i* = id + t*.
We will specify a Mackey functor for Z=2 by specifying the diagram
t*_________________
____________________________________________________
__________________________________________i*
M(Z=2) _____//M(e).oo_
i*
Example 2.5.
(a)The Mackey functor we will be most concerned with is the constant coefficient
Mackey functor Z_:
id_______________________________
_________________________________________________
_ßß____________________________________________2//_
Z ooid_Z.
Such a Mackey functor exists over any finite group G, and for any abelian
group in place of Z: the restriction maps are all identities, and the transf*
*er
maps M(G=H) ! M(G=K) are multiplication by the index [K : H].
(b)We define Z_opto be
id_______________________________
_________________________________________________
_ßß____________________________________________id/*
*/_
Z oo2__Z.
(c)The Burnside ring Mackey functor A_is the one for which A(G=H) is the Burn
side ring of H. For G = Z=2 this is
id_______________________________
_________________________________________________
ßß_____________________________________________i*//_
Z oo___Z Z.
i*
where i*(a, b) = a + 2b and i*(a) = (0, a).
Every Mackey functor M has an associated RO(G)graded cohomology theory
denoted V 7! HV (; M) (where V runs over all orthogonal Grepresentations),
which is uniquely characterized by the properties that
8 DANIEL DUGGER
(
o Hn(G=H; M) = M(G=H) ifn = 0,
0 otherwise,
(here n denotes the trivial representation of G on Rn), and
o the restriction maps H0(G=K; M) ! H0(G=H; M) induced by i: G=H !
G=K coincide with the maps i* in the Mackey functor.
The transfer maps of the Mackey functor will coincide with the transfer maps in
this cohomology theory (or with the pushforward maps in the associated homology
theory). Details are in [M , Chap. IX.5].
2.6. EilenbergMacLane spaces.
When M is a Mackey functor, the V th space in the spectrum for HM is called
an EilenbergMacLane space of type K(M, V ). Such spaces are (V  1)connected,
and have the properties that [SV +k^ G=H+ , K(M, V )] = 0 for k > 0 and the
Mackey functor G=H 7! [SV ^ G=H+ , K(M, V )] is isomorphic to M. See [Lw3 ,
Definition 1.4] for this characterization.
When G = Z=2 and V = Rp (R )q, we will usually adopt the motivic no
tation K(M, V ) = K(M(q), p + q). Likewise HV (; M) will be written as either
Hp+q,q(; M) or Hp+q(; M(q)), usually the former.
2.7. The theory H*,*(X; Z_).
In this section we set down the basic facts about the cohomology theory H*(;*
* Z_)
(which we'll sometimes write HZ_). We will need to know its coefficient groups
Hp,q(pt; Z_) and Hp,q(Z=2; Z_), their ring structures, and the transfer and res*
*triction
maps between them. These things have certainly been computed many times over
the years, although it's hard to find a precise reference. The corresponding fa*
*cts
about the theory H*(; A_) can be found in [Lw1 , Thms. 2.1,4.3], where they are
attributed to Stong. The necessary information about H*(; Z_) can be deduced
from these with a little bit of work, although it turns out to be much easier to
avoid H*(; A_) altogether. The corresponding information about H*(; Z_=2) is *
*in
[HK1 , Prop. 6.2], and again one can deduce the integral analogs with a little *
*bit
of work. An interesting computation of the positive part of the coefficient rin*
*g can
be found in [LLM , Thm. 4.1]; this gives explicit cycles representing each elem*
*ent.
In any case, the ultimate conclusions are listed in the theorem below. Although
the results are not new, we have included proofs in Appendix B for the reader's
convenience.
Theorem 2.8.
(a)The abelian group structure of H*,*(pt; Z_) is
8
>>>Z=2 if p  q is even and q p > 0;
>>:Z=2 if p  q is odd and q + 1 < p 0
0 otherwise.
These groups are shown in the following picture, where hollow circles denote
Z's and solid dots represent Z=2's (note that the paxis is the vertical one*
*):
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 9


p
6_

 q q q
3_
 2 2
2_ qy xqy q q

 y xy
1_ q q q q
_____________________________________________
____e__q___e___q___e___q___e_____ee_____e_______e_______e__________________*
*______________________________________________________________@
_____________________________________________________________________oe__*
*__________________________________0123120
q______q_______q_____________________________________________q


q q _

q q _


_


_
?
(An easy way to keep track of the grading is to remember that y 2 H1,1and
x 2 H0,2).
(b)The multiplicative structure is completely determined by the properties that
(i)It is commutative;
(ii)The solid lines in the above diagram represent multiplication by the cl*
*ass
y 2 H1,1;
(iii)The dotted lines represent multiplication by x 2 H0,2(but note that on*
*ly
a representative set of dotted lines have been drawn);
(iv)xff = 2.
In particular, the subring consisting of Hp,qwhere p, q 0 is the polynomial
algebra Z[x, y]=(2y).
(c)The ring H*,*(Z=2; Z_) is isomorphic to Z[u, u1], where u has degree (0, 1).
(d)The Mackey functor HZ__0,2nis Z_when n 0 and Z_opwhen n < 0 (see Exam
ple 2.5 for notation).
Remark 2.9. For computations it's often convenient to give every element of
H*,*(pt) a name in terms of x, y, and `. For instance, ff can be named as 2_x, *
*and
the class in degree (2, 7) can be named _`_xy2.
Remark 2.10. If q > 0 and you look in degree (1  q, q) and read vertically
upwards, you are seeing the groups H~*sing(RP q1). If you look in degree (q 
1, q) and read vertically downwards, you are seeing the groups H~*(RP q1). The
connection is explained in detail in Appendix B.
Remark 2.11. It's worth pointing out what aspects of the above picture are simi*
*lar
to the motivic setting, and which are not. In the motivic setting one has that
Hp,qmot(pt; Z_) = 0 for q < 0, but notice that this is not the case for the abo*
*ve
Z=2equivariant theory. This difference is tied to the fact that classical alge*
*braic
Ktheory is connective, whereas KOtheory is not. The BeilinsonSoul'e conjectu*
*re
is that Hp,qmot(pt; Z_) = 0 when q > 0 and p < 0, which is clearly satisfied in*
* our Z=2
world. The nonzero motivic cohomology groups of a point should correspond to
10 DANIEL DUGGER
the groups lying in the first quadrant of the above diagram, with the same vani*
*shing
line. The motivic groups lying along this line are the Milnor Ktheory groups.
The above result tells us everything about the EilenbergMacLane spaces
K(Z(n), 2n). Here are a couple of the main points:
Corollary 2.12.
(a)Nonequivariantly, K(Z(n), 2n) is a K(Z, 2n).
(b)The induced action of Z=2 on ß2n(K(Z(n), 2n)) = Z is multiplication by (1)n.
(c)The fixed set K(Z(n), 2n)Z2 has the homotopy type of either
K(Z, 2n) x K(Z=2, 2n  2) x K(Z=2, 2n  4) x . .x.K(Z=2, n) (n even,)
or
K(Z=2, 2n  1) x K(Z=2, 2n  3) x . .x.K(Z=2, n) (n odd).
Proof.A theorem of [dS] identifies K(Z(n), 2n) with AG(S2n,n), the free abelian
group generated by S2n,n. So both K(Z(n), 2n) and its fixed set are topological
abelian groups, hence products of EilenbergMacLane spaces. The homotopy groups
can be read off of H*,*(pt; Z_) and H*,*(Z=2; Z_). This proves (a) and (c).
Note that [S2n,n, K(Z(n), 2n)]e ~=[S2n,n^ Z=2+, K(Z(n), 2n)]* ~=H0,0(Z=2),
and we know the group action on the latter is trivial (because we know the Mack*
*ey
functor H_0,0). The group ß2nK(Z(n), 2n) may be written [S2n,0, K(Z(n), 2n)]e,
and this differs from the above in the replacement of S2n,nby S2n,0. On the for*
*mer
sphere, the automorphism coming from the Z=2 action has degree (1)n (complex
conjugation on Cn reflects n real coordinates). This proves (b).
Remark 2.13. The homotopy fixed set K(Z(n), 2n)hZ2 will also be a topological
abelian group, and hence a generalized EilenbergMacLane space. The spectral
sequence for computing homotopy groups of a homotopy limit collapses, and shows
that K(Z(n), 2n)hZ=2is either
K(Z, 2n) x K(Z=2, 2n  2) x K(Z=2, 2n  4) x . .x.K(Z=2, 0) (n even),
or
K(Z=2, 2n  1) x K(Z=2, 2n  3) x . .x.K(Z=2, 1) (n odd).
So the actual fixed set is a truncation of the homotopy fixed set. This observa*
*tion
reappears in section 7.
3. Equivariant Postnikovsection functors
In this section we define two types of equivariant Postnikov section functors,
denoted PV and PV , and list their basic properties.
To begin this section we work in the context of an arbitrary finite group G. *
*The
category TopG denotes the category of Gspaces which are compactlygenerated
and weak Hausdorff, with equivariant maps. We will eventually specialize to the
case G = Z=2, but for the present it is just as easy to work in greater general*
*ity.
There is a model category structure on TopG analagous to the usual one on Top.
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 11
3.1. Generalities. Recall that a space A is said to be small with respect to
closed inclusions if it has the property that for any sequence of closed inclus*
*ions
Z0 ,! Z1 ,! Z2 ,! . . .
the canonical map colimiTopG (A, Zi) ! TopG (A, colimZi) is an isomorphism. Ev
ery compact Hausdorff space is small in this sense. Let CA denote the cone on A,
and recall that [X, Y ] denotes unpointed homotopy classes of maps.
Let A be a set of wellpointed spaces, all of which are compact Hausdorff. The
pointedness can be ignored for the moment, but will be needed later. We will say
that a space Z is Anull if it has the property that the maps [*, Z] ! [ nA, Z]
(induced by nA ! *) are isomorphisms, for all n 0 and all A 2 A. This is
equivalent to saying that every map nA ! Z extends over the cone.
For a space X one can construct a new space PA (X) with the following proper
ties:
(1)There is a natural map X ! PA X;
(2)PA X is Anull;
(3)If Z is an Anull space and X ! Z is a map, then there is a lifting
X ______//Z<<___
 _______
 ______
fflffl____
PA X
and this lifting is unique up to homotopy.
The functors PA (X) are called nullification functors in [F ]. They are examples
of Bousfield localization functors, for which an excellent reference is [H , Ch*
*apters
3,4]. In our context we construct them as follows: For any space Y , let FA (Y *
*) be
defined by the pushout square
` n //
oeffAl________fflYfflffl_
___
 ______
` fflffln _fflffl__
oeC( A)_____//________FA Y,
where oe runs over all maps n A ! Y (for all A 2 A, and n 0). One then
considers the sequence of closed inclusions
X ,! FA X ,! FA FA X ,! FA FA FA X ,! . . .
and PA (X) is defined to be the colimit. It is routine to check, using basic ob*
*struc
tion theory, that this construction has the required properties.
Remark 3.2. We will often use the observation that if A A0 then one has a
canonical map PA X ! PA0X.
In terms of the localization of model categories (cf. [H ]), we are localizin*
*g TopG
at the set of maps {A ! *A 2 A} and PA is the localization functor. This uses
the fact that the objects in A are wellpointed. As a consequence, the functors
PA X have the standard properties one would expect from a localization functor.
We omit the proof of the following result: such properties can be found in [H ]*
* in
complete generality, or in [F ] for the category of spaces. In this case they a*
*re also
easy to prove directly by standard arguments.
12 DANIEL DUGGER
Proposition 3.3.
(a)Let X ! Y and X ! Z be maps, where X ! Y is a cofibration. If Z is Anull
and PA X ! PA Y is a weak equivalence, then there is a lift
X _____//Z>>__
 _______
 ______
fflffl___
Y
and this lift is unique up to homotopy.
(b)Let X : C ! TopG be a diagram. Then the natural map
PA (hocolimffXff) ! PA (hocolimPA Xff)
is a weak equivalence.
(c)If X ! Y ! Z is a homotopy cofiber sequence and PA X is contractible, then
PA Y ! PA Z is a weak equivalence.
3.4. The functors P and P . The most basic examples of nullification functors a*
*re
the ordinary Postnikov section functors (when G = {e}): given a nonequivariant
space X one forms PnX by killing off all homotopy groups above dimension n.
In the language of the previous section PnX = PAn X, where An is the set
{Sn+1, Sn+2, . .}.. In fact we would get a homotopy equivalent space by just ta*
*king
An to be {Sn+1}, but we have arranged things so that An+1 An because our
formalism then gives natural maps PAn+1X ! PAn X.
When one wants to introduce Postnikov section functors for Gspaces several
possibilities present themselves. One thing to note is that when we kill off al*
*l maps
from a space A, we would also like to be killing off maps from all spaces A ^ Z*
*. In
the nonequivariant setting this is automatic, because Z can be built from spher*
*es
and therefore A ^ Z is built from suspensions of A. In the equivariant setting *
*we
have to explicitly build this into the theory, by making sure that whenever we *
*kill
off a space A we also kill off A ^ G=H+ for all subgroups H.
If V is a representation of G then one Postnikov functor we can consider is P*
*A (X)
where A is the set {SV +n^ G=H+ n > 0, H G}. One can get the same result by
doing the following, and for functorial reasons it is somewhat better: Let
A~V = {SW ^ G=H+  W V + 1, H G},
and define PV X = PA~V(X). This definition guarantees that if V U then there
are natural maps PU X ! PV X.
On the other hand we can also do the following. Let
AV = {SW ^ G=H+  W V, H G},
and define PV X = PAV (X). Again, whenever V U there are natural maps
PU X ! PV X. Moreover, since ~AV AV one has maps PV X ! PV X.
Remark 3.5. The difference between PV and PV shows up in the following way.
Nonequivariantly, there are no nontrivial maps Sn ! Sk when k > n. As a
result, when one forms the Postnikov section PnX one doesn't change the homotopy
groups in low dimensions: [Sm , X] ! [Sm , PnX] is an isomorphism for m n. In
the equivariant theory, however, there can be many nontrivial maps SV ! SW
for V W ; so when forming a Postnikov section by killing off maps from large
spheres, one may actually be creating new maps from smaller spheres.
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 13
What is true, however, is that if V + 1 W _that is, if W contains V plus at
least one copy of the trivial representation_then all equivariant maps SV ! SW
are null. This leads one to the functors PV defined above, which are designed so
that they don't change homotopy classes of maps from SV and smaller spheres.
The general rule is that the functors PV are better behaved that PV : they are
easier to compute, and their properties (outlined below) closely resemble those*
* of
nonequivariant Postnikov sections. These are the same as the Postnikov functors
in [M , II.1].
Proposition 3.6 (Properties of P). If X is a pointed Gspace X and V is a G
representation, the following are true:
(a)The map X ! PV X induces an isomorphism of the sets [Sk,0^ G=H+ , ]* for
0 k dimV H, and an epimorphism for k = dimV H + 1.
(b)If W is a Grepresentation for which dim W H dim V H for all subgroups
H G, then [SW , X]* ! [SW , PV X]* is an isomorphism.
(c)The homotopy fiber of PV +1X ! PV X is an EilenbergMacLane space of type
K(ß_V +1X, V ).
(d)The homotopy limit of the sequence
. .!.PV +2X ! PV +1X ! PV X
is weakly equivalent to X.
(e)If V contains the regular representation of G, then the Postnikov section
PV (SV ) is an EilenbergMacLane space of type K(A_, V ) where A_ is the
Burnsidering Mackey functor.
Proof.The proof is completely standard, so we will only give a brief sketch. Su*
*ppose
that X is a space, SW ^G=J+ ! X is a map, and we attach the cone on this map to
construct a new space X1. The for any subgroup H G, XH1is obtained from XH
by attaching a cone on the map (SW ^ G=J+ )H ! XH . The domain of this map
is a wedge of spheres of dimension W H, and so XH ! XH1 is W Hconnected.
From these considerations (a) is immediate.
Part (b) follows from (a) and the fact that SW has an equivariant CWstructu*
*re
made up of cells Sk,0^ G=H+ where k dimW H. Part (d) is also immediate from
(a): the map X ! PV X becomes highly connected on all fixed sets as V gets larg*
*e.
For part (c), note first that for an arbitrary space X the object ß_VX may not
be a Mackey functor_it is instead a V Mackey functor as defined in [Lw3 , 1.2]*
*. A
characterization of EilenbergMacLane spaces is given in [Lw3 , 1.4], and it is*
* easy
to use parts (a) and (b) to check that the homotopy fiber we're looking at has *
*the
properties listed there.
Part (d) is an immediate consequence of (c) and the wellknown isomorphism of
Mackey functors ß_V(SV ) ~=A_(which follows from [M , IX.1.4,XVII.2]).
Suppose now that E is an equivariant spectrum, and let E denote the 0th space
of the corresponding spectrum. By applying the functors Pn we obtain a tower
. ._._____//P2E_________//P1E_________//_P0E______//*OOOOOO
  
  
  
K(ß_2E, 2) K(ß_1E, 1) K(ß_0E, 0)
14 DANIEL DUGGER
The homotopy spectral sequence for maps from X into this tower gives the classi*
*cal
equivariant AtiyahHirzebruch spectral sequence Hp(X; E_q) ) Ep+q,0(X) with a
suitable truncation. Here E_qdenotes the Mackey functor G=H 7! Eq(G=H). To
get the full spectral sequence one can look at the Postnikov towers for each EV*
* (the
V th space in the spectrum for E) and note that the resulting spectral sequen*
*ces
can be pasted together.
Unfortunately, for E = KR this spectral sequence is not the one we're looking
for. The above spectral sequence collapses when X = * and gives no information,
whereas the spectral sequence we're looking for is very nontrivial when X = *.*
* In
the context of algebraic Ktheory, the analog of the above spectral sequence is*
* the
BrownGersten spectral sequence of [BG ].
The functors PV don't have the property that the homotopy fiber of PV +1X !
PV X is necessarily an equivariant EilenbergMacLane space. For the record, here
are the basic properties of P . The proofs are the same as for Proposition 3.6.
Proposition 3.7 (Properties of P ). For any pointed Gspace X and any G
representation V , the following are true:
(a)The map X ! PV X induces an isomorphism of the sets [Sk,0^ G=H+ , ]* for
0 k < dimV H, and an epimorphism for k = dimV H.
(b)If W is a Grepresentation for which dim W H < dim V H for all subgroups
H G, then [SW , X]* ! [SW , PV (X)]* is an isomorphism.
The main reason we care about the functors PV is the following result:
Theorem 3.8. When G = Z=2 and V contains the trivial representation, the space
PV (SV ) has the equivariant weak homotopy type of the EilenbergMacLane space
K(Z_, V ).
The proof of this result is somewhat involved, and will be postponed until se*
*c
tion 8. However, we can give some intuition for why it's true. If V C one kno*
*ws
that [SV , SV ]* = Z Z (the Burnside ring of Z=2), and [SV , SV ]* ! [SV , PV (*
*SV )]*
is an isomorphism by Proposition 3.6(b). If one chooses the generators of [SV ,*
* SV ]*
appropriately, their difference factors through a `Hopf map' SV +R ! SV , where
R is the sign representation of Z=2. Since PV (SV +R) ' *, this difference be*
*comes
null in PV (SV ) (note that PV (SV +R) is not contractible). So the two copies*
* of
Z in [SV , PV (SV )]* become identified in [SV , PV (SV )]*, and this is ultima*
*tely why
PV (SV ) has the homotopy type of K(Z_, V ) rather than K(A_, V ).
We'd like to justify the claim that the difference of generators factors thro*
*ugh a
Hopf map. First look at the case V = C, so that SV = S2,1. Consider the degree
map [S2,1, S2,1]* ! Z Z which sends a map f to the pair (degf, degfZ=2) (the
degree, and the degree of the map restricted to the fixed set). This is injecti*
*ve, and
the image consists of pairs (n, m) for which n  m 0 mod 2. For the generato*
*rs
of [S2,1, S2,1]* we'll take the identity map, which has degree (1, 1), and the *
*complex
conjugation map, which has degree (1, 1).
Now consider the Hopf projection C2  {0} ! PC1, which we may write as
p: S3,2 ! S2,1. Smashing the inclusion S0,0 ,! S1,1with S2,1gives a map
j :S2,1,! S3,2, and the degree of the composition pj is (0, 2). So this compos
ite is homotopic to the sum of our two chosen generators. When V = C W , one
takes this same argument and smashes everything in sight with SW .
Another perspective on Theorem 3.8 is given in Section 8, where it is tied to*
* the
geometry of the infinite symmetric product construction.
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 15
4.The Postnikov tower for Z x BU
In this section we will consider the objects PnC(Z x BU). In motivic indexing
PnC would be written P(2n,n), and we will abbreviate this as just P2n. Our goal*
* is
the following
Theorem 4.1. Let fi : S2,1! Z x BU be a map representing the Bott element in
gKR0,0(S2,1), and let fin : S2n,n! Z x BU denote its nth power. Then
n
P2n(S2n,n) fi!P2n(Z x BU) ! P2n2(Z x BU)
is a homotopy fiber sequence.
Corollary 4.2. There is a tower of homotopy fiber sequences
. . ._____//P4(ZOxOBU)___//_P2(ZOxOBU)___//_P0(ZOxOBU)___//_*
  
  
  
K(Z(2), 4) K(Z(1), 2) Z
and the homotopy limit of the tower is Z x BU.
As explained in the introduction, one can prove the corollary using only the
functors P, and this is easier in the end. Writing P2n for PnC, we have the fol*
*lowing:
Proposition 4.3. The homotopy fiber of P2n(Z x BU) ! P2n2(Z x BU) is an
EilenbergMacLane space of type K(Z(n), 2n).
Proof.Let Fn denote the homotopy fiber. From Proposition 3.6(a) it follows im
mediately that the fixed set of Fn is (n  1)connected. Nonequivariantly P2n *
*has
the homotopy type of the ordinary Postnikov section functor, and so Fn is (2n1*
*)
connected as a nonequivariant space (here we are using that Z x BU has no odd
homotopy groups). So in equivariant language Fn is (nC  1)connected (cf. sec
tion 2.2). By construction we have that [S2n+k,n^ Z=2+, Fn]* = 0 = [S2n+k,n, Fn*
*]*
for all k > 0, and the long exact homotopy sequence shows that the Mackey funct*
*or
ß_2n,n(Fn) is isomorphic to ß_2n,n(Z x BU) ~=KR__2n,n(pt) ~=Z_. Lewis's charact*
*er
ization of EilenbergMacLane spaces [Lw3 , Def. 1.4 ] now shows that Fn is a
K(Z(n), 2n).
For the remainder of the section we let Fn denote the homotopy fiber
Fn ! P2n(Z x BU) ! P2n2(Z x BU).
The map fin :P2n(S2n,n) ! P2n(Z x BU) becomes null when we pass to the space
P2n2(Z x BU), and therefore it lifts to Fn (and the lifting is unique up to ho
motopy). Our task is to show that this lifting is a weak equivalence. Both the
domain and codomain are highly connected, and so we can use Lemma 2.3. Here
is a restatement for the special case where G = Z=2 and V = Cn:
Lemma 4.4. Let X and Y be pointed Z=2spaces with the properties that
(i)[Sk,0, X]* = [Sk,0, Y ]* = 0 for 0 k < n, and
(ii)[Z=2+ ^ Sk,0, X]* = [Z=2+ ^ Sk,0, Y ]* = 0 for 0 k < 2n.
16 DANIEL DUGGER
Then a map X ! Y is an equivariant weak equivalence if and only if it induces
isomorphisms
~= 2n+k,n 2n+k,n ~= 2n+k,n
[S2n+k,n, X]* ! [S , Y ]* and [Z=2+ ^ S , X]* ! [Z=2+ ^ S , Y ]*
for every k 0.
We know by Theorem 3.8 that P2n(S2n,n) is a K(Z(n), 2n)space and there
fore we know it's homotopy groups_these are precisely the groups Hp,q(pt; Z_) a*
*nd
Hp,q(Z=2; Z_). So it's easy to see that K(Z(n), 2n) satisfies the conditions i*
*n the
above lemma. The general strategy at this point would be to
(a)Show that Fn also satisfies the conditions of the lemma;
(b)Observe that [S2n+k,n, Fn]* = 0 = [Z=2+ ^ S2n+k,n, Fn]* for k > 0, for trivi*
*al
reasons;
(c)Show that the map P2n(S2n,n) ! Fn induces isomorphisms on [S2n,n, ]* and
[Z=2+ ^ S2n,n, ]*;
(d)Use Lemma 4.4 to deduce that P2n(S2n,n) ! Fn is a weak equivalence.
In fact this approach can be streamlined a bit by using the functors P as a c*
*rutch.
Lemma 4.5.
(a)Let X be a pointed Z=2space with the property that the forgetful*
* map
[S2n,n, X]* ! [S2n, X]e*is injective. Then the natural map P2nX ! P2nX
is a weak equivalence.
(b)P2n(Z x BU) ! P2n(Z x BU) is a weak equivalence.
Proof.For (a) we only have to show that [S2n+p,n+p, P2nX] = 0 for all p > 0;
that is, we must show that P2nX is null with respect to A(2n,n), not just A~(2n*
*,n)
(see section 3.4). Consider the basic Puppe sequence Z=2+ ! S0,0! S1,1!
Z=2+ ^ S1,0. Smashing with S2n,nyields
Z=2+ ^ S2n,n! S2n,n! S2n+1,n+1! Z=2+ ^ S2n+1,n.
Mapping this sequence into P2nX gives the top edge of the following diagram:
[S2n+1,n+1, P2nX]oo__[Z=2+ ^ S2n+1,n, P2nX] = 0


fflffl
[Z=2+ ^ S2n,n,OP2nX]oo___[S2n,n,OP2nX]OO
~= ~=
 
[Z=2+ ^ S2n,n, X]oo______[S2n,n,_X].
The rightmost group in the top row is zero just because of the definition of P*
*2n,
and Proposition 3.6(b) implies that the labelled vertical maps are isomorphisms.
The map in the bottom row may be identified with the forgetful map
[S2n, X]e [S2n,n, X],
and we have assumed that this is injective. It's now clear that [S2n+1,n+1, P2n*
*X]
must be zero.
Smashing the above Puppe sequence with S2n+p,n+pgives
S2n+p,n+p! S2n+p+1,n+p+1! Z=2+ ^ S2n+p+1,n+p.
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 17
By induction we know that P2nX is null with respect to the first and third spac*
*e,
so it is also null with respect to the second. This finishes (a).
Proving (b) is of course just a matter of checking that Z x BU has the proper*
*ty
specified in (a). So we must check that the forgetful map
Z = gKR 0,0(S2n,n) ! ~K0(S2n) = Z
is injective. But the map is easily seen to be an isomorphism, as fin is an exp*
*licit
generator for both the domain and target.
Proof of Theorem 4.1.We must show that j :P2n(S2n,n) ! Fn is a weak equiva
lence. The equivalences P2n(ZxBU) ! P2n(ZxBU) induce equivalences Fn ! Fn,
and we already know Fn ' K(Z(n), 2n). Lemma 4.4 now implies that j must be a
weak equivalence (one uses the fact that fin is a generator for KR2n,n(pt)).
Proof of Corollary 4.2.This is just a a restatement, together with the fact tha*
*t the
holim of the tower is Z x BU. The latter follows from Proposition 3.7(a,b).
5.Properties of the spectral sequence
If X is a Z=2space, then the associated homotopy spectral sequence for the
tower of Corollary 4.2 has the form
q_ pq,0
Hp,2(X; Z_) ) [S ^ X+ , Z x BU]*,
being confined to the quadrant p, q 0. This is an unstable version of the spe*
*ctral
sequence we're looking for. Producing the stable version is not difficult, as o*
*ne can
replace X by various suspensions Sa,b^ X and use the periodicity of Z x BU to g*
*et
a `family' of spectral sequences which patch together. We'll take another appro*
*ach
to this in the next section, and for now be content with analyzing the unstable*
* case.
5.1. Adams operations.
There is a map of Z=2spaces _k : Z x BU ! Z x BU inducing the operation _k
on KR0(X), constructed out of the ~i maps in the usual way. The functoriality of
the constructions P2n shows that _k induces a selfmap of the Postnikov tower f*
*or
Z x BU, and therefore we get an action of the Adams operations on the spectral
sequence. We must identify the action on the E2term:
Proposition 5.2. The induced map _k : Fn ! Fn coincides with the multiplication
by kn map K(Z(n), 2n) ! K(Z(n), 2n).
Proof.If fin : S2n,n! Z x BU is the nth power of the Bott element, then we know
the following diagram commutes:
S2n,n_____//Z x BU
.kn _k
fflffl fflffl
S2n,n_____//Z x BU.
This is just because we can compute
_k(fin) = (_kfi)n = (kfi)n = knfin.
18 DANIEL DUGGER
Applying P2n to the above diagram gives
S2n,n_____//P2n(S2n,n)___//P2n(Z x BU)
kn P2n(kn) P2n(_k)
fflffl fflffl fflffl
S2n,n_____//P2n(S2n,n)__//_P2n(Z x BU).
We have previously identified the map Fn ! P2n(Z x BU) with P2n(S2n,n) !
P2n(Z x BU), and so the argument may be completed by proving the following
lemma.
Lemma 5.3. Let k 2 Z and let k : S2n,n! S2n,ndenote the map obtained by
adding the identity to itself k times in the group [S2n,n, S2n,n]* (using the f*
*act that
S2n,nis a suspension). Then the localized map
P2n(k) : P2n(S2n,n) ! P2n(S2n,n)
may be identified with the map K(Z(n), 2n) ! K(Z(n), 2n) representing multipli
cation by k.
Proof.There are several ways one could do this. Write S for S2n,nand P for
P2n(S). We of course have the diagram
S _____//P
k  P(k)
fflfflfflffl
S ____//_P.
Using Proposition 3.3(a) it's easy to see that [P, P ]* ! [S, P ]* is an isomor*
*phism,
and the arguments in Section 8 show that [S, P ] ~=Z is generated by the locali*
*zation
map S ! P . This proves it.
5.4. The rational tower.
Grassmannians have nice Schubert cell decompositions, which make it easy to
compute H*,*(). One of course finds that H*,*(BU) = H*,*(pt)[c1, c2, . .].where
cihas degree (2i, i). If we regard cn as a map BU ! K(Z(n), 2n), then applying *
*P2n
gives P2n(BU) ! P2nK(Z(n), 2n) = K(Z(n), 2n) (the EilenbergMacLane space is
already A(2n,n)null). We claim the composite
n) cn
K(Z(n), 2n) = P2n(S2n,n) P(fi!P2n(Z x BU) ! K(Z(n), 2n)
is multiplication by (n  1)!. As in the last section, the argument comes down *
*to
knowing that cn(fin) is (n  1)! times the generator of ~H2n,n(S2n,n). This can*
* be
deduced via comparison maps to the nonequivariant groups, where the result is
wellknown (due to Bott, originally).
So we see that the inclusions of homotopy fibers K(Z(n), 2n) ! P2n(ZxBU) are
split rationally; hence the spectral sequence collapses rationally. If nH2n,n(*
*X) Q
is finitedimensional, then KR0(X) Q decomposes into eigenspaces of the Adams
operations.
5.5. Convergence.
The homotopy spectral sequence for a bounded below tower is automatically
conditionally convergent [Bd , Def. 5.10]. So if RE1 = 0 it converges strongly,*
* by
[Bd , 7.4] and the Milnor exact sequence.
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 19
5.6. Multiplicativity.
The proof of multiplicativity follows the same lines as the nonequivariant ca*
*se,
which is written up in detail in [D2 ]. We will only give an outline.
One starts by letting Wn be the homotopy fiber of Z x BU ! P2n2(Z x BU),
and these come with natural maps Wn+1 ! Wn. The square
Z x BU _______//P2n2(Z x BU)
 
 
fflffl fflffl
P2n(Z x BU) _____//P2n2(Z x BU)
gives us a map Wn ! Fn, where Fn is the homotopy fiber of the bottom map (which
we know is a K(Z(n), 2n)). Routine nonsense shows that Wn+1 ! Wn ! Fn is a
homotopy fiber sequence, and so we have a tower
K(Z(3),O6)O K(Z(2),O4)O K(Z(1),O2)O K(Z(0),O0)O
   
   
   
. . ._______//_W3__________//W2__________//_W1______//_W0 = Z x BU
with holimWn ' *. The spectral sequence for this tower is isomorphic to the
spectral sequence for our Postnikov tower: in fact, there is a map of towers P*
**(Zx
BU) ! W* which induces weak equivalences on the fibers.
At this point the goal becomes to produce pairings Wm ^ Wn ! Wn+m which
commute onthenose with the maps in the towers, and where W0^W0 ! W0 is the
usual multiplication on ZxBU. It is easy to produce pairings which commute up to
homotopy with the maps in the towers, and then an obstruction theory argument
shows that the maps can be rigidified. This is the standard argument, and has
been written up in detail in [D2 ]. The only thing which requires much thought
in the present context is carrying out the relevant equivariant obstruction the*
*ory,
but this is not hard in the end. We omit the details because the argument is not
particularly revealing.
The pairings Wm ^ Wn ! Wm+n now induce a multiplicative structure on the
homotopy spectral sequence in the usual way; the reader is again referred to [D*
*2 ].
5.7. The weight filtration and the flfiltration.
The weight filtration on KR0(X) = [X, Z x BU] is the one defined by the above
tower: F nKR0(X) is defined to be the image of [X, Wn] in [X, Z x BU], or equiv
alently as the subgroup of [X, Z x BU] consisting of all elements which map to 0
in [X, P2n2(Z x BU)]. This is a multiplicative filtration, and by Proposition *
*5.2
it has the property that if x 2 F nKR0(X) then _kx = knx (mod F n+1). If X
is a space for which H2n,n(X) = 0 for n o 0 we know that _k acts diagonally
on KR0(X) Q, with eigenvalues k0, k1, k2, etc. The tower shows that F n Q
coincides with the sum of the eigenspaces corresponding to ki, for i n.
In SGA6, Grothendieck introduced the flfiltration on algebraic K0, designed
to be an algebraic substitute for the topological filtration induced by the cla*
*ssical
AtiyahHirzebruch spectral sequence. For any rank 0 stable Real bundle , on a
Z=2space X, one has elements fli(,) 2 KR0(X) (see [Gr , Sec. 14] for a nice
exposition). The flfiltration is defined by letting Ffnlbe the subgroup of KR0*
*(X)
generated by all products fli1(,1)fli2(,2) . .f.lik(,k), where the ,i's are ran*
*k 0 stable
20 DANIEL DUGGER
bundles over X and i1 + i2 + . .+.ik n. (So it is the smallest multiplicative
filtration in which fli(,) is in F i). By playing around with the algebraic def*
*initions
of fli and _k, one can see that Ffnl Q also coincides with the sum of eigenspa*
*ces
of _k for the eigenvalues ki, i n (an explanation can be found in [Gr , Sec. *
*14]).
Proposition 5.8. For any Z=2space X one has FfnlKR0(X) F nKR0(X). If
H2n,n(X) = 0 for n o 0, this becomes an equality after tensoring with Q.
Proof.We have already discussed the agreement rationally, since both filtrations
give eigenspace decompositions for the Adams operations. To understand the inte
gral story, one regards fli as a map BU ! ZxBU (or as an element of KR0(BU)).
If E is a Real bundle of dimension i < n then one can see algebraically that
fln(E  i) = 0. So fln is null on BU(n  1), and hence factors through the
homotopy cofiber BU=BU(n  1). Both BU and BU(n  1) are weakly equiv
alent to Grassmannians, and one finds that HZ_*,*(BU) = HZ_*,*(pt)[c1, c2, . .].
(where ci has degree (2i, i)), and HZ_*,*(BU(n  1)) = HZ_*,*[c1, c2, . .,.cn1*
*]. So
it follows that HZ_*,*(BU=BU(n  1)) = HZ_*,*(pt)[cn, cn+1, . .].. In particul*
*ar,
HZ_2i,i(BU=BU(n1)) = 0 for i < n. Therefore the map fln :BU=BU(n1) ! Zx
BU lifts to Wn in the tower, and any element fln(E i) belongs to F nKR0(X).
6.Connective KRtheory
The final task is to stabilize the spectral sequence we produced in the previ*
*ous
section. That spectral sequence converged to KRp+q(X) only for p + q < 0, and
we'd like to repair this deficiency. This is not at all difficult, and proceeds*
* exactly
as in the nonequivariant case. What we will do is construct a öc nnective" ver*
*sion
of KRtheory, represented by a spectrum we'll call kr. There will be a homotopy
cofiber sequence
2,1kr fi!kr ! HZ_,
and the Bockstein spectral sequence associated with the map fi will give the st*
*abi
lized version of the spectral sequence we've been considering.
As in section 5.6, Wn denotes the homotopy fiber of Z x BU ! P2n2(Z x BU).
Proposition 6.1. There are weak equivalences Wn ! 2,1Wn+1, unique up to
homotopy, which commute with the Bott map in the following diagram:
Wn ________//_ 2,1Wn+1
 
 
fflffl fflffl
Z x BU ____//_ 2,1(Z x BU).
Proof of Proposition 6.1.Consider the natural map ff : Z x BU ! P2n(Z x BU),
and apply 2,1(). The fact that X 2 A(2n2,n1)) S2,1^ X 2 A(2n,n)shows
that 2,1P2n(Z x BU) is A(2n2,n1)null. By Proposition 3.3(a) this implies th*
*ere
is a lift
2,1(Z x BU)_______//_ 2,1P2n(Z x BU)
fflffl _____55_________
 _______________
fflffl____l___
P2n2( 2,1(Z x BU))
and this lift is unique up to homotopy.
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 21
Now let fi : ZxBU ! 2,1(ZxBU) be the Bott map, and consider the diagram
Wn __________//Z x BU_________//_P2n2(Z x BU)
fi Pfi
fflffl fflffl
2,1(Z x BU)____//_P2n2( 2,1(Z x BU))
id l
fflffl fflffl
2,1Wn+1 ____//_ 2,1(Z x BU)____//_ 2,1P2n(Z x BU)
It follows that there is a map on the homotopy fibers Wn ! 2,1Wn+1 making
the diagram commute. We need to show that this is a weak equivalence, and the
procedure is one which should be familiar by now: we use Lemma 4.4.
Using Proposition 3.6(a,b) and Lemma 4.5, one shows that
o [Sk,0, Wn]* = 0 for 0 k < n,
o [Z=2+ ^ Sk,0, Wn]* = 0 for 0 k < 2n, and
o the same is true with Wn replaced by 2,1Wn+1.
The definition of P2n2 yields that the maps
[S2n+k,n, Wn]* ! [S2n+k,n, Z x BU]* and
[Z=2+ ^ S2n+k,n, Wn]* ! [Z=2+ ^ S2n+k,n, Z x BU]*
are isomorphisms for k 0, using the fact that [S2n+k,n, P2n2(Z x BU)] = 0, e*
*tc.
Then the square
Wn __________//Z x BU
 
 'fi
fflffl fflffl
2,1Wn+1 ____//_ 2,1(Z x BU)
shows at once that Wn ! 2,1Wn+1 induces an isomorphism on [S2n+k,n, ]* and
[Z=2+ ^ S2n+k,n, ]* for k 0. By Lemma 4.4, Wn ! 2,1Wn+1 is an equivalence.
Definition 6.2. Let kr be the equivariant spectrum consisting of the spaces {Wn}
and the maps Wn ! 2,1Wn+1 given by the above proposition. The object kr is
called the connective KRspectrum.
The spectrum for 2,1kr has nth space equal to Wn+1, so the maps Wn+1 !
Wn give a `Bott map' 2,1kr ! kr. Corollary 4.2 identifies the homotopy fiber as
1,0HZ_, which is equivalent to the homotopy cofiber being HZ_. So we may form
the tower of homotopy cofiber sequences
2,1fi
. ._.___// 2,1krfi__//kr____//_ 2,1kr___//. . .
  
  
fflffl fflffl fflffl
2,1HZ_ HZ_ 2,1HZ_
The colimit of the spectra in the tower is clearly KR, and the homotopy inverse
limit is contractible (these follow from thinking about the spaces in the spe*
*ctra for
everything in the tower). This gives a stable versionqof the spectral sequence *
*we've
been considering: for any space X we have Hp,_2(X) ) KRp+q,0(X). It converges
22 DANIEL DUGGER
conditionally because the holim of the tower is contractible, and if RE1 = 0 th*
*en it
converges strongly by [Bd , Thm 8.10] (in the language of that result, the cond*
*ition
`W = 0' is easily checked to hold).
Remark 6.3. The Postnikov tower we've constructed_and its resulting spectral
sequence_can be used to completely determine the homotopy groups of the spaces
Pn(ZxBU), and hence of Wn as well. In other words, we can completely determine
the groups kr*,*(pt), and in fact the ring structure can also be deduced. At t*
*he
moment, however, the answer doesn't seem to admit a simple description_in this
sense it is somewhat like the ring H*,*(pt; Z_), only more complicated. It is *
*not
true that kr*,*(pt) ~=H*,*(pt)[v], as one might naively guess based on the non
equivariant case. The reason essentially comes down to the fact that there are *
*non
trivial differentials in the spectral sequence when X = pt (see below). The pap*
*er
[HK1 ] computes the much more complicated ring MR*,*(pt), and their methods
can be used to give kr*,*(pt) as well.
6.4. The spectral sequence for X = pt.
In the following diagram we draw the spectral sequence
Hp(pt; Z(q_2)) ) KRp+q(pt) = KOp+q(pt),
but using Adams indexing rather than the usual Serre conventions. In spot (a, b)
we have drawn Hb,a+b_2(pt), and the vertical line a = N gives the associated gr*
*aded
of KON . Said differently, the aaxis measures (p + q) and the baxis measures
p. 

b 
_6 q BM q
 q B q q
 q BBMBq q
6_ BM BM
 q BM B Bq q BMB
4_ q B Bq q B Bq
 q BBMqB q BBMBq
 BM BM
2_ q BMB Bq q BMB Bq
 q dB qB q B Bq q
q c q c q c q c _cc cB c Bc c
_____________________________________________________________________oeqqq*
*02460
BBM q q BBMqd _ a
Bq q BBMqB 
BBMB 
q BM q _
q B qB 
q BBMBq _
BBMB 
q BM q 
B Bq _
BBMBq 
B 
q _
?
There are several points to make:
(a)Using the multiplicative properties of the spectral sequence, one only has to
determine the two differentials labelled `d'_all the others can be deduced f*
*rom
these. Since we know the groups KO*(pt), it's clear that these two different*
*ials
have to exist. (It would be nice to have a more intrinsic explanation, howev*
*er).
(b)The spectral sequence collapses at the next page.
(c)The unstable spectral sequence of sections 4 and 5 is the part in the first *
*quad
rant. We can read off the action of the Adams operations on KOn(pt) for
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 23
n 0 directly from the diagram, the `weight lines' being along the antidiag*
*o
nals: KO0 is pure of weight 0, KO1 is pure of weight 1, KO2 and KO4
are of weight 2, KO8 is of weight 4, etc.
(d)Everything about the ring structure on KO*, as well as the comparison map
KO* ! K*, can be read off of this spectral sequence and the corresponding
spectral sequence where X = Z=2 (which lies entirely along the b = 0 line, a*
*nd
hence collapses). They can be deduced from our knowledge of H*,*(pt) and
H*,*(Z=2) provided by Theorem 2.8.
(e)The part of the spectral sequence in the first quadrant is known to topologi*
*sts
in another setting: it's the Adams spectral sequence for bo based on bu.
7. 'Etale analogs
The difference between algebraic Ktheory and 'etale Ktheory, or motivic co
homology and 'etale motivic cohomology, is very familiar in the motivic setting.
In this section we play with similar ideas in the Z=2 world. The analogs are we*
*ll
known, although the only source seems to be [MV , Section 3.3], which doesn't d*
*e
velop things in much detail. We use these ideas to give a proof of the classica*
*l fact
that KZ=2' KO and (Z x BU)hZ=2' Z x BO.
Let us return briefly to the setting where G is any finite group. By an equiv*
*ari
ant covering space E ! B we mean an equivariant map which, after forgetting
the Gactions, is a covering space in the usual sense. Given such a map we may
form its ~Cech complex ~C(E), which is the simplicial space
E oo___EoxBoE_oo___EoxBoE_xBoEo_oo___.o.o._oo_oo_
(where we have omitted the degeneracies for typographical reasons). In non
equivariant topology, the map hocolimnC~(E)n ! B is a weak equivalence (cf.
[DI, Cor. 1.3]). This is not true equivariantly, as the covering space G ! * sh*
*ows.
In this case the realization of the ~Cech complex is precisely EG, and EG ! * is
not an equivariant equivalence. We will see that in some sense this turns out t*
*o be
the only problem, though.
If Z is a Gspace there is a natural map of Gspaces
F (B, Z) ! holimnF (C~(E)n, Z)
(here F (X, Y ) is the usual mapping space, with its induced Gaction). We will*
* say
the space Z satisfies 'etale descent for the covering E ! B if this natural
map is an equivariant weak equivalence. This is the same as requiring that the
corresponding maps for the coverings G=H x E ! G=H x B all be nonequivariant
equivalences, where H ranges over the subgroups of G. If the phrase is not qual*
*ified,
then "'etale descent" means "'etale descent for all covering spaces".
Using results of [H , Chaps. 3,4], we may localize the model category TopG at
the maps hocolim~C(E) ! B where E ! B ranges over the elements of the set
{G=H x G ! G=H x *H < G}. This produces a new model category structure
which we'll denote TopetG. As the following result shows, the fibrant objects *
*are
precisely the spaces which satisfy 'etale descent for all covering spaces.
Proposition 7.1.
(a)If Z is any Gspace, then ZEG satisfies 'etale descent.
24 DANIEL DUGGER
(b)A map X ! Y is a weak equivalence in TopetGiff it is a nonequivariant weak
equivalence.
(c)For any space Z, the map Z ! ZEG is a fibrant replacement in TopetG.
Proof.Let X ! Y be a Gmap which is also a nonequivariant equivalence, and
assume that X and Y are cofibrant. Then X x EG ! Y x EG is an equivariant
equivalence, and therefore so is F (Y x EG, Z) ! F (X x EG, Z) for any Z. By
adjointness this map is the same as F (Y, ZEG ) ! F (X, ZEG ). In particular, *
*if
E ! B is an equivariant covering space then by taking X ! Y to be the map
hocolim~C(E) ! B (which is a nonequivariant equivalence by [DI, Cor. 1.3]) we
find that ZEG satisfies 'etale descent. This proves (a).
When forming TopetGwe are localizing at maps which are nonequivariant equiv
alences. It follows from this that every equivalence in TopetGis a nonequivari*
*ant
equivalence. For (b), we must show the other direction. Note that if Z is a fib*
*rant
object in TopetG, then Z ! ZEG is an equivariant equivalence (this is 'etale d*
*escent
for G ! *). If X ! Y is a map between cofibrant objects, we may consider the
diagram
~=
F (Y, Z)_~__//_F (Y, ZEG_)_//_F (Y x EG, Z).
  
  
fflffl~ fflffl~= fflffl
F (X, Z)____//_F (X, ZEG_)_//_F (X x EG, Z)
If X ! Y was a nonequivariant equivalence then X x EG ! Y x EG is an
equivariant equivalence, and so the right vertical map is an equivalence as wel*
*l.
It follows that F (Y, Z) ! F (X, Z) is an equivariant equivalence for every fib*
*rant
object Z in TopetG, and therefore X ! Y is an equivalence in TopetG.
Part (c) is an immediate consequence of (a) and (b).
We will call ZEG the 'etale localization (or Borel localization) of the space
Z, and we'll sometimes write it as Zet. Note that 'etale localization preserves*
* fiber
sequences and homotopy limits.
Everything from our above discussion generalizes directly to spectra as well.
We can talk about an equivariant spectrum E which satisfies 'etale descent_these
correspond to what are usually called `Borel cohomology theories' [M , p. 233].
If E is the RO(G)graded spectrum given by V ! EV , its 'etale localization (or
corresponding Borel theory) is the spectrum Eetgiven by V ! EEGV. Note that if
E was an spectrum then Eetis also an spectrum.
At this point we switch back to the Z=2setting, where we can write down the
following two results. The first is an immediate consequence of Corollary 4.2, *
*the
second of Theorem 1.5.
Proposition 7.2. There is a tower of homotopy fiber sequences
. . .____//[P4(ZOxOBU)]et__//_[P2(ZOxOBU)]et__//[P0(ZOxOBU)]et__//_*
  
  
  
K(Z(2), 4)et K(Z(1), 2)et Zet
and the homotopy limit of the tower is (Z x BU)et.
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 25
Proposition 7.3. There is a tower of homotopy cofiber sequences in Z=2spectra
of the form
2,1fi
. ._.___// 2,1kretfi_//_kret___//_ 2,1kret__//. . .
  
  
fflffl fflffl fflffl
2,1HZ_et HZ_et 2,1HZ_et
The homotopy colimit of the tower is KRet, and the homotopy limit is contractib*
*le.
We want to analyze the spectral sequence which comes from the above tower, so
to start with we need a knowledge of HZ_et. The theory HZ_ethas coefficient gro*
*ups
described as follows:
Proposition 7.4.
(a)HZ_*,*et(pt) = Z[x, x1, y]=(2y) where x has degree (0, 2) and y has degree *
*(1, 1).
(b)HZ_*,*et(Z=2) = Z[u, u1] where u has degree (0, 1).
(c)The Mackey functor HZ__0,2netis equal to Z_, for all n.
(d)The map HZ_*,*(Z=2) ! HZ_*,*et(Z=2) is an isomorphism. The map HZ_p,q(pt) !
HZ_p,qet(pt) is an isomorphism when p < 2q, and multiplication by 2 when p =*
* 0
and q < 0 is even. These are the only degrees in which its possible to have a
nonzero map.
The proof is given in Appendix B. q_
Proposition 7.3 gives rise to a spectral sequence HZ_p,2et(X) ) KRp+q,0et(X).
The spectral sequence for X = pt is drawn below, using the same indexing conven
tions as in section 6.4.

q q p_6q q q q
BM  BM BM
q BM B q qBMB q q BM B q q
q B Bq q _B Bq q B Bq q
q BBMqB q BBqMB q BBMBq q BBM
BBMB BBMB BBMB BBMB
q BM q q BM _q q BM q q BM q
q B qB q B Bq q B Bq q B Bq
q BBMqB q BBMBq _ q BBMqB q BBMBq
BBMB BBMB 2  BBMB BBMB
q B q q B q  q d B q q B q q
_____________________________________________________________________oeccc*
*ccccccc_0
0 2 4 6 (p + q)
_

_


?
Note that there is a map from the spectral sequence of Theorem 1.6 to the one
above (coming from the maps HZ_ ! HZ_etand KR ! KRet). Based on our
discussion in section 6.4 we therefore know that the differential labelled `d' *
*has to
exist_in fact, we know all the differentials in the first quadrant below the y *
*= x
line. Multiplicativity then allows us to deduce what's happening in the rest of*
* the
spectral sequence, and that is what is shown above. As we see, the spectral seq*
*uence
again converges to KO*; but this time we've actually gained some information:
Corollary 7.5. The natural map KR ! KRetis a weak equivalence_that is, KR
satisfies 'etale descent.
26 DANIEL DUGGER
In very fancy language, this says `the QuillenLichtenbaum conjecture holds for
KR'.
Proof.The map is of course a nonequivariant equivalence, so we only have to ana
lyze what happens on fixed sets_i.e., we study theqmap ff: KR*,0(pt) ! KR*,0et(*
*pt).
Consider the map of spectral sequences from HZp,_2(pt) ) KRp+q,0(pt) to the co*
*r
responding 'etale version. Using everything we know about both spectral sequenc*
*es,
as well as what Proposition 7.4(d) says about the map HZ_*,*(pt) ! HZ_*,*et(pt)*
*, it is
easy to see that ff is an isomorphism when * 0, and also when * = 4n. The fa*
*ct
that ff is a ring map then gives us the isomorphism in the remaining dimensions.
Corollary 7.6. If we consider Z x BU as a nonequivariant space with its Z=2
action, then (Z x BU)hZ=2' Z x BO. Similarly, KhZ=2' KO.
Proof.This is just a translation of the previous corollary. The 0th space in t*
*he
spectrum of KR is Z x BU, whereas the 0th space in the spectrum of KRetis
(Z x BU)EZ=2. Corollary 7.5 tells us that these spaces are equivariantly equiva*
*lent,
and therefore they have weakly equivalent fixed sets. The proof that KhZ=2' KO
follows the same lines, but takes place in the stable category.
Corollary 7.6 (and 7.5, which is equivalent) is of course well known_a recent
reference is [K ]. In the end our proof is only slightly different from the cl*
*assical
proof which analyzes ß*KhZ=2via the spectral sequence for ß* of a homotopy limi*
*t,
making use of the map KO ! KhZ=2 (in fact this spectral sequence has the same
form as the one drawn above). It may be worth summarizing our proof to exhibit
the similarities. The spectral sequence relating HZ and KR, when applied to
a point, gave us something converging to KO (the fixed set of KR). Because we
knew the homotopy groups of KO, we could analyze the differentials in this spec*
*tral
sequence. On the other hand we have a corresponding spectral sequence relating
HZetand KRet; when applied to a point, it converges to KhZ=2 (the fixed set of
KRet). There is a comparison map of spectral sequences, and our knowledge of the
differentials in the first lets us deduce the differentials in the second.
8. Postnikov sections of spheres
In this final section we prove Theorem 3.8, which identifies the Postnikov se*
*ction
PV (SV ) with the EilenbergMacLane space K(Z_, V ) over the group G = Z=2.
Theorem 1.1(a), which is all we really need in this paper, is the case where V *
*= Cn.
This section takes place entirely in the context of Z=2spaces.
Suppose that V contains a copy of the trivial representation. It follows fr*
*om
[dS] and Corollary A.8 of the present paper that the infinite symmetric product
SP 1(SV ) is a model for K(Z_, V ) (this actually works over any finite group, *
*not
just G = Z=2). Using this, we have:
Lemma 8.1. If V 1 then Sp1 (SV ) ! PV (Sp1 (SV )) is a weak equivalence.
Proof.We know by Theorem 2.8 that for any r, s 0
[SV +rR+sR, K(Z_, V )]*=Hrs,s(pt; Z_) = 0 and,
[Z=2+ ^ SV +rR+sR, K(Z_, V )]*=Hrs,s(Z=2; Z_) = 0
as long as r and s are not both zero. This shows that Sp1 (SV ) is AV null, wh*
*ich
implies that the map in the statement of the lemma is a weak equivalence.
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 27
Our goal is the following:
Theorem 8.2. If V 1 and SV ! Sp1 (SV ) is the obvious map, then
PV (SV ) ! PV (Sp1 (SV ))
is a weak equivalence. Therefore PV (SV ) ' K(Z, V ).
Note that the second statement follows immediately from the first, in light of *
*the
above lemma. The proof of the first statement is based on a geometric analysis *
*of
the infinite symmetric product construction, and involves the following steps:
(1)For each k 2 we produce a homotopy cofiber sequence
i j
SV ^ [V ~R]  0 = k ! SP k1(SV ) ! SP k(SV )
where R~is the reduced standard representation of the symmetric group k
(defined below) equipped with the trivial Z=2 action.
(2)We show that i j
PV SV ^ ([V ~R]  0)= k ' *.
The key ingredient for this is a geometric analysis of the Z=2fixed sets of*
* the
orbit space ([V ~R]  0)= k.
(3)From (1) and (2) it follows that
PV (SP k1(SV )) ! PV (SP k(SV ))
is a weak equivalence for every k 2, and passing to the limit yields the s*
*ame
for PV (SV ) ! PV (Sp1 (SV )).
8.3. Step 1. Consider the filtration of Sp1 (SV ) given by the finite symmetric
products:
SV SP 2(SV ) SP 3(SV ) . . .Sp1 (SV ).
Recall that the standard representation of the symmetric group k is the
space R = Rk where the group acts by permuting the standard basis elements.
This contains a trivial, onedimensional subrepresentation consisting of all ve*
*ctors
(r, r, . .,.r), and the reduced standard representation ~Ris the quotient of R
by this subrepresentation. We regard R and ~Ras Z=2representations by giving
them the trivial actions.
The following proposition was inspired by [JTTW , Thm. 2.3], which handled
the case k = 2:
Proposition 8.4. The inclusion SP k1(SV ) ,! SP k(SV ) sits in a homotopy
cofiber sequence of the form
SV ^ ([V ~R]  0)= k ! SP k1(SV ) ! SP k(SV )
where ~Rdenotes the reduced standard representation of k.
Proof.To save ink, write B = B(V ) and S = S(V ) for the unit ball and unit sph*
*ere
in V . We begin with the relative homeomorphism
~= V
(B, S) ! (S , *).
Applying SP kto these pairs gives a relative homeomorphism
~= k V k1 V
(SP k(B), Z= k) ! (SP (S ), SP (S ))
28 DANIEL DUGGER
where Z is the space
(S x B x . .x.B) [ (B x S x B x . .x.B) [ . .[.(B x . .x.B x S) Bk.
This says that there is a pushout square of the form
Z= k _____//_SP k1(SV )
fflffl fflffl
 
fflffl fflffl
SP k(B) _____//_SP k(SV ).
Since SP k(B) is clearly contractible, the desired cofiber sequence will follow*
* if we
can identify Z= k with SV ^ ([V ~R]  0)= k.
Z naturally includes into (V . . .V )  0 = [V R]  0, and the assignment
Z ! S(V . . .V ), v ! v_v
is a homeomorphism which is both Z=2 and k equivariant. So we may identify
the Z=2spaces Z= k and S(V R)= k.
Since R decomposes as R ~R(as both k and Z=2representations), we have
the corresponding decomposition V R ~=V [V ~R]. Lemma 8.5 below gives a
homeomorphism
i j
S V [V ~R] = k ~=S(V ) * [S(V R)= k],
where X * Y denotes the usual join of X and Y . It is a general fact (true in
any model category) that for pointed spaces X and Y , the join X * Y is weakly
equivalent to (X ^ Y ). Because V 1, both S(V ) and S(V ~R) have nonempty
Z=2fixed sets, and therefore can be made pointed. So we finally conclude that
Z= k ~=S(V ) * [S(V ~R)= k]' S1 ^ S(V ) ^ [S(V ~R)= k]
' SV ^ ([V ~R]  0)= k.
Lemma 8.5. Let V and W be orthogonal representations of Z=2. Let G be a finite
group acting Z=2equivariantly and orthogonally on W , and let G act on V trivi*
*ally.
Then there is a natural Z=2equivariant homeomorphism
~=
S(V ) * [S(W )=G] ! S(V W )=G
where the space on the left denotes the join.
Proof.A point in the lefthand space can be represented by a triple (v, t, [w])*
* where
v 2 S(V ), t 2 [0, 1], w 2 S(W ), and [w] denotespthe_Gorbit_ofpw._We leave it*
* to
the reader to check that the map (v, t, [w]) 7! [ 1  t. v t . w] is welld*
*efined
and a Z=2equivariant homeomorphism.
8.6. Step 2.
Proposition 8.7. Suppose V = Rp (R )q, where p 1. Let X = ([V R~]0)= k.
(a)The fixed set XZ=2 is pathconnected for k 3.
(b)When k = 2, XZ=2 ' RP p1q RP q1(where the second summand is inter
preted as ; when q = 0, and a point when q = 1). When q 1 there exists a
map S1,1! X which on fixed sets induces an isomorphism on ß0.
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 29
Proof.The argument involves producing explicit paths in the fixed sets. As it's
somewhat lengthy, we postpone it until the very end of the section.
Corollary 8.8. Consider the set A consisting of the objects
o Sn,0, n 1; o Sn,0^ Z=2+, n 1; o S1,1.
Then the nullification PA X at this set is contractible.
Proof.Let P X denote the nullification of X. To show P X is contractible we have
only to check that [Sn,0, X]* = 0 = [Sn,0^ Z=2+, X]* for all n 0. For n 1
this follows just from the definition of P X. We are therefore reduced to check*
*ing
n = 0, which is the statement that both P X and (P X)Z=2are pathconnected (as
nonequivariant spaces).
Clearly X is path connected. Attaching cones on maps cannot disconnect the
space, so P X must also be pathconnected.
Proposition 8.7 says that XZ=2 is pathconnected for k 3 (or k = 2 and q = *
*0),
and attaching cones on maps cannot disconnect the fixed set. So (P X)Z=2is again
pathconnected in this case.
When k = 2 and q 1 the proposition says that XZ=2has two path components,
but they are linked by an S1,1. Attaching a cone on this map will give a space *
*whose
fixed set is connected, and then reasoning as in the previous paragraph we find*
* that
P X will also have that property.
Corollary 8.9. The space PV (SV ^ ([V ~R]  0)= k) is contractible.
Proof.Let X = ([V ~R]  0)= k, and suppose we cone off a map S1,0! X to
make a space X0:
S1,0! X ! X0.
Smashing with SV gives a cofiber sequence
SV +1! SV ^ X ! SV ^ X0,
and since PV (SV +1) ' * it follows by Proposition 3.3(c) that
PV (SV ^ X) ~!PV (SV ^ X0).
In other words, we may cone off arbitrary maps S1,0! X without effecting the
homotopy type of PV (SV ^ X). The same reasoning shows we can cone off maps
S1,1! X, Sn,0! X, and Z=2+ ^ Sn,0! X (n 1) with the same result. So the
conclusion is that
PV (SV ^ X) ' PV (SV ^ P X),
where P X denotes the nullification considered in Corollary 8.8. But that corol*
*lary
says that P X is contractible, and so we're done.
8.10. Step 3.
Proof of Theorem 8.2.We are to show that the map SV ! Sp1 (SV ) becomes
a weak equivalence after applying PV . We'll simplify PV (X) to just P (X), and
SP k(SV ) to just SP k. Proposition 8.4 gives cofiber sequences
SV ^ ([V ~R]  0)= k ! SP k1! SP k,
and we have seen that
P (SV ^ ([V ~R]  0)= k) ' *.
30 DANIEL DUGGER
So Proposition 3.3(c) shows that P (SP k1) ! P (SP k) is a weak equivalence.
Hence, one has a sequence of weak equivalences
P (SV ) ~!P (SP 2) ~!P (SP 3) ~!. . .
and therefore P (SV ) ! hocolimkP (SP k) is a weak equivalence as well.
Now look at the composite of the two maps
P (SV ) ! P (hocolimSP k) ! P (colimSP k) = P (Sp1 ).
The middle object may be identified with hocolimkP (SP k) using Proposition 3.3*
*(b)
and some common sense, and so the first map is an equivalence. The second map
is a weak equivalence because hocolimSP k! colimSP kwas one. Hence, the
composite is also a weak equivalence.
8.11. Loose ends: The analysis of the fixed sets [(V ~R 0)= k]Z=2.
The one thing still hanging over our heads is the
Proof of Proposition 8.7.Recall that X = [V ~R]  0= k. Begin by decomposing
V as a sum of irreducibles V = U0 U1 . . .Un where U0 = 1 (or course the only
irreducible representations of Z=2 are R and R , but it's easiest to think in *
*slightly
more generality here). An element of X is represented by a coset [u0, . .,.un] *
*with
ui2 Ui ~Rand at least one ui nonzero.
We begin with part (a), which says that when k 3 the fixed set XZ=2 is path
connected. First note that if u = [u0, . .,.un] 2 XZ=2 and ui6= 0, then
t 7! [tu0, tu1, . .,.tui1, ui, tui+1, . .,.tun]
gives a path in XZ=2 from u to [0, 0, . .,.0, ui, 0, . .,.0]. It follows that i*
*t suffices to
prove the result when V is of the form 1 U, where U is a (possibly 0) irreduc*
*ible
representation. (Recall that the result requires V to contain 1, which is why *
*we
don't reduce to V = U.) Since our group is Z=2, we only have to worry about
V = R and V = R R = C.
The case V = R is trivial, because then XZ=2= X = (R~ 0)= k, and ~R 0 was
path connected because dimR~ 2 (recall k 3 here). So we are left to deal wi*
*th
V = C. In this case V ~Ris the complex reducedPstandard representation, which
we may identify with {(z1, . .,.zk) 2 Ck  zi= 0}. We will write [z1, . .,.zk*
*] for
the coset of (z1, . .,.zk) in (V ~R)= k.
Let A = {[r1, . .,.rk]  ri 2 R} XZ=2. Clearly A is pathconnected, as A ~=
(Rk1 0)= k. We will show that any element of XZ=2 can be connected by a path
to an element of A. If [z1, . .,.zk] 2 XZ=2 then there is a oe 2 k with the pr*
*operty
that
(zoe(1), . .,.zoe(k)) = (~z1, . .,.~zk).
By writing oe as a composite of disjoint cyclic permutations, it's easy to see *
*that
this can only happen if (z1, . .,.zk) has the form
(w1, ~w1, . .,.wl, ~wl, r1, . .,.rj)
up to permutation of the z's (where ri2 R). If all the wi's are real, then our *
*point
is already in A and we can stop. So we can assume that w1 62 R. Consider the pa*
*th
t 7! [w1 + f(t), ~w1+ f(t), tw2, tw~2, . .,.twl, tw~l, tr1, . .,.trj]
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 31
where
f(t) = 1_2(2Re(w1) + 2tRe(w2) + . .+.2tRe(wl) + tr1 + . .+.trj)
(so f(t) is the real number which makes the sum of the components zero in the
previous expression). It's easy to see that this describes a path in XZ=2 conne*
*cting
our original point with
[w1 + f(0), ~w1+ f(0), 0, . .,.0],
and this point has the form [bi, bi, 0, . .,.0] for some nonzero b 2 R. Next *
*we
consider the path
t 7! [t + b(1  t)i, t  b(1  t)i, 2t, 0, 0, . .,.0]
(and here we use the fact that k 3). This is a path in XZ=2 connecting
[bi, bi, 0, . .,.0] with [1, 1, 2, 0, . .,.0], the latter of which is in A. S*
*o this com
pletes the proof that XZ=2 is pathconnected when V = C, and we are done with
part (a).
Now we turn to part (b), which is the case k = 2. The reduced standard
representation of 2 is ~R= R with the 2action equal to multiplication by 1.
If V = Rp Rqthen X = [(Rp Rq)  0]= 2 ' RP p+q1. Using homogeneous
coordinates [r1, . .,.rp, rp+1, . .,.rp+q] on RP p+q1, the Z=2action is the o*
*ne which
changes the signs on the final q coordinates. Hence
XZ=2= {[r1, . .,.rp, 0 . .,.0]  ri2 R} q {[0, . .,.0, rp+1, . .,.rp+q]  ri*
*2 R};
the first set is isomorphic to RP p1, the second to RP q1.
When q 1, consider the map
[(R R )  0]= 1! X
[r, s]7! [r, 0, . .,.0, s, 0, . .,.0],
where the s is placed in the qth spot. It's easy to check that [(R R )0]= 1 '*
* S1,1,
and that on fixed sets the map induces a bijection on the sets of pathcomponen*
*ts.
This is what we wanted.
Appendix A. Symmetric products and their group completions
The goal of this section is to show that Sp1 (S2n,n) is equivariantly weakly
equivalent to AG (S2n,n). The proof is not at all difficult, but requires a few*
* lemmas.
I would like to thank Gustavo Granja for an extremely helpful conversation about
these results. In this section G is a fixed finite group.
Definition A.1.
(a)Let C be a category with products and colimits, and let M be an abelian mono*
*id
object in C. The group completion M+ is the coequalizer of the maps
M x M x M _______////_M x M
jjjjj44j (a, b)
jj
(a, b, c)XXXX
XXXX,,
(a + c, b +.c)
(b)If K is a pointed simplicial set (or topological space), define AG (K) =
Sp1 (K)+ .
32 DANIEL DUGGER
Remark A.2. In the above generality it is not true that M+ will be an abelian
group object in C, or even a monoid object (so the term `group completion' is
somewhat of a misnomer). But this is the case when C = Set, and therefore also
when C = sSet. It also holds when C = Top and M is `sufficiently nice'.
It's easy to see that AG is a functor, so that if K is a simplicial Gset (o*
*r a
Gspace) then AG (K) also has a Gaction. If ~Z[S] denote the free abelian group
on the pointed set S, where the basepoint is identified with the zero element, *
*one
may check that AG(K) is isomorphic to the simplicial set ~Z[K].
Proposition A.3. Let K be a pointed simplicial Gset with the property that KH
is pathconnected for every subgroup H G. Then the natural map Sp1 (K) !
AG (K) is an equivariant weak equivalence.
Proof.If M is a connected simplicial abelian monoid then [Q , Results Q1,Q2,Q4]
show M ! M+ induces an equivalence on integral homology. Since both M
and M+ are nilpotent spaces, the map is actually a weak equivalence. One can
check, with only a little trouble, that AG(K)H is isomorphic to [Sp1 (K)H ]+ , *
*and
that Sp1 (K)H is path connected. So Quillen's result implies that Sp1 (K)H !
AG (K)H is a weak equivalence for every subgroup H. This completes the proof.
The next step is to transport this result from Gsimplicial sets to Gspaces.*
* We
start with two simple lemmas:
Lemma A.4. Let K be a pointed simplicial set.
(i)There are natural homeomorphisms SP nK ! SP nK, for 1 n 1.
(ii)If M is a simplicial abelian monoid, then there is a natural homeomorphism
M+  ! M+ .
(iii)There is a natural homeomorphism AG (K) ! AG (K).
Proof.The essential point is that  commutes with colimits (being a left adjo*
*int)
and also finite products (making use of the fact that our topological spaces are
compactlygenerated and weak Hausdorff). Since SP n(n < 1) is constructed by
forming a finite product and then taking a colimit_namely, passing to norbits_
realization will commute with SP n. But then realization will also commute with
Sp1 , as Sp1 is defined as a colimit of the SP n's. This proves (i).
The proof of (ii) is in the same spirit: M+ is defined as a coequalizer of t*
*wo
products, so    will commute with this construction. Part (iii) is an immedi*
*ate
consequence of (i) and (ii).
Remark A.5. Again, since the above maps are all natural it follows that if K is
a Gsimplicial set then the maps are actually equivariant.
Lemma A.6. If K is a Gsimplicial set, the natural map KH  ! K factors
through the Hfixed set and gives a homeomorphism KH  ! KH .
Proof.The fact that the map factors though KH and that it is injective are
immediate. So the content is that a cell of K which is fixed by H must come f*
*rom
a simplex of KH . But this is obvious.
Proposition A.7. Let X be a Gspace of the form K for some simplicial Gset
K. If all the fixed sets XH are pathconnected, then the map Sp1 (X) ! AG (X)
is an equivariant weak equivalence.
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 33
Proof.What must be shown is that for any subgroup H G the map Sp1 (X)H !
AG (X)H is a weak equivalence of topological spaces. Now X = K, and by the
above lemmas we can commute the realization past the Sp1 , the AG , and the
fixed points. So we are left with showing that Sp1 (K)H  ! AG (K)H  is a we*
*ak
equivalence. This was Proposition A.3.
Corollary A.8. For any V which contains a copy of the trivial representation, t*
*he
map of Z=2spaces Sp1 (SV ) ! AG (SV ) is an equivariant weak equivalence.
Proof.It suffices to show that SV is Z=2homeomorphic to a space of the form K*
*.
It's not hard to verify this for V = R and V = R by writing down an explicit K1
and K2. For a general V = Rp (R )q we have
SV ~= (S1,0^ . .^.S1,0) ^ (S1,1^ . .^.S1,1)
~= K1 ^ . .^.K1 ^ K2 ^ . .^.K2 ~=K1 ^ . .^.K1 ^ K2 ^ . .^.K2.
Appendix B. Computations of coefficient groups
Here we give the proofs of Theorem 2.8 and Proposition 7.4, which compute
the coefficient rings of HZ_and HZ_et. As remarked earlier, these computations *
*are
routine among equivariant topologists_our only goal is to provide a reference f*
*or
the nonexpert.
B.1. HZ computations. For any pointed Z=2 spaces X and Y there is an
isomorphism [Z=2+ ^ X, Y ]* ! [X, Y ]e*obtained by restricting via the inclu
sion {0} ,! Z=2 . So for any equivariant spectrum E there are isomorphisms
Ep,q(Z=2) ! Epe(pt) where Ee is the nonequivariant spectrum obtained by for
getting the group actions. If E has a product, these isomorphisms give ring map*
*s.
It follows immediately that H*,*(Z=2) = Z[u, u1] where u has degree (0, 1) (in
effect, u is just a placeholder for the second index).
For H*,*(pt) we first recall that Hp,0(pt) is known by the definition of Eile*
*nberg
MacLane cohomology_it is 0 when p 6= 0, and Z when p = 0. For any space
X the groups Hp,0(X) and Hp,0(X) are Bredon cohomology and homology with
coefficients in Z, and in general one has Hp,0(X) = Hpsing(X=Z2; Z) (but the an*
*alog
for homology is not quite true).
When q > 0 we can now write
Hpq,q(pt) ~=~Hp,0(Sq,q) ~=~Hpsing(Sq,q=Z2; Z) ~=~Hpsing( RP q1).
(For the last isomorphism recall that Sq,qis the suspension of the sphere inside
Rq,q, which is the (q  1)sphere with antipodal action). So when q > 0 the gro*
*ups
H*,q(pt) are the reduced cohomology of RP q1, with a suitable shifting. See t*
*he
picture in Theorem 2.8.
When q > 0 we can also write
Hp+q,q(pt) ~=~Hp+q,q(S0,0) ~=~Hpq,q(S0,0; Z) ~=~Hp,0(Sq,q).
The second isomorphism uses equivariant SpanierWhitehead duality. Now, there
is a cofiber sequence S(Rq,q) ,! D(Rq,q) ! Sq,q, where the first two terms are
34 DANIEL DUGGER
the sphere and disk in Rq,q. The induced long exact sequence for H*,0shows that
~Ha,0(Sq,q) ~=~Ha1,0(S(Rq,q)) when a 6= 0, 1, and that there is an exact seque*
*nce
(B.1) 0 ! ~H1,0(Sq,q) ! H0,0(S(Rq,q)) ! Z ! ~H0,0(Sq,q) ! 0.
The space S(Rq,q) has free action, and so ~Ha1,0(S(Rq,q)) ~=~Hsinga1(S(Rq,q)=*
*Z2) =
~Hsinga1(RP q1). Hence Hp+q,q(pt) ~=~Hsingp1(RP q1) when p 6= 0, 1.
The center map in (B.1) may be seen to coincide with the map H0,0(Z=2) !
H0,0(pt) induced by Z=2 ! pt. This is the same as the transfer map i* in the
Mackey functor Z, which is the x2 map Z ! Z. So Hq1,q(pt) ~=H~1,0(Sq,q) = 0
and Hq,q(pt) ~=H~0,0(Sq,q) ~=Z=2. We have now seen that when q > 0 the groups
H*,q(pt) are the reduced singular homology groups of RP q1, read downwards from
the group in degree (q  1, q). Again, the reader is referred to the picture th*
*at goes
with Theorem 2.8.
At this point we have computed the additive groups H*,*(pt) and H*,*(Z=2), so
we turn our attention to the maps between them. Consider the cofiber sequence
Z=2+ ! S0,0! S1,1and the induced long exact sequence
*
. . .H0,2n1(pt) H0,2n(Z=2) i H0,2n(pt) H1,2n1(pt) . . .
When n 0 then H0,2n1(pt) = 0 = H1,2n1(pt), and so i* is an isomorphism.
When n < 0 we know enough to conclude that cokeri* ~=Z=2, so i* is multiplicati*
*on
by 2. A Z=2Mackey functor with both groups equal to Z is completely determined
by its restriction map i*, so we can deduce that H_0,2n= Z_when n 0 and
H_0,2n= Z_opwhen n < 0.
The map i*: H*,*(pt) ! H*,*(Z=2) is a map of rings, and we know the target is
Z[u, u1]. This allows us to determine the subring H0,2*(pt). Also, the commuta
tivity of the usual diagram
AG(SV ) ^ AG(SW )_____//AG(SV ^ SW )
t AG(t)
fflffl fflffl
AG(SW ) ^ AG(SV )_____//AG(SW ^ SV ),
shows that H*,*(X) is gradedcommutative in a certain sense, for any X. For
X = pt we know that the groups H*,*(pt) are either Z=2's or else located in even
degrees, so the ring is commutative onthenose.
It is not hard to see that S1,1! AG(S1,1) ' K(Z(1), 1) is a weak equivalence *
*(we
know the homotopy groups of the target and its fixed set, so this can be checked
directly). Let y denote the composite S0,0,! S1,1! K(Z(1), 1). The cofiber
sequence S0,0,! S1,1! Z=2+ ^ S1,0gives us a long exact sequence on H*,*in
which one of the maps is multiplication by y. Analysis of this long exact seque*
*nce
lets us determine all the multiplicationbyy's shown in the diagram in Theorem*
* 2.8.
At this point we have determined almost all of the ring structure on H*,*(pt)*
*. If
`n denotes the class in H0,2n1(pt) ~=Z=2 and x the generator of H0,2(pt) ~=Z,*
* we
have only to show that x . `n+1 = `n. Let E be the spectrum defined by the cofi*
*ber
sequence 0,2HZ ! HZ ! E, where the first map denotes multiplication by x.
Using what we have already proven, one computes that En,0(pt) = 0 if n 6= 0,
E0,0(pt) = Z=2, and En,0(Z=2) = 0 for all n. So E is the EilenbergMacLane
cohomology theory for the Mackey functor E_0,0, and the nature of this Mackey
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 35
functor lets us conclude that En,0(X) ~= Hnsing(XZ2; Z=2). So when n > 0 we
have E0,n(pt) ~=~En,0(Sn,n) ~=~Hnsing(S0) = 0. It follows that multiplication *
*by x
gives an isomorphism H0,n2(pt) ! H0,n(pt) when n 2. This completes the
analysis of the ring structure on H*,*(pt)_all products can be deduced from the
ones we've computed together with commutativity and degree considerations.
B.2. HZetcomputations. Recall that the spaces in the spectrum for HZ_etare
K(Z, V )EZ=2. From this it follows that HZ_! HZ_etis a nonequivariant equivalen*
*ce,
and so H*,*(Z=2) ! H*,*et(Z=2) is an isomorphism of rings.
Remark 2.13 gives the homotopy type of K(Z(n), 2n)hZ=2, and from this we
immediately compute the groups Hp,qet(pt) where p, q 0 and p 2q. The point *
*is
that
Hp,qet(pt) = H2q(2qp),qet(pt) = ~H2q,qet(S2qp,0) = [S2qp,0, K(Z(q), 2q)h*
*Z=2].
Using the cofiber sequence S0,0,! S1,1! Z=2+ ^ S1,0now lets us deduce
Hp,qet(pt) in the two ranges (p q) and (p 1). In a moment we will show that
for all n > 0 one has H0,2net(pt) = Z, H0,2n+1et(pt) = 0, and the restriction*
* map
H0,2net(pt) ! H0,2net(Z=2) is an isomorphism. Using these facts, this same co*
*fiber
sequence will show that Hp,qet(pt) vanishes when both p < 0 and q < 0.
The above cofiber sequence induces an exact sequence
* 0,n+1
0 = H1,n+1et(Z=2) H1,n+1et.yH0,net H0,n+1et(Z=2) i Het .
When n = 1 we already know i* is the identity, and H1,0et= 0; so H0,1et= 0. Wh*
*en
n = 2 we find the exact sequence
0 Z=2 H0,2et Z 0,
so H0,2etis either Z or Z Z=2. But we also know that
H0,2et~=~H2,0et(S2,2) ~=~H2,0(EZ=2 +^~S2,2)=~H2sing(EZ=2 +^Z2S2,2)
~=~H2,0sing((S1a)+ ^Z2S2,2),
where S1adenotes the circle with antipodal action, and S1a,! EZ=2 is the obvious
inclusion. The cofiber sequence (Z=2)+ ,! (S1a)+ ! Z=2+ ^ S1,0gives a diagram
of spaces
Z=2 +^Z2S2,2 ____//_(S1a)+ ^Z2S2,2_//(Z=2+ ^ S1,0) ^Z=2S2,2___S3
 mmmmmm
 mmmm
fflfflvvmm
S2,2=Z2
where the top row is a cofiber sequence and the two maps to the bottom row squa*
*sh
S1a(and Z=2) to a point. Applying H2singto this diagram now gives
H3(S3) oo___ZOoo___?Ooo___0@@
2

Z
where ? denotes H2sing((S1a)+ ^Z2S2,2). We have so far determined that this gro*
*up
is either Z or Z Z=2, and so the only possibility is Z. So we have learned th*
*at
H0,2et= Z, and the map ? ! Z in the diagram must be an isomorphism. The
36 DANIEL DUGGER
diagram now shows that H0,2! H0,2etis multiplication by 2, because this is the
map Z !?. In the square
H0,2 _________//_H0,2et
 
 
fflffl fflffl
H0,2(Z=2)_____//H0,2et(Z=2)
we know all the groups are Z, the top and left maps are multiplication by 2, an*
*d the
bottom map is an isomorphism; so the right vertical map is also an isomorphism.
Using an induction, the above arguments actually show that H0,2n+1et= 0,
H0,2net= Z, and H0,2net! H0,2netis an isomorphism, for all n 1. This compl*
*etes
our determination of the groups H*,*et, and of the Mackey functors H_0,2net.
The ring structure on H0,*et(pt) can now be determined by comparing with the
known structure of the rings H0,*(pt) and H0,*(Z=2) (the latter is also H0,*et(*
*Z=2)).
The multiplicationbyy's are deduced from the long exact sequences induced by
S0,0,! S1,1! Z=2+ ^ S1,1, just as for H*,*. This completes the proof of Proposi
tion 7.4.
References
[AM] S. Araki and M. Murayama, ficohomology theories, Japan. J. Math 4 (1978)*
*, no. 2, pp.
361416.
[Ar] S. Araki, Forgetful spectral sequences, Osaka J. Math. 16 (1979), pp. 173*
*199.
[AH] M. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, Proc.*
* Symp.
Pure Math. 3, pp. 738.
[At] M. Atiyah, Ktheory and reality, Quart. J. Math. Oxford (2) 17 (1966), pp*
*. 367386.
[Be] A. Beilinson, Height pairing between algebraic cycles, in Ktheory, Arith*
*metic and Ge
ometry, Lect. Notes in Math. 1289, Springer (1987), pp. 126.
[BL] S. Bloch and S. Lichtenbaum, A spectral sequence for motivic cohomology, *
*preprint,
1995. Available at http://www.math.uiuc.edu/Ktheory/0062.
[Bd] M. Boardman, Conditionally convergent spectral sequences, in Homotopy inv*
*ariant al
gebraic structures (Baltimore, MD, 1998), 4984, Contemp. Math. 239, Amer*
*. Math.
Soc., Providence, RI, 1999.
[BK] A. Bousfield and D. Kan, Homotopy limits, completions, and localizations,*
* Lect. Notes
in Math. 304, SpringerVerlag, 1972.
[BG] K. Brown and S. Gersten, Algebraic Ktheory and generalized sheaf cohomol*
*ogy, Lecture
Notes in Math., vol. 341, Springer, 1973, pp. 266292.
[dS] P. dos Santos, A note on the equivariant DoldThom theorem, to appear in *
*J. Pure.
Appl. Alg.
[D1] D. Dugger, A Postnikov tower for algebraic Ktheory, MIT PhD thesis, 1999.
[D2] D. Dugger, Multiplicative structures on homotopy spectral sequences I, II*
*, preprint, 2003.
[DI] D. Dugger and D. Isaksen, Hypercovers in topology, to appear in Math. Zei*
*t.
[F] E. Dror Farjoun, Cellular Spaces, Null Spaces, and Homotopy Localization.*
* Lect. Notes
in Math. 1622, SpringerVerlag, 1996.
[FS] E. Friedlander and A. Suslin, The spectral sequence relating motivic coho*
*mology to al
gebraic Ktheory, Ann. Sci. 'Ecole Norm. Sup. (4) 35 (2002), no. 6, 7738*
*75.
[Gr] D. Grayson, Weight filtrations in algebraic Ktheory, in Motives, Proc. S*
*ymp. Pure
Math. 55 (1994), pp. 207244.
[H] P. Hirschhorn, Model Categories and Their Localizations, Mathematical Sur*
*veys and
Monographs, vol. 99, Amer. Math. Soc., 2003.
[HK1] P. Hu, I. Kriz, Realoriented homotopy theory and an analogue of the Adam*
*sNovikov
spectral sequence, Topology 40 (2001), no. 2, 317399.
[HK2] P. Hu, I. Kriz, Some remarks on Real and algebraic cobordism, Ktheory 22*
* (2001), no.
4, 335366.
AN ATIYAHHIRZEBRUCH SPECTRAL SEQUENCE FOR KRTHEORY 37
[JTTW]I. James, E. Thomas, H. Toda, G. W. Whitehead, On the symmetric square of*
* a sphere,
J. Math. Mech. 12 (1963), pp. 771776.
[J] J.F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445553.
[K] M. Karoubi, A descent theorem in topological Ktheory, Ktheory 24 (2001)*
*, no. 2,
109114.
[LLM] H. B. Lawson, P. LimaFilho, M. Michelsohn, Algebraic cycles and the clas*
*sical groups
Part I, Real cycles, Topology 42 (2003), no. 2, 467506.
[Lw1] L. G. Lewis, The RO(G)graded equivariant ordinary cohomology of complex *
*projec
tive spaces with linear Z=p actions, appearing in Algebraic topology and *
*transformation
groups, ed. tom Dieck, Lecture Notes in Math., vol. 1361, Springer, 1988.
[Lw2] L. G. Lewis, The equivariant Hurewicz map, Tran. Amer. Math. Soc., 329 (1*
*992), no.
2, pp. 433472.
[Lw3] L. G. Lewis, Equivariant EilenbergMacLane spaces and the equivariant Sei*
*fertvan
Kampen suspension theorems, Topology Appl., 48 (1992), no. 1, pp. 2561.
[LMS] L. G. Lewis, J. P. May, and M. Steinberger, Equivariant stable homotopy t*
*heory, Lecture
Notes in Math., vol. 1213, Springer, 1986.
[LF] P. LimaFilho, On the equivariant homotopy of free abelian groups on Gsp*
*aces and
Gspectra, Math. Zeit. 224 (1997), pp. 567601.
[M] J.P. May, et al, Equivariant Homotopy and Cohomology Theory, CBMS volume *
*91,
American Mathematical Society, 1996.
[MV] F. Morel and V. Voevodsky, A1homotopy theory for schemes, Inst. Hautes '*
*Etudes Sci.
Publ. Math., No. 90 (2001), 45143.
[Q] D. Quillen, On the group completion of a simplicial monoid, appearing as *
*an appendix
to Filtrations on the homology of algebraic cycles (Friedlander and Mazur*
*), Mem. Amer.
Math. Soc. (1994), no. 529.
[V1] V. Voevodsky, The Milnor conjecture, preprint, 1996. Available from the K*
*theory
archive at http://www.math.uiuc.edu/Ktheory/0170.
[V2] V. Voevodsky, Open problems in motivic stable homotopy theory, I, preprin*
*t, 2000.
Available at http://www.math.uiuc.edu/Ktheory/0392.
[V3] V. Voevodksy, A possible new approach to the motivic spectral sequence, a*
*ppearing in
Recent progress in homotopy theory (Baltimore, MD, 2000), 371379, Contem*
*p. Math.
293, Amer. Math. Soc., Providence, RI, 2002.
[V4] V. Voevodksy, On the zeroslice of the sphere spectrum, preprint, 2002. A*
*vailable at
http://www.math.uiuc.edu/Ktheory/0612.
[W] S. Waner, G  CW(V ) complexes. Unpublished manuscript.
Department of Mathematics, University of Oregon, Eugene, OR 97403
Email address: ddugger@math.uoregon.edu