GENERALIZED GROUP CHARACTERS AND COMPLEX
ORIENTED COHOMOLOGY THEORIES
MICHAEL J. HOPKINS, NICHOLAS J. KUHN, AND DOUGLAS C. RAVENEL
July 9, 1999
Abstract.Let BG be the classifying space of a finite group G. Given a
multiplicative cohomology theory E*, the assignment
G 7-! E*(BG)
is a functor from groups to rings, endowed with induction (transfer) map*
*s.
In this paper we investigate these functors for complex oriented cohomol*
*ogy
theories E*, using the theory of complex representations of finite group*
*s as a
model for what one would like to know.
An analogue of Artin's Theorem is proved for all complex oriented E*: *
*the
abelian subgroups of G serve as a detecting family for E*(BG), modulo to*
*rsion
dividing the order of G.
When E* is a complete local ring, with residue field of characteristic*
* p and
associated formal group of height n, we construct a character ring of cl*
*ass
functions that computes 1_pE*(BG). The domain of the characters is Gn;p,
the set of n-tuples of elements in G each of which has order a power of *
*p. A
formula for induction is also found. The ideas we use are related to the*
* Lubin
Tate theory of formal groups. The construction applies to many cohomology
theories of current interest: completed versions of elliptic cohomology,*
* E*n-
theory, etc.
The nth Morava K-theory Euler characteristic for BG is computed to be
the number of G-orbits in Gn:p. For various groups G, including all symm*
*etric
groups, we prove that K(n)*(BG) concentrated in even degrees.
Our results about E*(BG) extend to theorems about E*(EGxG X), where
X is a finite G-CW complex.
Contents
1. Introduction 2
1.1.A generalized Artin's Theorem
1.2.Morava K-theory Euler characteristics
1.3.Generalized characters
1.4.A formula for induction
1.5.When is K(n)*(BG) concentrated in even degrees?
1.6.Historical Remarks
____________
1991 Mathematics Subject Classification. Primary 55N22; Secondary 20C99, 55N*
*91, 55R35.
All three authors were partially supported by the National Science Foundatio*
*n.
1
2 HOPKINS, KUHN, AND RAVENEL
2. Complex oriented descent and equivariant bundles 11
2.1.A useful way to construct equalizers
2.2.Cohomology of flag bundles
2.3.Equivariant flag bundles
3. Rational equivariant stable homotopy and Artin's Theorem 14
3.1.Rational equivariant stable homotopy theory
3.2.Complex oriented equivariant stable homotopy
3.3.Proof of Proposition 3.2
4. Complex oriented Euler characteristics 19
4.1.Additive functions
4.2.Complex oriented additive functions
4.3.Morava K-theory Euler characteristics
5. Formal groups and E*(BA) 24
5.1.Formal groups and their height
5.2.The structure of E*[[x]]=([pr](x))
5.3.The group of torsion points
5.4.Cohomology of abelian groups
6. Generalized characters 29
6.1.Hopf algebra isomorphisms and a discriminant calculation
6.2.The rings Lr(E*) and L(E*)
6.3.Defining the generalized characters
6.4.Proof of Theorem C
6.5.Induction
7. Good groups 39
7.1.Good groups and the wreath product theorem
7.2.The non-abelian groups of order p3
References 43
1.Introduction
Let BG be the classifying space of a finite group G. Given a multiplicative
cohomology theory E*, the assignment
G 7-! E*(BG)
is a functor from groups to rings, endowed with induction (transfer) maps. In t*
*his
paper we investigate these functors for complex oriented cohomology theories E*,
particularly p-complete theories with an associated formal group of height n.
We briefly remind our readers of the terms in this last sentence. A multiplic*
*ative
cohomology theory E* is complex oriented if there exists a class x 2 E2(C P 1) *
*that
restricts to a generator of the free rank one E* = E*(pt)-module "E2(C P 1). Su*
*ch
a class x is called a complex orientation of E. An orientation allows for the c*
*on-
struction in E*-theory of Chern classes for complex vector bundles. Furthermore,
the behavior of these Chern classes under the tensor product of bundles is dete*
*r-
mined by an associated formal group law over the ring E*. When localized at a
GENERALIZED CHARACTERS 3
prime p, such formal group laws are classified by `height'. Under the completen*
*ess
hypothesesnwe will be assuming, a height n formal group law will force an eleme*
*nt
vn 2 E2-2p to be invertible, and thus we may informally refer to such theories*
* as
vn-periodic.
Our work was inspired by complex K-theory, which, when localized at a prime
p, is v1-periodic. In this case, Atiyah [Ati61] showed that K*(BG) is isomorphi*
*c to
the completion of the complex representation ring R(G), in the topology induced
by the ideal of virtual representations of degree 0. The ring R(G) can, of cour*
*se,
be studied via group characters, leading to many well known results: the cyclic
subgroups form a detecting family, the rank of R(G) is the number of conjugacy
classes, etc.
It often turns out that the ring E*(BG) can also be studied with characters.
For suitable vn-periodic theories E*, these characters assign to each conjugacy
class of commuting n-tuples of p-elements of G, an element of a ring we associa*
*te
to E*. Furthermore, one can extend these constructions to detect elements in
E*(EG xG X) for finite G-CW complexes X, where now the `character ring' also
depends on the abelian fixed point data of X. As a result, almost anything that
can be said about representation rings and equivariant K-theory has an analogue
for E*(BG) and E*(EG xG X).
Versions of our basic theorems date from 1986 and 1987, and many readers will
be aware of earlier drafts of this paper dating from 1989 and 1992. We thank
such readers for their patience, and hope they will appreciate our more accurate
arguments, improved organization, and slightly strengthened theorems.
A standing convention in this paper is that we are working in categories of g*
*raded
objects. Thus rings are graded, tensor products of graded objects are graded in
the standard way, ideals in a ring are assumed to be generated by homogeneous
elements, homomorphisms preserve grading, etc.
1.1. A generalized Artin's Theorem. Our first theorem is valid for all complex
oriented theories, and highlights the privileged role played by the abelian gro*
*ups.
Let A(G) be the category having objects the abelian subgroups of G, and with
morphisms from B to A being the G-maps from G=B to G=A. (Thus this is a
full subcategory of the standard orbit category.) Given a G-space X, a morphism
G=B -! G=A in A(G) induces maps G xB X -! G xA X and XA -! XB in the
usual way.
If E* is a generalized cohomology theory, the G-maps G xA X -! X induce
a map from E*(EG xG X) to the limit lim E*(EG xA X). Furthermore, the
A2A(G)
G-maps G=A x XA -! G xA X induces a map from lim E*(EG xA X) to the
R A2A(G)
end A2A(G)E*(BA x XA ).1
____________
1A G-map ff : G=A1 -!G=A2 induces ff*R: E*(BA1x XA1) -!E*(BA1x XA2) and ff* :
E*(BA2x XA2)Q-!E*(BA1x XA2). The end A2A(G)E*(BA x XA ) is then defined to be *
*the
subring of A2A(G)E*(BA x XA ) consisting of those elements (xA) in this produ*
*ct such that
ff*(xA1) = ff*(xA2) for all ff : G=A1 -!G=A2.
4 HOPKINS, KUHN, AND RAVENEL
Theorem A. Let E be a complex oriented cohomology theory. For any finite group
G, and finite G-C.W. complex X, each of the natural maps
Z
E*(EG xG X) -! lim E*(EG xA X) -! E*(BA x XA )
A2A(G) A2A(G)
becomes an isomorphism after inverting the order of G. In particular,
_1_E*(BG) -! lim _1_E*(BA)
|G| A2A(G) |G|
is an isomorphism.
This is an analogue of Artin's theorem [Ser67, Chapter 9]: For any finite gro*
*up
G, the natural map
_1_R(G) -! lim _1_R(C)
|G| C2C(G)|G|
is an isomorphism, where C(G) is the full subcategory of A(G) having the cyclic
subgroups as objects.
In the case of ordinary cohomology, Theorem A offers no information since
"H*(BG) is all |G|-torsion. However K*(BG) is known to be torsion free, and
MU*(BG) is presumed to often be.
In proving this theorem, we use the most fundamental idea from the theory of
complex oriented cohomology: the notion of complex oriented descent (a.k.a. the
splitting principle). The other ingredient is a modest amount of equivariant st*
*able
homotopy, together with related ideas from the theory of Mackey functors.
Section 2 contains what we need about complex oriented descent. Theorem A
is proved in section 3 as a special case of a more general theorem, Theorem 3.3.
This theorem has other interesting corollaries. For example, it implies that th*
*e map
between equivariant bordism rings
MU*G-! lim MU*A
A2A(G)
also becomes an isomorphism after inverting the order of G.
1.2. Morava K-theory Euler characteristics. Now we describe the computa-
tion that led to this project. Fixing a prime p, let K(n)* denote the nthMorava*
* K-
theory at p [Rav86 ]. This is a complex oriented theory with coefficients K(n)**
* equal
to the graded field Fp[vn; v-1n]. The third author noted in [Rav82 ] that K(n)**
*(BG)
is a finite dimensional vector space over K(n)*, and asked for its dimension. T*
*he
second author [Kuh87 ] computed this in the special case when G has abelian p-
Sylow subgroups, with the answer involving certain orbits of n-tuples of elemen*
*ts
in a fixed Sylow subgroup.2
Our partial answer to the general question is the following. Let Gn;pdenote
the set of n-tuples of commuting elements each of which has order a power of
p. The group G acts on Gn;pby conjugation: if ff = (g1; : :;:gn), define g . f*
*f =
(gg1g-1; : :;:ggng-1).
____________
2Let WG(A) = NG(A)=A, where the abelian group A is a p-Sylow subgroup. Then*
* the
dimension of K(n)*(BG) is the number of WG(A)-orbits in An. It is an exercise w*
*ith the Sylow
theorems to check that this agrees with Theorem B.
GENERALIZED CHARACTERS 5
Theorem B (Part 1). The Morava K-theory Euler characteristic
OGn;p= dimK(n)even(BG) - dimK(n)odd(BG)
is equal to the number of G-orbits in Gn;p.
The proof of Theorem B involves counting these orbits by means of M"obius
functions on the partially ordered set of abelian subgroups of G. We also gener*
*alize
our computation to a computation of
OGn;p(X) = dimK(n)even(EG xG X) - dimK(n)odd(EG xG X);
for any finite G-C.W. complex X.
Our M"obius functions are defined using the usual Euler characteristic as fol*
*lows.
For all abelian subgroups A G and finite G-C.W. complexes X, an integer CA(X)
is defined by downward induction on A by the equation
X
CB(X) = O(XA ):
AB
B abelian
Our formula for OGn;p(X) is then
Theorem B (Part 2).
X |A|
OGn;p(X) = ___|A(p)|nCA(X):
AG |G|
Here the sum is over the abelian subgroups of G, and A(p)denotes the p-Sylow
subgroup of A.
The function OGn;pon finite G-CW complexes is an example of an `additive
invariant' in the sense of tomDieck [Die87, p.227]. Our function satisfies an e*
*xtra
condition due to complex oriented descent. In x4 we develop the general theory
of such complex oriented additive functions. Theorem B then follows from this,
together with one bit of special information about K*(n): if A is an abelian gr*
*oup,
K(n)*(BA) is an |A(p)|n dimensional K(n)*-vector space concentrated in even
degrees.
1.3. Generalized characters. Theorem A can be interpreted in terms of char-
acters - at the cost of adhering more structure to E*.
To motivate our result, recall that classical characters for finite groups ar*
*e defined
in the following situation. Let L be the smallest characteristic 0 field contai*
*ning
all roots of unity, and, if G is a finite group, let Cl(G; L) be the ring of cl*
*ass
functions on G with values in L. The units in the profinite integers ^Zact on L*
* as
the Galois group over Q. Observing that G = Hom (^Z; G), the set of continuous
homomorphisms, one sees that ^Zxalso acts naturally on G, and thus on Cl(G; L):
given OE 2 ^Z, g 2 G, and O 2 Cl(G; L), one lets (OE . O)(g) = OE(O(OE-1(g))). *
* The
6 HOPKINS, KUHN, AND RAVENEL
character map is a ring homomorphism3
O : R(G) -!Cl(G; L)^Z;
and this induces isomorphisms
O : L R(G) ' Cl(G; L)
and
O : Q R(G) ' Cl(G; L)^Z:
(See [Ser67, Theorem 25] for this last statement.)
Let E* be a complex oriented theory, with associated formal group law F as-
sociated to a fixed orientation x 2 E2(C P 1). Suppose the graded ring E* and F
satisfy
o E* is local with maximal ideal m, and complete in the m-adic topology.
o The graded residue field E*=m has characteristic p > 0.
o p-1E* is not zero.
o The mod m reduction of F has height n < 1 over E*=m. (See x5.1.)
We define L(E*), the analogue of L, in the following way. The inverse system
. .-.!(Z=pr+1)n -! (Z=pr)n -! . . .
induces a direct system of E*-algebras
. .-.!E*(B(Z=pr)n) -!E*(B(Z=pr+1)n) -!: :;:
and we let E*cont(BZnp) denote the colimit. A continuous homorphism from the
n-fold product of the p-adic integers to the circle ff : Znp-! S1 will induce a*
* map
ff* : E*(C P 1) -! E*cont(BZnp) and we let c1(ff) = ff*(x) 2 E2cont(BZnp): Now *
*let
L(E*) = S-1E*cont(BZnp) where S is the set of c1(ff) corresponding to nonzero
homorphisms ff. Note that the continuous automorphism group Aut(Znp) acts on
L(E*) as a ring of E*-algebra maps.
The analogue of Cl(G; L) will be Cln;p(G; L(E*)) defined to be the ring of fu*
*nc-
tions O : Gn;p-! L(E*) stable under G-orbits. Noting that
Gn;p= Hom (Znp; G);
one sees that Aut(Znp) acts on Gn;p, and thus on Cln;p(G; L(E*)) as a ring of E*
**-
algebra maps: given OE 2 Aut(Znp), ff 2 Gn;p, and O 2 Cln;p(G; L(E*)), one lets
(OE . O)(ff) = OE(O(OE-1(ff)))
More generally, if X is a finite G-CW complex, let
a
Fixn;p(G; X) = XIm(ff):
ff2Hom(Znp;G)
This is a space with commuting actions of G and Aut(Znp). Let
Cln;p(G; X; L(E*)) = L(E*) E* E*(Fixn;p(G; X))G :
This is again an E*-algebra acted on by Aut(Znp).
____________
3We apologize for our excessive use of the symbol O. We use it to denote bot*
*h group characters
and Euler characteristics, as dictated by traditions in representation theory a*
*nd topology. In
context, we hope there is no confusion.
GENERALIZED CHARACTERS 7
We define our character map
n)
OGn;p: E*(EG xG X) -!Cln;p(G; X; L(E*))Aut(Zp
as follows. Using a Kunneth isomorphism
E*(B(Z=pm )n x XIm(ff)) ' E*(B(Z=pm )n) E* E*(XIm(ff))
available in our situation (see Corollary 5.11), a homomorphism ff 2 Hom (Znp; *
*G)
induces
E*(EG xG X) -!E*cont(BZnp) E* E*(XIm(ff)) -!L(E*) E* E*(XIm(ff)):
This will be OGn;p(ff), the component of OGn;pindexed by ff.
Our main theorem is then
n) -1 * *
Theorem C. The invariant ring L(E*)Aut(Zp = p E , and L(E ) is faithfully
flat over p-1E*. The character map OGn;pinduces isomorphisms
OGn;p: L(E*) E* E*(EG xG X) ' Cln;p(G; X; L(E*))
and n
OGn;p: p-1E*(EG xG X) ' Cln;p(G; X; L(E))Aut(Zp):
In particular, there are isomorphisms
OGn;p: L(E*) E* E*(BG) ' Cln;p(G; L(E*))
and n
OGn;p: p-1E*(BG) ' Cln;p(G; L(E*))Aut(Zp):
Theorem C applies to many cohomology theories:
o The `competions of complex cobordism' introduced by Morava in [Mor78 ].
o The `In-adically completed' version of E(n) studied by A. Baker and Wur-
gler [BW89 ]. They consider ^E(n), the Bousfield localization of E(n) w*
*ith
respect to K(n), where E*(n) = Z(p)[v1; : :;:vn; v-1n], as a module over
BP *= Z(p)[v1; v2; : :]:. They show that ^E(n)* is the completion of E*(*
*n)
with respect to the ideal In = (p; v1; : :;:vn-1).
o The `integral lifts' of the Morava K-theories studied by Morava in [Mor8*
*8 ].
The coefficients are W (Fpn)[u; u-1] where W (Fpn) denotes the Witt vect*
*ors
for the finite field, and u has degree -2.
o The completion of elliptic cohomology [LRS ] at any maximal ideal (see
[Bak98 ], and [Hop89 ]).
o The theories En studied by Hopkins and Miller with coefficients
W (Fpn)[[w1; : :;:wn-1]][u; u-1]
(wihas degree 0). These spectra have been shown to admit E1 -ring struc-
tures, thus one has a good theory of power operations which our characte*
*rs
can be used to explore.
Section 5 contains the prerequisites we need about formal group laws and E*(B*
*A)
for A abelian. Properties of L(E*) are then developed in section 6, along with a
proof of Theorem C.
8 HOPKINS, KUHN, AND RAVENEL
1.4. A formula for induction. It is useful to have a formula for the character
of an `induced' cohomology class. Recall that, for H G, X a G-space, and any
cohomology theory E*, there is a transfer map [Ada78 , Chapter 4]
Tr: E*(EH xH X) -! E*(EG xG X):
Theorem D. Let x be an element of E*(EH xH X). Then
X
OGn;p(ff)(Tr*(x)) = OHn;p(g . ff)(x):
gH2(G=H)Im(ff)
In this formula, ff : Znp-! G is a homomorphism, and g . ff means to follow t*
*his
with conjugation by g.
This formula generalizes the classic situation [Ser67, p.30], and is proved a*
*t the
end of section 6.
As a simple application, Theorem D can be used to compute the kernel of the
stable Hurewicz map
ss0S(BG) -!MU0(BG);
up to finite index. This example appears as Example 6.16, and is related to work
of Stretch [Stre81] and Laitinen [Lai79] on the Segal Conjecture.
1.5. When is K(n)*(BG) concentrated in even degrees? In contemplating
Theorems A and C, it is natural to wonder if there is any |G|-torsion in E*(BG)*
* for
the theories E* of interest. Suppose, in particular, E* is a p-complete integra*
*l lift
of K(n)*, that is E*(X; Z=p) = K(n)*(X).4 Then an argument with Bocksteins
shows that E*(BG) will be p-torsion free if and only if K(n)*(BG) is concentrat*
*ed
in even degrees. Similarly, a more elaborate argument with Bockstein-like spect*
*ral
sequences shows that if K(n)*(BG) is concentrated in even degrees then E*n(BG) *
*is
torsion free, where En is the important theory mentioned above. (See e.g. [Stri*
*98]
for this type of argument.)
In the classic case of K-theory, along with proving that K0(BG) is the comple-
tion of the representation ring, Atiyah showed that K1(BG) = 0. Many readers of
this paper will know that, inspired by this, the authors originally conjectured*
* that
K(n)odd(BG) = 0 for all n and G. This conjecture was disproved by I. Kriz [Kri9*
*7]
in the case p = 3; n = 2, and with G the 3-Sylow subgroup of GL4(Z=3), a group
of order 36. (Kriz and K. Lee now have examples for all odd p [KL98 ].)
From the beginnings of this project, the authors knew that a critical part of
Atiyah's inductive argument failed to generalize to the n 2 case: if G is a p-
group, K0(BG) is a permutation Aut(G)-module. This is also the point that Kriz
exploits as his example arises from an extension
1 -!H -! G -!Z=p -!1
in which E*(BH) is not a permutation Z=p-module, where E* is (roughly) an
integral lift of K(2).
This permutation module problem suggested that perhaps K(n)*(BG) could be
proven to be concentrated in even degrees if G is built up out of extensions re*
*lated
to permutations. We have a result along these lines.
____________
4It would also suffice to have the spectrum E=p be a K(n)-module, necessaril*
*y free, on even
dimensional classes.
GENERALIZED CHARACTERS 9
We define a finite group G to be good if K(n)*(BG) is generated as a K(n)*-
module by transfers of Euler classes of complex representations of subgroups. In
particular, if G is good then K(n)*(BG) is concentrated in even degrees.
Theorem E. i)Every finite abelian group is good.
ii)If G1 and G2 are good then so is their product G1 x G2.
iii)G is good if its p-Sylow subgroup is good.
iv)If G is good, then so is Z=p o G, the wreath product arising as the extensi*
*on
1 -!Gp -!Z=p o G -!Z=p -!1:
This will be proved in section 7. The first of these statements is well known*
*, and
the second and third are easy to verify. It is the last statement that makes ou*
*r notion
interesting. In particular, since the p-Sylow subgroup of any symmetric group i*
*s a
product of iterated wreath products of Z=p, we have proved that symmetric groups
k are all good.
In Proposition 7.10, we will show that calculations by Tezuka-Yagita [TY ] im*
*ply
that all groups of order p3 are good. In recent years, the list of known good g*
*roups
has been expanded by various people, see e.g. [Kri97, Tan95]
J. Hunton [Hun90 ] independently has shown that the symmetric groups have no
odd dimensional Morava K-theory by defining a variant of our notion good and
then proving an analogue of the last statement of our theorem.
1.6. Historical Remarks. Since this paper has taken so long to be put in final
form, it is perhaps appropriate to comment on its history.
The idea that the correct domain for our characters should be Gn;pwas inspired
by the 1985 work [Kuh87 ]. At the January 1986 A.M.S. meeting in New Orleans,
the third author conjectured that the rank of K(n)*(BG) was equal to the number
of G-orbits in Gn;p. Soon after, the first and third authors realized that Lubi*
*n-
Tate theory together with a Vandermonde determinant argument (appearing here
in the proof of Proposition 6.2) led to some sort of version of Theorem C. We s*
*oon
derived the formula for the transfer, and a first proof of Theorem B was discov*
*ered
by the first author. Our work, as of mid 1986, was publicly presented in an inf*
*ormal
evening talk by Hopkins on July 31, 1986 at the algebraic topology conference in
Arcata, CA.
It is fair to say that, at that time, the necessary hypotheses regarding comp*
*letion
had not been accurately sorted out, nor had adequate attention been played to t*
*he
flatness of the extension of scalars to L(E*).
In 1986, it seemed a big presumption to assume the coefficient ring E* was
complete. Today this seems much less a problem, as numerous interesting results
exploiting this hypothesis have been proven, and interesting examples developed,
starting with [BW89 , Mor88]. Thus, during the first year of the project, E* w*
*as
just assumed to be local, but not necessarily complete, and we attempted to pro*
*ve
theorems about E*(BG)^, where ^ is algebraic completion at the maximal ideal.
This seems to lead into a thicket of questions about the exactness of completion
in non Noetherian settings, and during 1986-87, we became convinced that the
hypothesis of completeness was hard to avoid.
10 HOPKINS, KUHN, AND RAVENEL
The formalities of our character rings from the Mackey functor point of view *
*were
investigated by the second author during a 1986-87 visit to Cambridge Universit*
*y.
Theorem A was discovered as a consequence, and the first formal presentation
of this theorem was at the Oxford Topology Seminar of June 8, 1987.5 Related
observations made at this time are more fully discussed in [Kuh89 ].6
Armed with Theorem A, it was initially unclear if one really needed to prove
the flatness of L(E*) as an E*-module - and one doesn't, if one is content with
describing E*(BG) and not E*(EG xG X). (See Remark 6.11.) However the
demands of subsequent work by Hopkins and his younger colleagues at M.I.T.
made it clear that a good understanding of this ring was important. By 1992,
the algebro-geometric point of view had become more conceptually important, and
has been crucial in subsequent work by Ando, Hopkins, Strickland, and others.
Our natural description of the spectrum of E*(BA) (Proposition 5.12) reflects t*
*his
changing point of view.
In recent years, the proofs of Theorems A and B have been significantly alter*
*ed,
leading to the current nearly axiomatic presentations. The refinement of Theorem
C using Aut(Znp) dates from the early 1990's.
The formulation and proof of Theorem E is due to the third author around 1988.
Expository articles about aspects of this work have been written by all of us:
[Hop89 , Kuh89, HKR92 ].
From the beginning of this project, it has been noticed that, if Y = EGxG X, *
*E*
is `vn-periodic' and D* is `vn-1-periodic', then there seems to be a close rela*
*tionship
between 1
E*(Y ) and D*(Y S ):
Similarly, if G is a p-group and T nis the n-torus, there is a natural isomorph*
*ism
of L*(E*) algebras
n *
L(E*) E* E*(BG) ' H*(BGT ; L(E )):
When n = 2, this fits well with work of Taubes [Tau ] and Bott-Taubes [BT ] on
the elliptic genus. However, it has yet to be explained in a satisfactory mann*
*er,
and, to get to the heart of the matter, there is not yet a good explanation of *
*what
basic geometric structures, analogous to vector bundles, our characters are giv*
*ing
us information about. Perhaps in the next millennium we will learn more.
____________
5This was a talk by Kuhn joint with J.F.Adams (!).
6This includes an analogue of Theorem C for equivariant K-theory: there is a*
* character map
Y
O : C KG(X) -!( C K(Xg))G
g2G
that is an isomorphism for all finite G-CW complexes X. Though this formula als*
*o appeared in
a letter from Kuhn to G.Segal dated December 4, 1986, it was widely advertised *
*by Hirzebruch
as a `recent formula of Atiyah and Segal' after the publication of [AS89]. But *
*precedence for such
formulae seems to be due tom Dieck: in his 1979 book, he gave the closely relat*
*ed formula [Die79,
p.198]
_1_K (X) ' (Y S-1R(C) K(XC ))G;
|G| G C C
where the product runs over the cyclic subgroups of G, and SC is the set of Eul*
*er classes of
non-trivial ireducible C-modules.
GENERALIZED CHARACTERS 11
Acknowledgements. Many people deserve thanks for their interest in this project.
In particular, we thank the late J. F. Adams for supplying the second author wi*
*th
lemmas in stable rational equivariant homotopy used in Section 3 and Andreas
Dress for conversations on similar subjects, and Andrew Baker, Peter Landweber,
and the referee of an earlier draft of this paper for steering us toward accura*
*te
hypotheses in Theorem C. The influence of Jack Morava on our work is also clear;
he was certainly on the lookout for character theoretic interpretions of E*(BG)*
* as
early as any of us.
2.Complex oriented descent and equivariant bundles
2.1. A useful way to construct equalizers. At various times in this paper, we
describe objects as equalizers. Without exception, they all arise using a gene*
*ral
categorical procedure we describe in this subsection.
The situation is the following. One has
o Two categories C and A.
o Two functors F : C -! C and H : Cop -!A.
o Two natural transformations p(X) : F(X) -! X and r(X) : H(F(X)) -!
H(X).
These are required to satisfy the following two properties:
(2.1) For allX 2 C; p(X) O p(F(X)) = p(X) O F(p(X)) : F(F(X)) -!X:
(2.2) For allX 2 C; r(X) O H(p(X)) = 1 : H(X) -!H(X):
Proposition 2.3.In this situation, there is an equalizer diagram
H(X) -!H(F(X)) H(F(F(X))):
Proof.Applying H to (2.1) shows that the two composites are equal. To finish the
proof, we show that if ff : A -!H(F(X)) is any morphism in A satisfying
H(p(F(X))) O ff = H(F(p(X))) O ff : H(F(X)) -!H(F(F(X)));
then ff factors uniquely through H(p(X)); more precisely,
ff = H(p(X)) O fi; where fi = r(X) O ff : A -!H(X):
To check this, we compute:
H(p(X)) O r(X) O ff= r(F(X)) O H(F(p(X))) O ff
= r(F(X)) O H(p(F(X))) O ff
= ff:
Here the first equality is a consequence of the naturality of r, the second is *
*true by
the hypothesis on ff, and the third follows from (2.2). __|_|
12 HOPKINS, KUHN, AND RAVENEL
2.2. Cohomology of flag bundles. Let be an m-dimensional complex vec-
tor bundle over a space B, and let F () ! B be the bundle of complete flags.
Corresponding to the m canonical line bundles over F () are m classifying maps
ffi: F () -!C P 1.
If E* is complex oriented, then a complex orientation x 2 E*(C P 1) determines
m classes xi= ff*i(x) 2 E*(F ()).
Proposition 2.4.
(1) There is an identification
E*(F ()) = E*(B)[x1; : :;:xm ] /(oei({xj}) - ci())
where oei is the ith elementary symmetric function.
(2) The E*(B)-module E*(F ()) is free of rank m!.
(3) If 0 is the bundle over B0 induced by a map B0! B then the map
E*(B0) E*(B)E*(F ()) ! E*(F (0))
is an isomorphism.
(4) There are natural isomorphisms
E*(F ()) E*(B)E*(F ()) ' E*(F () xB F ()):
Proof.Statement (1) is classical (e.g. compare with [Ati67, Prop.2.7.13]), and *
*(2)
follows as a matter of pure algebra. Then (3) follows from (2), and (4) is the *
*special
case of (3) applied to the map F () -!B itself. __|_|
Proposition 2.5.The following sequence is an equalizer:
E*(B) ! E*(F ()) E*(F () xB F ()):
Proof.We apply Proposition 2.3 to the following situation:
The category C is the category whose objects are pairs (B; ) where is a comp*
*lex
vector bundle over B. A morphism from (B0; 0) to (B; ) consists of a map f :
B0-! B, together with a bundle isomorphism 0' f*(). The category A will be
the category of E*-modules.
F(B; ), will be the pair (F (); p*()), where p : F () -! B is the projection.
H(B; ) = E*(B).
The natural transformation `p' of Subsection 2.1 will be induced in the obvio*
*us
way by p : F () -!B. To define `r', first note that the composite
* ss
E*(B) p-!E*(F ()) -! E*(F ())=(x1; : :;:xm )
is an isomorphism by the previous proposition. We then define r : E*(F ()) -!
E*(B) to be the inverse of this natural isomophism, precomposed with ss. __|_|
We end this subsection by noting that we can apply these results inductively *
*to
F () xB X - ---! X
?? ?
y ?y
F () - ---! B
GENERALIZED CHARACTERS 13
to get similar assertions with F () replaced by an iterated fiber product of fl*
*ag
bundles, or more generally, a disjoint union of iterated fiber product of flag *
*bundles.
2.3. Equivariant flag bundles. Now let G be a finite group.7
Proposition 2.6.Let be an m dimensional equivariant complex vector bundle
over a G-space X. Let Y ! B be either the map
i) EG xG F () ! EG xG X, or
ii)F ()A ! XA where A is an abelian subgroup of G.
Then, for any complex oriented theory E*, we have
(1) The E*(B)-module E*(Y ) is free of rank m!.
(2) The following sequence is an equalizer:
E*(B) ! E*(Y ) E*(Y xB Y ):
Proof.In case (i), Y is the bundle of complete flags in EG xG , and the result
follows from the previous subsection.
The more delicate case is (ii). Note that we can assume that G = A and X = XA*
* .
Thus we need to analyze F ()A , where is an equivariant n dimensional bundle
over a trivial A-space X. Let L1; : :;:Lk denote the distinct irreducible A-mod*
*ules.
In [Seg68:2, Prop.2.2], G. Segal noted that will admit a decomposition
Mk
' Li i
i=1
for uniquely defined nonequivariant bundles i over X.
Let ihave dimension mi. Since A is abelian, each of the Liis one dimensional,
and thus m1+ . .+.mk = m. By inspection, it then follows that, as spaces over X,
a
F ()A ' F (1) xX F (2) xX . .x.XF (k)
with the disjoint union running over all partitions of m into k subsets of card*
*inality
m1, : :,:mk. Noting that the number of such partitions is ___m!__m1!:::mk!and t*
*hat
(____m!____m)m1! : :m:k! = m!;
1! : :m:k!
the proposition_now follows from the previous subsection.
|_|
The following consequence of our analysis is a generalization of the well kno*
*wn
fact that the Euler characteristic of U(m)=T m is m!. It will be used in our pr*
*oof
of Theorem B.
Corollary 2.7.Let be an m dimensional equivariant complex vector bundle over
a finite G-CW complex X. If A is any abelian subgroup of G, then O(F ()A ) =
m!O(XA ).
Finally, we will use without further comment:
____________
7More generally, the results and proofs of this subsection remain valid if G*
* is any compact Lie
group.
14 HOPKINS, KUHN, AND RAVENEL
Proposition 2.8.Let be an equivariant complex vector bundle over a finite G-
CW complex X. Then F () also has the homotopy type of a finite G-CW complex.
Proof.By induction on the cells of X, we may assume that X = G=H for some
H G. In this case F () is a compact smooth G-manifold, and thus admits the
structure of a G-CW complex (necessarily finite) by [Ill74]. __|_|
Remark 2.9.Results in this section easily generalize to `relative' versions inv*
*olving
pairs X0 X of G-spaces. This leads to the obvious relative versions of our main
theorems.
3. Rational equivariant stable homotopy and Artin's Theorem
If E is a generalized cohomology theory, one can regard the assignment
X 7-! "E*(EG+ ^G X)
as an equivariant cohomology theory, defined on CG , the stable category of bas*
*ed
finite G-CW complexes ([LMS86 ]). Although we make only elementary use of this
deep fact, it is conceptually very illuminating. Here we begin with some gener*
*al
facts about rational equivariant cohomology theories8, and then use the extra h*
*y-
pothesis that E* is complex oriented. Although many of these general facts are *
*in
the literature as parts of more general machines (see [LMS86 , especially page *
*271],
[Die79], and [Ara82]), we develop what we need from minimal prerequisites. For
general background material on the stable category of finite G-CW complexes see
also [Ada82 ] and [Die87].
3.1. Rational equivariant stable homotopy theory. Recall that the Burnside
ring A(G) is the Grothendieck group associated to the monoid of finite G-sets, *
*with
addition coming from disjoint union and multiplication from cartesian product.9
Additively, A(G) is the free abelian group on the isomorphism classes of transi-
tive G-sets. Intuitively, A(G) is the object containing the algebra of double c*
*oset
formulae. There is an isomorphism of rings [Die87, xII.8]
A(G) ' {S0; S0}G ;
where {X; Y }G denotes the stable equivariant homotopy group. Thus A(G) acts
on any equivariant cohomology theory, and, more generally, on any additive con-
travariant functor defined on CG .
There is a character theory for A(G). Given H < G, let OH : A(G) ! Z be
defined by OH (S) = |SH |.
Lemma 3.1 ([Die79, page 3]).The map
Y Y
OH : A(G) -! Z
(H) (H)
is an inclusion, and becomes an isomorphism after inverting |G|.
____________
8What matters here is that the order of G is inverted.
9The Burnside ring A(G), used mainly in this section, should not be confused*
* with the category
A(G).
GENERALIZED CHARACTERS 15
Here, the product runs over the conjugacy classes (H) of subgroups of G.
It follows that 1 2 A(G) Z[|G|-1] can be written as the sum of orthogonal
idempotents
X
1 = eH ;
(H)
where OK (eH ) is 1 if K is conjugate to H, and 0 otherwise.
For an additive contravariant functor
"h: CG -! Z[|G|-1]-modules;
this decomposition gives a natural splitting
"h(X) ' Y eH "h(X):
(H)
The following is a key observation, for which we would like to thank J. F. Ad*
*ams.
Proposition 3.2.The natural G-map
G=H+ ^ XH ! X
induces an isomorphism
eH "h(X) ' eH "h(G=H+ ^ XH )WG(H):
Here Y+ denotes the union of Y with a disjoint basepoint, and the Weyl group
WG (H) is the quotient of the normalizer NG (H) of H in G by H, acting on the
right of G=H+ ^ XH via (gH; x) . (nH) = (gnH; n-1x).
We postpone the proof of Proposition 3.2 until the end of the section.
3.2. Complex oriented equivariant stable homotopy. Fix an embedding
G U(m), and let F be the flag manifold U(m)=T m. The main result of this
section is the next theorem, which includes Theorem A as a special case.
Theorem 3.3. Let h be a contravariant functor from the category of (unbased)
G-CW complexes to Z[|G|-1]-modules. Suppose that h satisfies
(1) h(X) -!h(X x F ) is a monomorphism for all X.
(2) There exists a contravariant additive functor
"h: CG -! Z[|G|-1]-modules
extending h: "h(X+ ) = h(X).
Then, for all G-CW complexes X, each of the maps
Z
h(X) -! lim h(G xA X) -! h(G=A x XA )
A2A(G) A2A(G)
is an isomorphism.
16 HOPKINS, KUHN, AND RAVENEL
To see that Theorem A follows, let
h(X) = E*(EG xG X) Z[|G|-1];
with E* a complex oriented cohomology theory. Then Proposition 2.6 shows that
condition (1) applies, and condition (2) holds by letting
"h(X) = "E*(EG+ ^X) Z[|G|-1]:
G
Proof of Theorem 3.3.We begin by showing that, under the hypotheses on h, there
are natural isomorphisms
Y
(3.4) h(X) ' eA h(G=A x XA )WG(A)
(A)
with the product running only over the conjugacy classes of abelian subgroups of
G.
Since h extends to "h, there are natural commutative diagrams
Q
h(X) ----! (H)eH h(G=H x XH )WG(H)
?? ?
y ?y
Q
h(X x F ) ----! (H)eH h(G=H x XH x F H)WG(H)
with both horizontal maps isomorphisms, and the products running over all con-
jugacy classes of subgroups. As the left vertical map is a monomorphism by as-
sumption, so are each of the components of the right one. But F H = ; unless H
is abelian, so we conclude that only the terms in these products corresponding *
*to
abelian subgroups will be nonzero.
Now we apply the method of Proposition 2.3 to the following two situations. In
both cases, C will be the category of G-CW complexes, and H(X) = h(X). For X
a G-CW complex, let a
F1(X) = G xA X;
A2A(G)
and let a
F2(X) = G=A x XA :
A2A(G)
The apparent G-maps F2(X) -! F1(X) -! X define `p' in each case. To see that
there exists a natural retraction r(X) : h(Fi(X)) -!h(X) for i = 1; 2, we note *
*that
the isomorphism (3.4) will factor through h(p(X)) in each case.
Applying Proposition 2.3, we conclude that for i = 1; 2, we have equalizer di*
*a-
grams
h(X) -!h(Fi(X)) h(Fi(Fi(X))):
To complete the proof of the theorem, one checks that when i = 1 the equalizer *
*of
the two maps on the right is
lim h(G xA X);
A(G)
GENERALIZED CHARACTERS 17
and when i = 2 the equalizer is
Z
h(G=A x XA ):
A(G)
__|_|
Remarks 3.5.(1) The standard properties of of the transfer lead to an argument
like the one above establishing (3.4) to show that for any cohomology theory E,
eH . E*(BH x XH )WG(H) Z[|G|-1] is zero unless H is a p-group. (This is the
effect of working with A(G)-modules whose action extends to A^(G)) Thus the
inverse limit in Theorem A need be taken only over abelian p-groups.
(2) Note that our proof of Theorem A shows that the inverse limit is an E*-
module direct summand of the product. Thus E*(BG) Z[|G|-1] is a flat (or
projective) E* Z[|G|-1]-module for all G if it is true for abelian G. For exam*
*ple,
MU*(BG) Z[|G|-1] is thus a flat MU*-module by Landweber's observation in
[Lan71].
3.3. Proof of Proposition 3.2. We couldn't quite find this in the literature. It
seems likely that a combination of results in Chapter 5, Section 6 of [LMS86 ] *
*would
yield the proposition. The following proof is based on an unpublished argument *
*of
Adams.
Lemma 3.6. The fixed point map
Y
fGX;Y: {X; Y }G Z[|G|-1] -! {XH ; Y H}WG(H) Z[|G|-1]
(H)
is an isomorphism for all Y and all finite G-complexes X.
Here {X; Y }G denotes the equivariant stable maps, and {XH ; Y H}WG(H) denotes
the WG (H)-invariants of the WG (H)-module {XH ; Y H} of nonequivariant maps.
Proof.Suppose first that X = Sn and Y = Sm with trivial G-action. Segal
observed in [Seg71] that
Y
{Sn; Sm }G = ssSn-m(BWG (H)+ )
(H)
Thus when m = n, the map fGSn;Smreduces to the isomorphism of Lemma 3.1, and
when m 6= n, both the domain and range of fGSn;Smare 0.
Next one observes that in each variable, both the domain and the range of
fGX;Ybehave in the same way with respect to induction and G-cofibrations. For
induction, we have that [Ada82 ]
{G+ ^H X; Y }G ' {X; Y }H ' {X; G+ ^H Y }G ;
and, perhaps less obviously, that as WG (K)-spaces.
a 0
(G+ ^H X)K = WG (K0)+ ^WH(K0) XK ;
(K0)
where the union is over conjugacy classes in H of subgroups conjugate to K in G.
For cofibrations, note that if X ! Y ! Z is a G-cofibration then XH ! Y H ! ZH
18 HOPKINS, KUHN, AND RAVENEL
is a WG (H)-cofibration. The logic now goes as follows: fGSn;Smis known to be an
isomorphism for all G; m, and n. Thus fGG=H+^Sn;Smis an isomorphism for all G,
H G, m, and n. Thus fGX;Smis an isomorphisms for all groups G, finite G-
complexes X, and m (a 5-lemma argument). Thus fGX;G=H+^Smis an isomorphism
for all G, H G and finite G complexes X, and m. Thus fGX;Yis an isomorphism
for all G and all finite G- complexes, X and Y . Finally, an arbitrary Y is the*
* direct
limit of its finite subcomplexes, and homotopy groups of maps from a finite com*
*plex
commutes with direct limits, showing that fGX;Yis an isomorphism in general. _*
*_|_|
Letting X = Y in Lemma 3.6, we get a decomposition
X
1X = eH;X 2 {X; X}G Z[|G|-1]:
(H)
That this notation is redundant is shown by
Lemma 3.7. eH;X = eH ^ 1X .
Proof. The point here is that if f and g are G-maps, then (f^g)K = fK ^gK . Th*
*us
(eK ^ 1X )K = eKH^ 1XK , which is zero if K =2(H) and the identity if K = H. _*
*_|_|
As a corollary of this we have
Lemma 3.8. Let "hbe a functor as in Proposition 3.2 and let f; g 2 {Y; Z}G
Z[|G|-1]. If fH ' gH then
eH f* = eH g* : eH "h(Z) ! eH "h(Y ):
Lemma 3.9. With "has in Proposition 3.2, the map
ss ^ 1 : G=H+ ^ X ! X
induces an isomorphism
eH "h(X) ' eH "h(G=H+ ^ X)WG(H):
Proof.Repressing Z[|G|-1] from the notation, let t 2 {S0; G=H+ }G be any map
such that tH 2 {S0; WG (H)+ } is the sum of the maps sending the nonbasepoint
of S0 to the pointsPof (G=H)H = WG (H). Then (ss O t)H = |WG (H)| 2 {S0; S0}
and (t O ss)H = w2WG(H) w 2 {WG (H)+ ; WG (H)+ }. Hence the preceding lemma
implies that ss ^ 1 and t ^ 1 induce natural maps
ss
eH "h(X) ---------!---------eH "h(G=H+ ^ X)
t
satisfying the same formulae. The lemma follows. __|_|
Proof of Proposition 3.2T.he map G=H+ ^ XH ! X, sending (gH; x) to gx,
factors as
G=H+ ^ XH - j!G=H+ ^ X ss^1-!X;
where j(gH; x) = (gH; gx). Note that j is a map of left G-spaces, and is a map *
*of
right WG (H)-spaces if WG (H) acts on G=H+ ^ XH by
(gH; x) . (nH) = (gnH; n-1x):
GENERALIZED CHARACTERS 19
On H-fixed point sets, jH : WG (H)+ ^ XH ! WG (H)+ ^ XH is the homeomor-
phism (w; x) 7! (w; wx). Thus we can apply the last two lemmas to conclude that
j and ss ^ 1 induce isomorphisms
eH h(X) -'!eH h(G=H+ ^ X)WG(H) -'! eH h(G=H+ ^ XH )WG(H);
as needed. __|_|
4.Complex oriented Euler characteristics
Let X be a finite G-complex. In this section we prove Theorem B: the Morava
K-theory Euler characteristic
OGn;p(X) = dimK(n)even(EG xG X) - dimK(n)odd(EG xG X)
will be expressed in terms of the ordinary Euler characteristics of the fixed p*
*oint
spaces XA , with A G abelian. When X is a point, our computation specializes
to show that OGn;p, the Euler characteristic of K(n)*(BG), equals
X |A|
___|A(p)|nA(G)(A);
AG |G|
where, for all abelian subgroups A G, the integer A(G)(A) is defined by down-
ward induction on A 2 A(G) by the equation
X
A(G)(B) = 1:
AB
B abelian
We then use elementary group theory to show that this sum equals the number
of G-orbits in Gn;p.
4.1. Additive functions. Let M be an abelian group.
Definition 4.1 (Compare with [Die87, p. 227]).A function
X 7! O(X) 2 M
associating to each finite G-CW complex X, an element of M is additive if it
satisfies the following conditions:
(1) If X and Y are G-homotopy equivalent then O(X) = O(Y ).
(2) If Z = X [ Y and W = X \ Y then
O(W ) + O(Z) = O(X) + O(Y ):
(3) O(OE) = 0.
Since any finite G-CW complex can be built using cells of the form G=H x Dn,
we have
Lemma 4.2. An additive function O is determined by the values it takes on the
G-sets G=H for all subgroups H G.
A slightly more refined statement is
20 HOPKINS, KUHN, AND RAVENEL
Lemma 4.3. The function which associates to each finite G-set X, the class of
X in the Burnside ring A(G), extends to a unique additive function Ouniv. This
additive function is universal in the sense that
ae oe
Hom (A(G); M) ! additivevfunctionsawithlues in M
f 7!f O Ouniv
is a bijection.
From now on, unless otherwise stated, additive functions will take values in *
*the
abelian group Z[|G|-1].
Example/Definition 4.4.i)For a subgroup K G, let OK (X) be the Euler char-
acteristic of the fixed point space XK . If K0 is conjugate to K then OK = OK0.
ii)Let H be the additive function defined by downward induction on H by
X
H = OK :
KH
This also depends only on the conjugacy class of H.
We remark that in the proof of Theorem 4.8 below, we will use that the additi*
*ve
functions OK are also multiplicative:
OK (X x Y ) = OK (X)OK (Y ):
Lemma 4.5. Let K run through a set of representatives for the conjugacy classes
of subgroups of G. Then each of the sets
{OK } and {K }
is a basis of the Z[|G|-1]-module of additive functions.
Proof.To show that the set {OK }is a basis amounts, by Lemma 4.3, to showing
that the homomorphisms
A(G) ! Z[|G|-1]
X 7! |XK |
form a basis of
Hom (A(G); Z[|G|-1]):
This is precisely the content of Lemma 3.1. The assertion that {K } is a basis *
*now
follows easily from the definition. __|_|
Any additive function O can thus be written as a linear combination of the H ,
with coefficients which depend only on the values O(G=H). The next proposition
makes this more precise.
Proposition 4.6.If O is an additive function then
X 1
O = ________O(G=H)H
(H)|WG (H)|
X
= _1_|G| |H|O(G=H)H :
H
GENERALIZED CHARACTERS 21
Proof.By linearity we need only check the first equality when O = OK , which is
easy to do. First note that
X
OK (G=H) = |(G=H)K | = |WG (H)|;
H0KH0~H
thus
X 1 X X
________|(G=H)K |H = H
(H)|WG (H)| (H)HKH002(H)
X
= H
KH
= OK
as needed. For the second equality, recall that |G|=|NG (H)| is the number of H0
conjugate to H. __|_|
4.2. Complex oriented additive functions. Recall that, if is a complex vector
bundle over X, F () denotes the associated bundle of complete flags. The follow*
*ing
definition is motivated by complex oriented descent.
Definition 4.7.An additive function O is complex oriented if for every n dimen-
sional equivariant complex vector bundle over X,
O(F ()) = n! . O(X):
Our complex oriented version of Lemma 4.5 is
Theorem 4.8. An additive function O is complex oriented if and only if it is a
linear combination of the functions OA with A abelian.
Proof.We first note that Corollary 2.7 precisely says that if A G is abelian t*
*hen
OA is complex oriented. P
For the converse, suppose that O = H aH OH is complex oriented. Choose
an n-dimensional faithful representation V . Since the representation is faith*
*ful,
each isotropy subgroup in the G-space F (V ) will be abelian. Thus if H G is
nonabelian then F (V )H = OE and so OH (F (V )) = 0. We then compute, for all X,
X
n! aH OH (X)= n!O(X)
H
= O(X x F (V ))
X
= aH OH (X x F (V ))
HX
= aH OH (X)OH (F (V ))
H X
= n! aA OA (X);
A
where the last sum is only over abelian subgroups. It follows from the linear i*
*nde-
pendence of the OH that aH = 0 if H is not abelian. __|_|
Now we will prove an analogue of Proposition 4.6.
22 HOPKINS, KUHN, AND RAVENEL
Definition 4.9.For an abelian subgroup A G, the additive function CA is
defined by downward induction on A by
X
CB= OA :
AB
B abelian
In particular CA(pt.) = A(G)(A), the Moebius function of the beginning of this
section.
Proposition 4.10.If O is a complex oriented additive function then
X
O = _1_|G||A|O(G=A)CA:
A
Proof.Observe that if Y is a G-space with all isotropy groups abelian, then
aeC
H (Y ) = H (Y ) if H is abelian
0 otherwise:
We apply this to the case Y = X x F (V ), where V is a faithful n-dimensional
representation. Using 4.6, we have, for all X,
n!O(X)= O(X x F (V ))
X
= 1__|G||H|O(G=H)H (X x F (V ))
H
X
= 1__|G||A|O(G=A)CA(X x F (V ))
A
X
= n!_|G||A|O(G=A)CA(X):
A
__|_|
4.3. Morava K-theory Euler characteristics.
Proposition 4.11 (See also [Rav82 ]).If X is a finite G-CW complex , then
K(n)*(EG xG X)
is a finite dimensional vector space over K(n)*. Thus OGn;p(X)is a complex orie*
*nted
additive function. Moreover, for an abelian subgroup A G, with p-Sylow subgroup
A(p),
OGn;p(G=A) = |A(p)|n:
Proof.By replacing X with X x F (V ) and using complex oriented descent, it can
be assumed that the cells of X are of the form G=A x Dk with A abelian. An
induction over the skeleta reduces to the case X = G=A. But
K(n)*(EG xG G=A) = K(n)*(BA);
so the result follows from the well known calculation of K(n)*(BA) . (See Corol-
lary 5.10 below.) __|_|
GENERALIZED CHARACTERS 23
Proposition 4.10 immediately implies the next theorem, which was also stated
as Theorem B (Part 2) in the introduction.
Theorem 4.12. For any finite G-CW complex X,
X
OGn;p(X) = _1_|G||A||A(p)|nCA(X);
A
the sum being over the set of abelian subgroups A G. In particular (taking X to
be a point),
X
OGn;p= _1_|G||A||A(p)|nA(G)(A):
A
Now we count the G-orbits in Gn;p= Hom (Znp; G). For convenience, let = Znp.
Lemma 4.13. | Hom (; G)=G| = _1_|G|| Hom (Z x ; G)|.
Proof.After choosing a generator of Z, the fiber of the restriction mapping
Hom (Z x ; G) ! Hom (; G)
over a map ff can be identified with the centralizer, CG (ff), of the image of *
*ff. Thus
X
| Hom (Z x ; G)|= |CG (ff)|
ff:!G
X |CG (ff)|
= |G| (_______)
ff:!G |G|
= |G| . | Hom (; G)=G|
The last equality comes from the isomorphism of G-sets:
a
Hom (; G) ' G=CG (ff):
ff2Hom(;G)=G
__|_|
Lemma 4.14. There is an equality
X
| Hom (Z x ; G)| = | Hom (Z x ; A)|A(G)(A):
A
Proof.Note that the image of ff 2 Hom (Z x ; G) is abelian. Thus
X
| Hom (Z x ; G)|= 1
Xff X
= A(G)(A)
ffImage(ff)A
X X
= A(G)(A)
A Imagffe(ff)A
X
= | Hom (Z x ; A)|A(G)(A):
A
__|_|
24 HOPKINS, KUHN, AND RAVENEL
Now note that if A is abelian,
| Hom (Z x Znp; A)| = |A| . |A(p)|n:
Thus the last two lemmas combine to yield
P |A|
Corollary_4.15.| Hom (Znp; G)=G| = A ___|G||A(p)|nA(G)(A).
|_|
This formula, together with Theorem 4.12, yields Theorem B (Part 1).
5.Formal groups and E*(BA)
5.1. Formal groups and their height. The complex orientation of E determines
a formal group law F over E*: the orientation
x 2 E2(C P 1)
gives rise to isomorphisms
E*(C P 1)= E*[[x]]
E*(C P 1 x CP 1)= E*[[x1; x2]];
and the formal sum is the image of x under the map classifying the tensor produ*
*ct
of line bundles:
x1+ x2 = F (x1; x2)=*(x) 2 E*[[x1; x2]]
F
: CP 1 x CP 1 ! C P 1:
In algebro-geometric terms, the orientation x is a coordinate on F . In ordina*
*ry
geometric terms, the orientation is the first Chern class of a line bundle, and*
* the
formal group law is the formula for the first Chern of a tensor product of line
bundles.
We recall some standard notation [Rav86 ]. Given an integer m, the m-series of
F is the formal power series
z__m_"____-
[m](x) = x +: :+:x2 E*[[x]]:
F F
The m-series is an endomorphism of F :
[m](x +y) = [m](x) +[m](y):
F F
More generally, if E* is a complete, local ring with maximal ideal m, and a pri*
*me
p is in m, then, by continuity, one can define [m](x) 2 E*[[x]] for any m 2 Zp,*
* and
thus Zp acts on F as a ring of endomorphisms.
Suppose we are in this last situation, so that the mod m reduction, F0, of F *
*is
a formal group over a (graded) field of characteristic p > 0. Then F0 is more or
less determined by a single invariant, its height (see e.g. [Rav86 , Thm.A.2.2.*
*11]).
The height of F0 is the degree of the isogeny "multiplication by p", and can be
defined as follows. If the p-series of F0 is identically 0, we say that the hei*
*ght is 1.
Otherwise, [p](x) can be written uniquely in the form
n
[p]F0(x)=f(xp )
f0(0)6=0 2 E*=m;
GENERALIZED CHARACTERS 25
for some n < 1, and we define the height of F0 to be n.
From now on it will be assumed that the height of F0 is an integer n < 1.
5.2. The structure of E*[[x]]=([pr](x)). Fundamental to our work is an under-
standing of the E*-algebra E*[[x]]=([pr](x)). We begin with a very general lemm*
*a,
a form of the Weierstrass Preparation Theorem.
Lemma 5.1. Let R be a graded commutative ring, complete in the topology de-
fined by the powers of an ideal I. Suppose ff(x) 2 R[[x]] satisfies ff(x) uxd *
*mod
(I; xd+1) with u 2 R a unit. Then
i) (Euclidean algorithm) Given f(x) 2 R[[x]], there exist unique
p(x) 2 R[x] and q(x) 2 R[[x]]
such that
f(x) = p(x) + ff(x)q(x)
with degp(x) d - 1.
ii)The ring R[[x]]=(ff(x)) is a free R-module with basis
{1; x; : :;:xd-1}:
iii)(Factorization) There is a unique factorization
ff(x) = ffl(x)g(x)
with ffl(x) is a unit, and g(x) is a monic polynomial of degree d.
The number d is called the Weierstrass degree of g(x).
Proof.Statement iii) follows easily from ii). This in turn follows from i), whi*
*ch is
well known; see e.g. [Lan78, pages 129-131]. (The assumption made there that R
is local is not used in the proof.) __|_|
We apply this to [pr](x) 2 E*[[x]]. Our assumption on the height is that
n pn+1
[p](x) uxp mod (m; x )
with u 2 E* a unit10, which implies that, for all r,
rn prn+1
[pr](x) urxp mod (m; x )
where ur is an power of u (depending on r), and thus a unit in E*.
Proposition 5.2.E*[[x]]=([pr](x)) is a free E* algebra, with basis
rn-1
{1; x; : :;:xp }:
The formal group law induces a cocommutative coproduct on E*[[x]]=([pr](x)), ma*
*k-
ing E*[[x]]=([pr](x)) into a Hopf algebra over E*.
____________
10In our graded setting, this unit u will have degree 2-2pn. If E* came equi*
*pped with a BP*
orientation, then u will be the image of vn 2 BP*, thus one refers to E* as `vn*
*-periodic'.
26 HOPKINS, KUHN, AND RAVENEL
Proof.The first statement follows immediately from the previous lemma. The
second then follows: note that
E*[[x1]]=([pr](x1)) E* E*[[x2]]=([pr](x2)) -!E*[[x1; x2]]=([pr](x1); [pr](x2*
*))
is an isomorphism. Thus we can define the coproduct
E*[[x]]=([pr](x)) -!E*[[x]]=([pr](x)) E* E*[[x]]=([pr](x))
to be the map correponding to the formal sum. __|_|
There is a inverse system of Hopf algebras over E*
(5.3) . .-.!E*[[x]]=([pr+1](x)) -!E*[[x]]=([pr](x)) -!. .:.
induced by sending x 2 E*[[x]]=([pr+1](x)) to x 2 E*[[x]]=([pr+1](x)), and a di*
*rect
system of Hopf algebras over E*
(5.4) . .-.!E*[[x]]=([pr](x)) -!E*[[x]]=([pr+1](x)) -!. .:.
induced by sending x 2 E*[[x]]=([pr](x)) to [p](x) 2 E*[[x]]=([pr+1](x)).
5.3. The group of torsion points.
Definition 5.5.Given a graded E*-algebra R and integer r 0, let
prF (R)
be the group of E*-algebra homomorphisms
E*[[x]]=([pr](x)) ! R;
where the group structure is induced by the coalgebra structure on E*[[x]]=([pr*
*](x)).
The inverse system (5.3) induces a direct system of inclusions
. .!.prF (R) ! pr+1F (R) ! . .;.
and we let p1F (R) denote the colimit. We will regard p1F (R) as a subset of R
via evaluation on x.
Classically, the group p1F (R) arises in the following way. Given a local hom*
*o-
morphism
E* ! R
of complete, graded, local rings, let F (R) be the group whose underlying set i*
*s the
set of homogeneous elements of degree 2 in the maximal ideal of R, and whose
sum is the formal sum x +F y. In this case, the group prF (R) will just be the
subgroup of F (R) consisting of elements killed by pr, and p1F (R) will be the
torsion subgroup11. The point of our definition is that one can define p1F (R) *
*for
an arbitrary E*-algebra R.
The following theorem is a variation of a theorem of Lubin and Tate [LT65 ].
Theorem 5.6. Let L be an E*-algebra that is also an algebraically closed graded
field of characteristic 0. Then
____________
11As Zp acts as a ring of endomorphisms on F(R), the torsion in F(R) will al*
*l be p-torsion.
GENERALIZED CHARACTERS 27
i) For each r > 0, the group prF (L) is isomorphic to (Z=pr)n.
ii)The group p1F (L) is isomorphic to (Qp=Zp)n.
Proof.By the definition of height,
r.n
[pr](x) fflxp + . . .modm; ffl a unit:
The elements of prF (L) are the roots of [pr](x) in L. By 5.1 there are prn of *
*these
roots, counted with multiplicity. The proof will be finished if we show that al*
*l these
multiplicities are 1, for then prF (L) will be an abelian p-group with exactly *
*prn
distinct elements of order dividing pr.
To show the multiplicities are 1, we will show that [pr]0(x) has no zeros in *
*L.
We need to introduce the logarithm of F [Haz78, Rav86]. This is the unique power
series
logF(x) 2 L[[x]]
satisfying
logF(x)=x + : : :
logF(x +F y)=logF(x) + logF(y):
If x is given degree 2, the logarithm of F is homogeneous of degree 2. Note that
the derivative of the logarithm has the form
log0F(x) = 1 + . . .
and hence is a unit in L[[x]].
Taking the derivative of
logF([pr](x)) = prlogF(x)
gives
log0F([pr](x)) . [pr]0(x) = prlog0F(x);
from which it follows that
[pr]0(x) = pr . (a unit in L[[x]])
has no zeroes, since L has characteristic 0. This completes the proof. __|_|
5.4. Cohomology of abelian groups. Much of the material in this section can
be found in [RW80 ].
For a finite (or profinite) abelian group A, let
A* = Hom (A; S1)
be the character group.
Lemma 5.7. Suppose that A is cyclic of order m, and let
x 2 E2(BA)
be the first Chern class of a generator of the character group A*. The ring
E*(BA)
is isomorphic to
E*[[x]]=([m](x)):
28 HOPKINS, KUHN, AND RAVENEL
Proof.The Gysin sequence of the fibration
S1 ! BA ! CP 1
is a long exact sequence
. .!.E*[[x]] .[m](x)----!E*[[x]] ! E*BA ! . .:.
Lemma 5.1 and our standing assumption that E* is complete imply that [m](x) is
not a zero divisor. __|_|
Corollary 5.8.Write m = spr, with (s; p) = 1. The E*-module E*(BZ=m) is
free of rank prn.
Proof.This follows from Proposition 5.2. __|_|
Lemma 5.9. Let Y be a space with the property that E*(Y ) is a finitely genera*
*ted
free module over E*. Then for any space X, the map
E*(Y ) E* E*(X) ! E*(Y x X)
is an isomorphism.
Proof.Think of the map in question as a transformation of functors of X. Both
sides convert pushout squares into Meyer-Vietoris sequences, and both sides con*
*vert
infinite wedges into products. Since both functors agree when X is a point, the
transformation is an isomorphism. __|_|
Corollary 5.10.Suppose that A is a finite abelian group with p-Sylow subgroup
A(p). Then the E*-module E*(BA) is free of rank |A(p)|n.
Proof.Write A as a product of cyclic groups, and apply Corollary 5.8 and Lemma
5.9 __|_|
Corollary 5.11.If A is a finite abelian group, and X is any space, then the map
E*(BA) E* E*(X) ! E*(BA x X)
is an isomorphism. __|_|
We end this section with a natural description of the spectrum of E*(BA).
Proposition 5.12.Let A be a finite abelian group and R an E*-algebra. The
natural transformation (of functors of pairs (A; R))
(5.13) Hom E*-alg(E*(BA); R) -!Hom (A*; p1F (R))
defined by
f 7-! {O 7! f(c1(O))}
is an isomorphism.
GENERALIZED CHARACTERS 29
Proof.First suppose that A is cyclic of order m. If m is prime to p, both the
domain and range have one element. If m = pr, choose a generator O of A*, and l*
*et
x 2 E*(BA) be the first Chern class of O. By Proposition 5.7, E*(BA) can then
be identified with E*[[x]]=([pr](x)). Define a map
Hom (A*; prF (R)) ! Hom E*-alg(E*(BA); R)
by
g 7-! g(O):
This is easily checked to be an inverse to (5.13). The result for arbitrary fin*
*ite A
now follows since both sides of (5.13)convert finite sums of abelian groups into
finite products. __|_|
Note that, under the correspondence of the proposition, the identity map on
E*(BA) corresponds to a group homomorphism
OEuniv: A* -!p1 F (E*(BA)):
6. Generalized characters
In this section, we begin to relate group cohomology to certain rings of func*
*tions,
ultimately proving the basic properties of L(E*) and related rings Lr(E*), where
L(E*) is the E*-algebra defined in the introduction. The proof of Theorem C then
quite easily follows.
Let = Znp, with quotient groups r = (Z=pr)n. Then * ' (Qp=Zp)n, and
*r' (Z=pr)n is the subgroup of * of elements of order pr.
6.1. Hopf algebra isomorphisms and a discriminant calculation. If A is
a finite abelian group and R is an E*-algebra, then RA , the ring of R-valued
functions on A, will be a Hopf algebra (over E*). The next lemma will be used*to
construct maps to Hopf algebras of this elementary form, in particular to Rr .
Lemma 6.1. Let A be a finite abelian group, and R a graded E*-algebra. The set
of homomorphisms
OE : A ! prF (R)
are naturally in one to one correspondence with the set of Hopf algebra maps (o*
*ver
R)
R[[x]]=([pr](x)) ! RA :
Proof.Unraveling the definitions, a map OE : A ! pnF (R) corresponds to the Hop*
*f-
algebra homomorphism
R[[x]]=([pm ](x)) ! RA
whose a-component is the R-algebra extension of the E*-algebra homomorphism
corresponding to OE(a). __|_|
Proposition 6.2.The following conditions on a homomorphism
OE : *r! prF (R)
are equivalent:
30 HOPKINS, KUHN, AND RAVENEL
i) for all ff 6= 0 2 *r, OE(ff) is a unit,12
ii)the Hopf algebra homomorphism
*
R[[x]]=([pr](x)) ! Rr
is an isomorphism.
The following two conditions are also equivalent:
i0)for all ff 6= 0 2 *r, the element OE(ff) is not a zero-divisor.
ii0)the Hopf algebra homomorphism
*
R[[x]]=([pr](x)) ! Rr
is a monomorphism.
Proof.With respect to the basis of powers of x of the domain, and the obvious
basis of the range, the matrix of the Hopf algebra homomorphism
*
R[[x]]=([pr](x)) ! Rr
is the Vandermonde matrix of the set OE(*r). The result therefore follows from *
*the
first assertion of the next lemma. __|_|
In the following lemma, if a and b are elements of R, we will write a ~ b if *
*a = ffl.b
where ffl is a unit in R. We also let denote the discriminant of the set OE(*r*
*), i.e.
Y
= (OE(ffi) - OE(ffj)):
ffi6=ffj2*r
Lemma 6.3. Let OE : *r! prF (R) be a homomorphism, as in the last lemma.
i)
Y |*r|
~ OE(ff) :
ff6=02*r
ii)If the Hopf-algebra map
*
(6.4) R[[x]]=([pr](x)) ! Rr
is a monomorphism, then
Y
OE(ff) ~ pr;
ff6=02*r
so
rn
~ prp :
____________
12and thus OE is an isomorphism.
GENERALIZED CHARACTERS 31
Proof.The formula
x -y = (x - y) . ffl(x; y) ffl(x; y) 2 E[[x; y]]x
F
gives
Y Y
(OE(ffi) - OE(ffj))~OE(ffi- ffj)
Y Y
= OE(ff)
ffi-ffj=ffff6=0
Y |*r|
= OE(ff):
ff6=0
Now write
[pr](x) = g(x)ffl(x)
with g a monic polynomial of degree prn and ffl(x) 2 E[[x]]x . If the map (6.4)*
* is a
monomorphism, then, over R, there is a factorization
Y
g(x) = (x - OE(ff)):
ff2*r
Comparing coefficients of x gives
Y __
OE(ff) = ffl(0) . pr: |_|
ff6=02*r
6.2. The rings Lr(E*) and L(E*). If R is an E*-algebra, let Lr(R) be the set
of all group homomorphisms
OE : *r! prF (R)
satisfying either of the conditions i) or ii) of Proposition 6.2.
Proposition 6.5.The functor Lr is representable by a ring Lr(E*) that is finite
and faithfully flat over p-1E*.
Proof.Let OEuniv: *r! prF (E*(Br)) be the homomorphism corresponding to
the identity map of E*(Br), and let S E*(Br) be the multiplicatively closed
subset generated by the OE(ff) with ff 6= 0. The functor Lr is represented by *
*the
ring
Lr(E*) = S-1E*(Br);
which is clearly flat over E*. Let Dr(E*) be the image of E*(Br) in Lr(E*). The
ring Dr(E*) is finite over E*, being a quotient of E*(Br). By Lemma 6.2, the
Hopf algebra homomorphism
*
Dr(E*)[[x]]=([pr](x)) ! Dr(E*)r
is a monomorphism. It follows from Lemma 6.3 that
Lr(E*) = p-1Dr(E*);
so Lr(E*) is finite and flat over p-1E*. To check that Lr(E*) is faithfully fla*
*t, we
need to find, for each homogeneous prime ideal p of p-1E*, a homogeneous prime
32 HOPKINS, KUHN, AND RAVENEL
q Lr(E*) extending p. Choose a homomorphism f, with kernel p, from p-1E*
to an algebraically closed, graded field L. By Theorem 5.6, the group prF (L) *
*is
isomorphic to *r. A choice of an isomorphism determines a map
^f: Lr(E*) ! L
extending f. The prime ideal ker^fis the desired q. __|_|
The group Aut(r) acts naturally on Lr(E*), since it acts on the functor Lr.
Proposition 6.6.The ring of invariants is just p-1E*.
Proof.To prove this, it suffices to find a faithfully flat p-1E*-algebra R, and*
* show
that the ring of Aut(r) invariants in R E* Lr(E*) is R.
For any p-1E*-algebra R, the ring R E* Lr(E*) represents the functor that
assigns to each R-algebra S the set of Hopf algebra isomorphisms (over S)
*
S[[x]]=([pr](x)) -! Sr :
Now choose a faithfully flat R with the property that there is a Hopf algebra
isomorphism
*
R[[x]]=([pr](x)) Rr
(for example, R = Lr(E*)). Then, R E* Lr(E*) will represent the functor that
assigns to each R-algebra S the set of Hopf algebra automorphisms (over S)
* *
Sr -! Sr :
By the last part of the next lemma, we conclude that R E* Lr(E*) and RAut(r)
are isomorphic, as they represent the same functor. But it is obvious that the *
*ring
of Aut(r)-invariants in RAut(r) is just R. __|_|
Lemma 6.7. Let R be a ring.
i) Let A and B be finite sets. The functor (on the category of R-algebras)
A B
S 7! S-algebra homomorphisms: S ! S
is represented by RHomSet(B;A).
ii)Let A and B be finite abelian groups. The functor
A B
S 7! Hopf algebra homomorphisms (over S): S ! S
is represented by RHomAb(B;A).
iii)Let A be a finite abelian group. The functor
A A
S 7! Hopf algebra automorphisms (over:S)S ! S
is represented by RAutAb(A).
GENERALIZED CHARACTERS 33
Proof.Let's first construct a natural transformation. Associate to
RHomSet(B;A)! S
the map
eval. B
RA -R--! RBxHom(B;A) RHom(B;A) ! SB :
The map SA ! SB is then extended by linearity. Note that this transformation
is natural in A and B. It is an isomorphism if A is the one element set. It is
therefore an isomorphism in general since both functors carry disjoint unions i*
*n A
to cartesian products, and we have proved the first part of the lemma.
The other two parts now follow from part i) by naturality in A and B. __|_|
A homomorphism
OE : *r-!prF (R)
satisfying one of the conditions of Lemma 6.2 restricts to a homomorphism
OE : *r-1-!pr-1F (R)
satisfying the same condition. It follows that there are natural maps of E*-alg*
*ebras
Lr-1(E*) -!Lr(E*):
Furthermore, this map will be Aut(r) equivariant, where the action on the domain
is via the projection Aut(r) -!Aut (r-1).
We let L(E*) be the colimit colimrLr(E*). L(E*) will be acted on by the group
Aut(), and comes equipped with a canonical isomorphism of groups
OEuniv: * -!p1 F (L(E*)):
Corollary 6.8. i)L(E*) represents the functor that assigns to each E*-algebra
R, the set of isomorphisms of groups
OE : * -!p1 F (R)
such that OE(ff) is a unit for all ff 6= 0 2 *r.
ii)L(E*) is faithfully flat over p-1E*.
iii)The ring of Aut() invariants is just p-1E*.
6.3. Defining the generalized characters. In this subsection, we elaborate on
the character ring constructions given in the introduction.
Choose r large enough so that all p torsion in G has order dividing pr. Then
Hom (r; G) = Gn;p, so that
a
Fixn;p(G; X) = XIm(ff);
ff2Hom(r;G)
where X is a finite G-CW complex, and Fixn;p(G; X) is as in the introduction.
This fixed point space is a space with commuting actions of G and Aut(r).
A homomorphism
ff : r -!G
induces a map
ff : Br x XIm(ff)-!EG xG X
34 HOPKINS, KUHN, AND RAVENEL
and thus a map
ff* : E*(EG xG X) -!E*(Br x XIm(ff)) -!Lr(E*) E* E*(XIm(ff)):
Taking the product over ff yields a map
OGn;p: E*(EG xG X) -!Lr(E*) E* E*(Fixn;p(G; X)):
The codomain of OGn;padmits an action of G x Aut(r): G acts via its action on
Fixn;p(G; X), and Aut(r) acts diagonally on each of the factors in this tensor
product.
Lemma 6.9. This map lands in the G x Aut(r) invariants.
Proof.This compatibility is a consequence of geometric facts which have nothing
to do with the cohomology theory E*.
The invariance under the Aut(r) action follows from the commutative diagrams,
for all ff 2 Hom (r; G) and OE 2 Aut(r),
Br x XIm(ffOOE)-OEx1---!Br x XIm(ff)
? ?
ffOOE?y ff?y
EG xG X _______ EG xG X:
To see the invariance under the action of G, suppose that f : H ! G is a map
of finite groups, X a G-space, and consider the commutative diagram
BH x XH -1xg.---!BH x XgHg-1
? ?
f?y gfg-1?y
EG xG X ----! EG xG X;
in which the bottom row comes from the map
(G; X) ! (G; X)
(t; x)7!(gtg-1; gx):
It is_well known that is that the bottom map is homotopic to the identity [Seg6*
*8:1].
|_|
Because of the lemma, we conclude that OGn;pinduces
OGn;p: E*(EG xG X) -!Cln;p(G; X; Lr(E*))Aut(r)
where
Cln;p(G; X; Lr(E*)) = Lr(E*) E* E*(Fixn;p(G; X))G :
Also note that it is clear that these maps are compatible as r varies, and, ind*
*eed,
the codomain is independent of r for all r large enough.
6.4. Proof of Theorem C. Consider the following properties of a functor C = C*
from the category C of pairs (G; X) to the category of graded abelian groups:
GENERALIZED CHARACTERS 35
Meyer-Vietoris. On the category of pushout squares
W ! X
# #
Y ! Z;
in C, there is a natural connecting homomorphism
ffi : Cn(W ) ! Cn+1(Z) n 2 Z
giving rise to a long exact Meyer-Vietoris sequence
. .!.C*(Z) ! C*(Y ) C*(X) ! C*(W ) ffi!C*+1(Z) ! . .:.
Induction. For any H G and H-space Y , the natural map
(H; Y ) -!(G; G xH Y )
induces an isomorphism
C*(G; G xH Y ) C*(H; Y ):
Descent. Let F be the bundle of flags in an equivariant complex vector bundle o*
*ver
X. The sequence
X F F xXF
gives rise to an equalizer diagram
C*(G; X) ! C*(G; F ) C*(G; F xXF ):
Lemma 6.10. Let o : C ! D be a natural transformation between homotopy func-
tors satisfying the above three properties. Suppose that o also commutes with t*
*he
connecting homomorphisms of the Meyer-Vietoris sequences. If o(A; pt) is an iso-
morphism for all abelian groups A, then o(G; X) is an isomorphism for all finite
groups G, and all finite G-CW complex X.
Proof.First use descent to conclude that o(G; X) is an isomorphism if o(G; X x
F (V )) is an isomorphism, where F (V ) is the manifold of complete flags in a *
*faithful
complex representation of G. We may therefore assume that the only subgroups
of G which fix a point of X are abelian. Next run an induction on the dimension
of X, applying the Meyer-Vietoris sequences to an equivariant cell decompostion,
to show that o(G; X) is an isomorphism if o(G; G=H x Dn) is an isomorphism for
all n 0 and all H G which fix a point of X. We have reduced to showing
that o(G; G=A x Dn) is an isomorphism for all abelian A G and all n 0. By
homotopy invariance, this is the same as showing that o(G; G=A) is an isomorphi*
*sm,
and by the induction property, this is equivalent to showing that o(A; pt) is an
isomorphism for all abelian A. This completes the proof. __|_|
Proof of Theorem C.The proof will be complete once it is established that the
functors
L(E*) E* E*(EG xG X) and L(E*) E* E*(Fixn;p(G; X))G
have the above three properties, the transformation OGn;pcommutes with the con-
necting homomorphisms of the Meyer-Vietoris sequences, and finally that OAn;pis
an isomorphism when X is a point.
36 HOPKINS, KUHN, AND RAVENEL
Meyer-Vietoris. The Meyer-Vietoris sequences come from the usual Meyer-Vietoris
sequences of the pushout squares
EG xG W ----! EG xG X W Im(ff)----! XIm(ff)
?? ? ? ?
y ?y and ?y ?y :
EG xG Y ----! EG xG Z Y Im(ff)----! ZIm(ff)
The connecting homomorphisms clearly commute with OGn;p. The sequences are
exact because
(1) L(E*) is flat over E*
(2) The order of G is a unit in L(E*), so passage to G-invariants is exact.
Induction. Since the map
(H; Y ) ! (G; G xH Y )
gives rise to a homotopy equivalence
EH xH Y ' EG xG G xH Y;
the functor L(E*) E* E*(EG xG X) has the induction property.
The induction property for L(E*) E* E*(Fixn;p(G; X))G follows from the ob-
servation that there is a natural homeomophism of G-spaces
G xH Fixn;p(H; Y ) ' Fixn;p(G; G xH Y ):
See [Kuh89 ] for more about the properties of such fixed point functors.
Descent. The descent property follows from Proposition 2.6, the flatness of L(E*
**),
and the fact that passage to G-invariants is exact.
Finally, we need to verify that
OAn;p: L(E*) E* E*(BA) -!L(E*)Hom(;A)
is an isomorphism for every finite abelian group A.
Since both domain and range of the character map convert products of abelian
groups into tensor products, it suffices to consider the case when A is cyclic.*
* In
particular, it is convenient to let A = (Z=pr)*. In this case, Hom (; A) is can*
*onically
isomorphic to *r, E*(BA) is canonically isomorphic to E*[[x]]=([pr](x)), and OA*
*n;p
identifies with the canonical isomorphism
*
L(E*)[[x]]=([pr](x)) ' L(E*)r :
This_completes the proof of Theorem C.
|_|
Remark 6.11.Here is a second proof of the isomorphism
OGn;p: L(E*) E* E*(BG) ' Cln;p(G; L(E*))
that doesn't use the flatness of L(E*): Theorem 3.3 applies to the functor
h(X) = L(E*) E* E*(EG xG X);
GENERALIZED CHARACTERS 37
thus
L(E*) E* E*(BG) ' lim L(E*) E* E*(BA):
A2A(G)
But since it is clear that Gn;p= colim An;p, we have
A2A(G)
Cln;p(G; L(E*)) ' lim Cln;p(A; L(E*)):
A2A(G)
One is reduced to checking the character map is an isomorphism in the abelian
group case, and one proceeds as before.
6.5. Induction. In this subsection we establish Theorem D, the induction formula
for characters.
We will use four properties of the transfer associated to a finite covering W*
* ! Z
[Ada78 , Chapter 4]:
(1) the transfer associated to the identity map is the identity map;
`
(2) if W1 W2 ! Z is a disjoint union of finite coverings, then the transfer m*
*ap
E*(W1) E*(W2) ! E*(Z)
is the sum of the transfer maps associated to the coverings W1 ! Z and W2 ! Z;
(3) the transfer E*(Z) ! E*(W ) is a map of E*(Z)-modules;
(4) if
W1 ----! W
?? ?
y ?y
Z1 ----! Z
is a fiber square, then the diagram
E*(W1) ---- E*(W )
? ?
Tr?y ?yTr
E*(Z1) ---- E*(Z)
commutes.
Lemma 6.12. (Compare with [Die72, Satz 4].) If A r be a proper subgroup,
the composite
E*(BA) Tr-!E*(Br) ! Lr(E*)
is zero.
Proof.Recall that Lr(E*) = S-1E*(Br), where S is the image of the nonzero
elements of *runder the canonical map
OEuniv: *r-!E*(Br):
Choose a non-trivial ff in the kernel of *r! A*, and let x = OEuniv(ff) 2 E*(Br*
*).
By construction, x restricts to 0 in E*(BA), thus multiplication by x annihilat*
*es
the image of the transfer by property (3). But x becomes a unit in Lr(E*). __|*
*_|
38 HOPKINS, KUHN, AND RAVENEL
Corollary 6.13.Suppose that Y is a trivial r-space, and that J is a finite r-set
with
Jr = ;:
Then the composite
(6.14) E* (Er xr (J x Y))-Tr!E* (Br x Y )! Lr(E*) E* E*(Y )
is zero.
Proof.By property (2) we reduce to the case
J = r=A A 6= r:
Properties 3, 4 and Corollary 5.11 show in this case, that (6.14) is just the t*
*ensor
product (over E*) of the identity map of E*(Y ) with
E*(BA) Tr-!E*(Br) ! Lr(E*):
But this map is zero by the lemma. __|_|
Proof of Theorem D.Let H be a subgroup of G, X a G-space, and x an element
of E*(EH xH X) = E*(EG xG (G xH X)). If ff 2 Hom (; G) factors through
-!r, we can calculate the OGn;p(ff)(Tr*(x)) 2 L(E*)E*E*(XIm(ff)) by applying
property (4) to the pullback diagram
Er xr (G=H x XIm(ff)) ----! EG xG (G xH X)
?? ?
y ?y
Br x XIm(ff) ----! EG xG X;
which is the composite of the pullback diagrams
Er xr (G xH X) ----! EG xG (G xH X)
?? ?
y ?y
Br xr X ----! EG xG X:
and
Er xr (G=H x XIm(ff)) ----! Er xr (G xH X)
?? ?
y ?y
Br x XIm(ff) ----! Er xr X
Let J be the complement of (G=H)Im(ff)in G=H, so that
Jr = ;:
The space Er xr (G=H x XIm(ff)) decomposes into the disjoint union of
a
Br x {gH} x XIm(ff) and Er xr J x XIm(ff):
gH2(G=H)Im(ff)
GENERALIZED CHARACTERS 39
The image of x under the composite
E*(EG xG (G xH X)) -! E*(Br x {gH} x XIm(ff))
-Tr!E*(B Im(ff)
r x X )
-! Lr(E*) E* E*(XIm(ff))
is OHn;p(g . ff)(x)). The composite
f Tr * f * * Im(ff)
E* Er xr J x X -! E Br x X ! Lr(E ) E* E (X )
is zero by Corollary 6.13. Theorem D now follows from property (2) of the trans-
fer. __|_|
Remark 6.15.The key idea in this proof is our use of property (3) in the proof *
*of
Lemma 6.12. This can be similarly used to get a quick derivation of a formula of
tom Dieck [Die72]. Let Cm = Z=m. Then, for any k, m, and complex oriented
theory E* with orientation x, one has
Tr(1) = [mk](x)=[m](x) 2 E*(BC*mk)
where
Tr : E*(BC*m) -!E*(BC*mk)
is the transfer associated to C*m C*mk. The proof goes as follows. Firstly, one*
* can
assume E* = MU*, so that E*(BCmk ) = E*[[x]]=([mk](x)). Arguing as in Lemma
6.12, one deduces that Tr(1) will satisfy: Tr(1) is annihilated by multiplicati*
*on by
[m](x), and Tr(1) k mod x. But the only element in E*[[x]]=([mk](x)) satisfyi*
*ng
these two properties is [mk](x)=[m](x).
Example 6.16.As an illustration of our induction formula, we compute the map
A(G) -!ss0(BG) -!MU*(BG) -!E*(BG) -!Cln;p(G; L(E*));
i.e. the Hurewitz map from the Burnside ring of G to our ring of characters. Gi*
*ven
a virtual finite G-set S, and ff 2 Gn;p, the formula is
OGn;p(ff)(S) = |SIm(ff)|:
To verify this, it suffice to assume that S = G=H. In this case, the image of S*
* in
E*(BG) will be Tr(1), and Theorem D shows that
OGn;p(ff)(Tr(1)) = |(G=H)Im(ff)|:
In particular, we deduce that
Ker{A(G) -!MU*(BG)} {S | |SA | = 0 for all abelianA G}:
But Theorem A implies that the other inclusion is true up to finite index.
7.Good groups
In this section, we fix a prime p and n > 0, and study the question of whether
K(n)*(BG) is concentrated in even degrees via the notion of good groups.
40 HOPKINS, KUHN, AND RAVENEL
7.1. Good groups and the wreath product theorem.
Definition 7.1.(1)For a finite group G, an element
x 2 K(n)*(BG)
is good if it is a transferred Euler classes of a complex subrepresentation of *
*G, i.e.,
a class of the form TrGH(e(ae)) where ae is a complex representation of a subgr*
*oup
H < G, and e(ae) 2 K(n)*(BH) is its Euler class (i.e., its top Chern class).
(2) G is good if K(n)*(BG) is spanned by good elements as a K(n)*-module.
Note that, in general, TrGH(e(ae)) 6= e(IndGH(ae)) . For good G, K(n)*(BG) ca*
*n-
not be described only in terms of representations of G itself. If G is good th*
*en
K(n)*(BG) is of course concentrated in even dimensions.
Elementary properties of good elements and good groups are summarized in the
next propostion.
Proposition 7.2. i)Every finite abelian group is good.
ii)If x1 2 K(n)*(BG1) and x2 2 K(n)*(BG2) are both good, then so is x1x x2 2
K(n)*(B(G1xG2)). Thus if G1 and G2 are good, then so is their product G1xG2.
iii)G is good if its p-Sylow subgroup is good.
iv)If f : H ! G is a homomorphism and x 2 K(n)*(BG) is good, then f*(x) is
a linear combination of good elements in K(n)*(BH).
v) If x and y are good elements of K(n)*(BG) then their cup product xy is a sum
of good elements.
Proof.For statement (i), we note that if G is abelian then the results in subse*
*ction
5.4 make it clear that K(n)*(BG) is generated by Euler classes of representatio*
*ns
of G itself.
Statement (ii) is a consequence of the behavior of the transfer and Euler cla*
*sses
with respect to products: if x1 = TrG1H1(e(ae1)) and x2 = TrG2H2(e(ae2)), then *
*x1xx2 =
TrG1xG2H1xH2(e(ae1 ae2)).
Statement (iii) follows from the fact that Tr : K(n)*(BG(p)) -! K(n)*(BG) is
onto.
To prove statement (iv), suppose x = Tr(e(ae)) where ae is a representation of
K < G. Then there is a pullback diagram of the form
Q
Q fff
BHff ---------! BK
# #
f
BH ---------! BG
where each Hffis a subgroup of H. By naturality of the transfer,
X
f*(x) = Tr(e(f*ff(ae)):
ff
Finally, statement (v) is a consequence of (ii) and (iv): xxy 2 K(n)*(B(GxG))
is good_and xy = *(x x y) where : G ! G x G is the diagonal map.
|_|
GENERALIZED CHARACTERS 41
The next theorem is the main result of this section.
Theorem 7.3. If a finite group G is good, then so is the wreath product W =
Z=p o G.
To prove the Theorem 7.3, we study the extension
1 -!Gp -!W -ss!Z=p -!1
and the associated spectral sequence {E*;*r(BW )} with
(7.4) E*;*2(BW ) = H*(Z=p; K(n)*(BGp)) ) K(n)*(BW ):
Z=p acts on Gp by permuting the factors. Our first observation is that the in-
duced action of Z=p on K(n)*(BGp) makes K(n)*(BGp) into a permutation module.
This follows from the fact that the Kunneth isomorphism
(7.5) K(n)*(BG)p ' K(n)*(BGp)
is a Z=p-module map. This is formal if p is odd, as then K(n) is a commutative *
*ring
spectrum. If p = 2, K(n) is no longer commutative; however a formula of W"urgler
[W"ur86, Prop.2.4] measures the deviation, and we conclude that (7.5) will stil*
*l be
a map of Z=p-modules because of our hypothesis that K(n)*(BG) is concentrated
in even degrees.
Thus, as a module over Z=p, we have a decompostion
K(n)*(BGp) = F T
where F is a free Z=p-module and T has trivial Z=p-action. Moreover
ae Z=p
Hi(Z=p; F ) = F0 forif=o0ri > 0
and
H*(Z=p; T ) = H*(BZ=p) T:
We recall that E0;*2(BW ) is isomorphic to K(n)*(BGp)Z=p via the restriction
ResWGp: K(n)*(BW ) -!K(n)*(BGp). Via ss*, the spectral sequence {E*;*r(BW )}
is a module over the Atiyah-Hirzebruch spectral sequence {E*;*r(BZ=p)} that
converges to K(n)*(BZ=p).
Lemma 7.6. Every element in E0;*2(BW ) is a permanent cycle represented by a
linear combination of good elements.
Assuming this lemma, it follows that for all r 2, there are isomophisms of
differential graded K(n)*-vector spaces
(E*;*r(Z=p) T ) F Z=p-!E*;*r(BW ):
Thus we conclude that K(n)*(BW ) is spanned by products of the good elements
of Lemma 7.6 and the image of ss* : K(n)*(BZ=p) -! K(n)*(BW ). By parts (i),
(iv), and (v) of Proposition 7.2, W is good, and we have proved Theorem 7.3.
It remains to prove the lemma.
We begin by being more explicit about a decomposition of Z=p-modules
K(n)*(BGp) = F T:
42 HOPKINS, KUHN, AND RAVENEL
Choose a basis {xi} for K(n)*(BG). Let F have basis {xi1x . .x.xip} where
the subscripts i1; : :i:pare not all the same. Note that F Z=pwillPthen be span*
*ned
by elements of the form N(y), where, if y 2 K(n)*(BGp), N(y) = oe2Z=poe .y. L*
*et
T have basis {P (xi)}, where P (x) = x x . .x.x 2 K(n)*(BGp) for x 2 K(n)*(G).
Lemma 7.6 then follows from the next two lemmas.
Lemma 7.7. If y 2 K(n)*(BGp) is good, there is a good element z 2 K(n)*(BW )
so that ResWGp(z) = N(y).
Lemma 7.8. If x 2 K(n)*(BG) is good, there is a good element z 2 K(n)*(BW )
so that ResWGp(z) = P (x).
Proof of Lemma 7.7 .As Gp is normal in W , we have ResWGp(TrWGp(y) = N(y).
Thus one can let z = TrWGp(y). __|_|
Proof of Lemma 7.8 .Suppose x is the transferred Euler class Tr(e(ae)) for ae a*
* com-
plex representation of some subgroup H < G. The representation ae . . .ae of Hp
extends to a representation ^aeof W = Z=p o H and e(^ae) restricts to P (e(ae)).
There is a commutative diagram
K(n)*(BHp) -ResK(n)*(B(Z=p o H))
#Tr #Tr
K(n)*(BGp) -Res K(n)*(BW ):
Hence we have
Res(Tr(e(^ae)))=Tr(Res(e(^ae)))
= Tr(P (e(ae)))
= P (Tr(e(ae)))
= P (x):
So we can take z = Tr(e(^ae)). __|_|
7.2. The non-abelian groups of order p3. Next we recall the results of Tezuka-
Yagita [TY ] and explain how their work shows that G is good for each nonabelian
group of order p3. For each prime there are two such groups, and in each case t*
*here
is an extension
1 -! Z=p -! G -! (Z=p)2 -! 1:
Let c 2 G be a generator of the subgroup of order p, which is the center C of G,
and let a; b 2 G be elements which map to generators of the quotient group, whi*
*ch
is the abelianization of G. In each case we have relations c = [a; b] and cp =*
* 1.
For p odd we can take bp = 1 and ap = 1 orc; these two groups are denoted by E
and M respectively. For p = 2 we can take a2 = c and b2 = 1 orc; these are the
dihedral group D8 and the quaternion group Q8 respectively.
Consider the abelian subgroup A < G generated by b and c. In the quaternion
case it is cyclic of order four generated by b, and in the other three cases it*
* is
elementary abelian of rank 2. Define a one-dimensional representation OE of A by
OE(b) = i in the quaternion case and OE(c) = e2ssi=p, OE(b) = 1 in the other ca*
*ses. Let
ae be the representation of G induced by OE and let c1; : :c:p2 K(n)*(BG) be its
Chern classes.
GENERALIZED CHARACTERS 43
Let 1 and 2 be two multiplicative generators of the representation ring of the
quotient and let y1; y2 2 K(n)*(BG) denote the images of their Euler classes.
Then we have
Theorem 7.9 ([TY ]).Let G be a nonabelian group of order p3. Then K(n)*(BG)
is multiplicatively generated by the classes
y1; y2; c1; : :;:cp
defined above. A similar statement holds for BP *(BG) and there is an isomorphi*
*sm
K(n)*(BG) ' K(n)* BP* BP *(BG):
Moreover, the generators c1; : :;:cp can be replaced by any other elements x1; *
*: :;:xp
such that xi and ci have the same restriction in BP *(BC).
We assume this calculation and note that y1, y2, and cp are Euler classes of
representations of G. By Corollary 7.6, if we can find elements x1; : :;:xp-1 a*
*s in
the theorem that are also transferred Euler classes, it will follow that G is g*
*ood.
Let v 2 BP *(BA) be the Euler class of OE and u 2 BP *(BC) its restriction.
The total Chern class of ae restricts to (1 + u)p, i.e. ci restricts to a unit *
*multiple
of pui for 1 i p - 1. Similarly, by the double coset formula we see that
xi = Tr(vi) 2 BP *(BG) restricts to pui 2 BP *(BC). Since vi is an Euler class,
y1; y2; x1; : :;:xp-1; cp are good elements that generate BP *(BG). We have pro*
*ved
Proposition 7.10.Every group of order p3 is good in the sense of 7.1.
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Department of Mathematics, Massachusetts Institute of Technology, Cambridge,
MA 02139
E-mail address: mjh@math.mit.edu
Department of Mathematics, University of Virginia, Charlottesville, VA 22903
E-mail address: njk4x@virginia.edu
Department of Mathematics, University of Rochester, Rochester, NY 14627
E-mail address: drav@math.rochester.edu