CHROMATIC CHARACTERISTIC CLASSES
IN ORDINARY GROUP COHOMOLOGY
DAVID J. GREEN, JOHN R. HUNTON, AND BJÖRN SCHUSTER
Abstract.We study a family of subrings, indexed by the natural numbers,
of the mod p cohomology of a finite group G. These subrings are based on
a family of vn-periodic complex oriented cohomology theories and are con-
structed as rings of generalised characteristic classes. We identify the*
* varieties
associated to these subrings in terms of colimits over categories of ele*
*mentary
abelian subgroups of G, naturally interpolating between the work of Quil*
*len
on var(H*(BG)), the variety of the whole cohomology ring, and that of Gr*
*een
and Leary on the variety of the Chern subring, var(Ch(G)). Our subrings *
*give
rise to a `chromatic' (co)filtration, which has both topological and alg*
*ebraic
definitions, of var(H*(BG)) whose final quotient is the variety var(Ch(G*
*)).
Introduction
This paper develops a structure within the mod p cohomology ring H*(BG) =
H*(BG; Fp) of a finite group G which we show to arise both topologically and
algebraically. In general, the determination and description of the ring H*(BG)*
* is
a hard problem and we build upon two separate approaches, due respectively to
Quillen [18] and to Thomas [22], incorporating also ideas from the chromatic po*
*int
of view in homotopy theory (in particular [10]) and the theory of coalgebraic r*
*ings
(Hopf rings) and the homology of infinite loop spaces [8, 11, 12, 19, 23]. We u*
*tilise
the work of Leary and the first author [9] on categories associated to subrings*
* of
H*(BG), which first hinted at the possibility of a theorem such as our result (*
*0.1)
below.
Recall that Quillen showed [18] that the ring H*(BG) could be described up to
F-isomorphism in terms of its elementary abelian subgroup structure. For its mo*
*st
elegant formulation, suppose k is an algebraically closed field of characterist*
*ic p and
for an Fp algebra R denote by var(R) the variety of algebra morphisms from R to
k, topologised with the Zariski topology. Quillen's result describes var(H*(BG))
in terms of a colimit colimAvar(H*(BV )) over a certain category A of elemen-
tary abelian subgroups V of G. In particular recall that, for such V , the vari*
*ety
var(H*(BV )) is isomorphic to V k where V is viewed as an Fp vector space.
Fp
On the other hand, Thomas has shown [22] that a frequently significant as well
as computable subring of H*(BG) is the Chern subring, Ch(G), essentially that
part of the cohomology ring given most immediately by the complex representa-
tion theory of G (specifically, it is the subring generated by all Chern classe*
*s of
irreducible representations). Leary and the first author [9] have completed thi*
*s pic-
ture by describing var(Ch(G)) as colimA(1)var(H*(BV )), the colimit over a furt*
*her
category A(1)of elementary abelians. The categories A and A(1)have the same
____________
Date: 20 Aug 2001.
1
2 D. J. GREEN, J. R. HUNTON, AND B. SCHUSTER
objects but in general they have different morphisms and the natural surjection
var(H*(BG)) i var(Ch(G)) is not usually a homeomorphism.
However, Atiyah's theorem [2] linking the (completed) representation theory of
G with K*(BG), the complex K-theory of BG, allows a reinterpretation of Ch(G)
in terms of (unstable) maps from BG to the spaces in the -spectrum for K-
theory: the homology images of such maps can be identified with Chern classes,
the generators of the Chern subring. See section 1 for details. Replacing K-the*
*ory
by any other representable generalised cohomology theory E*(-), this constructi*
*on
allows for the definition of `E-type characteristic classes' in H*(BG) based on*
* the
spaces in the -spectrum for E. These generate a subring, ChE (G) say, of H*(BG*
*).
In many instances (including, by the work of Bendersky and the second author [8*
*],
those when E is Landweber exact [16]) there are inclusions of rings
Ch(G) ChE (G) H*(BG).
Our main results concern a family of such subrings, defined by a family of co*
*ho-
mology theories E. Specifically, we are interested in the subrings given by a f*
*amily
of vn-periodic cohomology theories such as the Johnson-Wilson [14] theories E(n*
*),
n 2 N. However, to obtain our best results we concentrate on the subrings given
by their In-adic completions [E(n), as given by Baker and Würgler [6], and these
subrings we denote ChE[(n)(G). The use of complete theories has the advantage of
enabling us to apply the results of Hopkins, Kuhn and Ravenel [10]. In section 3
we define certain categories A(n)of elementary abelian subgroups of G allowing *
*us
to prove
Theorem 0.1. Let G be a finite group and k an algebraically closed field of ch*
*ar-
acteristic p. Then there is a homeomorphism of varieties
colimvar(H*(BV )) -! var(Ch [ (G)) .
A(n) E(n)
The categories A(n)appeared in [9] where, as mentioned, the first case A(1)
was proved to represent the classical Chern subring. It was also suggested that
there may be a possible link between the A(n)and certain vn-periodic spectra; t*
*his
theorem provides such a link and supplies a topological construction for subrin*
*gs
of H*(BG) associated to the A(n).
The same result almost certainly holds on replacing the spectra [E(n)by Morava *
*E-
theory or in fact by any Landweber exact spectrum satisfying the main hypotheses
of [10]; the recent work of Baker and Lazarev [5] supplies the technical constr*
*uctions
needed for adapting the proof given below for [E(n)(the essential element being*
* the
analogue of the Baker-Würgler tower [7]). It is less clear that the analagous r*
*esult
for E(n) will hold as these spectra fail the completion hypotheses of [10].
A number of basic results on the ChE[(n)(G) can be proved.
Theorem 0.2. 1.The varieties var(Ch [E(n)(G)) form a (co)filtered space
var(H*(BG)) i . .i.var(Ch E"(n+1)(G)) i var(Ch [E(n)(G)) i . .i.k
2. For each n > 1 there is a finite group G for which var(Ch [E(n)(G)) is dis*
*tinct
from var(Ch E"(n+1)(G)), and hence ChE[(n)(G) from ChE"(n+1)(G);
3. For n at least the p-rank of G, var(Ch [E(n)(G)) = var(H*(BG));
CHROMATIC CHARACTERISTIC CLASSES 3
4. var(Ch [E(1)(G)) = var(Ch(G)), the Chern subring of H*(BG);
5. For each n there is an inclusion of rings Ch(G) ChE[(n)(G).
Given Theorem 0.1, the statements (1) and (3) follow immediately from the
definition of the A(n)while (2) and (4) follow from results of [9]. A more gene*
*ral
version of statement (5) is proved in section 2.
For simplicity we shall assume, at least after section 1, that all our rings *
*R, such
as ChE (G), are defined as subrings of the even degree part of H*(BG), and hence
are commutative. In fact, as noted in [9], this is not restrictive: if p 6= 2 t*
*he odd
degree elements of H*(BG) are nilpotent and any homomorphism Heven(BG)
R !f k extends trivially for any odd degree element; if p = 2 such a map would
extend uniquely on an odd degree element x to the unique square root of f(x2).
The rest of the paper is organised as follows. We begin by defining our E-type
characteristic classes in section 1, linking this construction to that of the c*
*lassical
Chern subring in the case E = K. We also introduce here some of the basic notat*
*ion
and language of coalgebraic algebra [12, 19] in order to discuss the homology of
spaces in the -spectrum for E.
In section 2 we recall and extend the main result of [9] and prove that, at l*
*east for
E Landweber exact, var(ChE (G)) has a description as a colimit of var(H*(BV ))
as V runs over some category, AE say, of elementary abelian subgroups V of G.
Here we also demonstrate that for such E there is an inclusion Ch(G) ChE (G).
Most of the rest of the paper is devoted to the proof of the Theorem 0.1, i.e*
*, to
the identification of the category AE . In section 3 we introduce the cateogry *
*A(n)
together with an intermediate category CE . We show that if E both is Landweber
exact and satisfies the assumptions of the character isomorphism of [10] then it
follows fairly easily that CE = A(n). The harder part is to show that AE = CE
and we prove that this will follow if certain properties of the `unstable Hurew*
*icz
homomorphism'
H: E*(X) = [X, E*] ! Hom (H*(E*), H*(X))
can be established. Our most general result, which for E = [E(n)specialises to
Theorem 0.1, is stated as Theorem 3.4 and the complete set of hypotheses needed
on a spectrum E for its application are set out in (3.2).
Proof that these hypotheses are satisfied by [E(n)is given in section 4 which
examines in detail the homology of the spaces in its -spectrum, building on Ba*
*ker
and Würgler's description [7] of [E(n)as a homotopy limit and Wilson's calculat*
*ion
of the Hopf ring for Morava K-theory [23].
We conclude with some example computations in section 5.
The construction of the subrings ChE (G) is firmly based in the world of unst*
*able
homotopy through the use of the infinite loop spaces representing E-cohomology.
We rely extensively on the techniques of coalgebraic algebra for our proofs and
computations; see [12, 19] for introductory material on this field. We also re*
*ly
on the work of Wilson [23] as the major computational input, but we note one
more recent aspect of our work. The results of [10] needed in section 3 demand
we work with completed spectra E and consequently with the spaces in their -
spectra. These are truly huge spaces whose homologies have only recently become
4 D. J. GREEN, J. R. HUNTON, AND B. SCHUSTER
accessible; see [8, 11, 13] for the methods of handling such spaces and for the*
* results
we use below.
Acknowledgements We are grateful for the suggestions and advice of Andy Baker,
Dave Benson, John Greenlees, Ian Leary, Erich Ossa and Steve Wilson. The second
author thanks the Bergische Universität Wuppertal for financial support and the
University of Leicester for sabbatical leave while much of this work was comple*
*ted.
1. E-type characteristic classes
In this section we define the class of subrings considered and show that the
classical Chern subring arises as a special case. Here and in the next section*
* we
can take G to be any compact Lie group; later we shall need to restrict to fini*
*te
groups. We begin with a very general construction.
Definition 1.1.Suppose {Xi} is a family of spaces and FG is a set of maps of the
form f :BG ! Xi. Let ChFG be the subring of H*(BG) generated by all elements
of the form f*(x) as f runs over the elements of FG and x over the homogeneous
elements of all the H*(Xi).
Example 1.2. Take {Xi} to be the set containing just one space, BU. Let F
be the set of maps BG ! BU given by irreducible representations G ! U(n) on
embedding U(n) in the infinite unitary group U and taking classifying space. The
subring in this instance is the classical Chern subring Ch(G) [22].
Remark 1.3. It is not hard to show that the Chern subring, as given in the last
example, can equivalently be defined by taking FG as the maps given by all repr*
*e-
sentations of G, or even as the maps given by all elements of the representatio*
*n ring
R(G). Clearly enlarging FG will only if anything enlarge the subring of H*(BG)
it defines, so it suffices to show that elements of R(G) give no new elements t*
*han
are already in Ch(G) as defined in (1.2). For example, if æ1: G ! U(n) and
æ2: G ! U(m) are two representations, the H-space structure on BU induced by
U(n) x U(m) ! U(n + m) gives, in cohomology, the commutative diagram
* Bæ*
H*(BU) H*(BU) -Bæ1---2--!H*(BG) H*(BG)
x? ?
? ?y
*
H*(BU) -B(æ1+æ2)------!H*(BG)
P P
sending an element x 2 H*(BU) to Bæ*1(x0).Bæ*2(x00) in H*(BG) where x0 x00
denotes the image of x in H*(BU) H*(BU). Thus the representation æ1+æ2 fails
to give anything in H*(BG) that was not already in the subring defined by æ1 and
æ2. The rest of the proof is similar.
To set up notation that will be useful later, recall that H*(BU) is the polyn*
*omial
ring on generators c1, c2, . .,.the universal Chern classes, where ci2 H2i(BU).*
* The
pull-back Bæ*(ci) of a representation æ: G ! U is the called the ithChern class*
* of
æ, also written ci(æ), and is an element of H2i(BG).
For the remainder of this paper we assume E*(-) is a representable cohomology
theory on the category of CW complexes with coefficients E* concentrated in even
dimensions. In particular, there is a representing -spectrum with spaces Er, r*
* 2 Z,
CHROMATIC CHARACTERISTIC CLASSES 5
and, for all CW complexes X, natural equivalences between Er(X) and the set of
homotopy classes of maps from X to Er. The following is the main definition of
this paper.
Definition 1.4.Let E be as just described and let G be a compact Lie group. Let
FG be the set of all homotopy classes of maps BG ! E2r, allowing all r 2 Z.
The E-Chern subring, ChE (G), is defined as the subring ChFG of H*(BG). Also,
define an E-type characteristic class of G to be an element f*(x) 2 H*(BG) for
some f :BG ! E2r and some homogeneous x 2 H*(E2r).
Remark 1.5. Note that it does not matter in this definition whether we take the
whole spaces E2r or just the connected components of their basepoints, usually
denoted E02r. This follows since all components of the E2r are the same, with
E2r= ß0(E2r) x E02r= E2rx E02r.
The following result, building on Remark 1.3, shows that this definition incl*
*udes
that of the classical Chern subring.
Proposition 1.6.Let G be a compact Lie group. Then the K-Chern subring
ChK (G) is equal to Ch(G), the classical Chern subring.
Proof.As any element of the representation ring R(G) gives rise to a homotopy
class of maps BG ! BU, it is immediate by (1.3)and (1.5)that every element
of Ch(G) is also an element of ChK (G) since the only even graded spaces in the
-spectrum for K-theory are of the homotopy type of Z x BU. However, not every
class of map BG ! BU necessarily arises as the image under B(-) of something
from the representation ring. Thus we must check further that ChK (G) Ch(G).
Recall from the work of Atiyah and Segal [3] (or Atiyah [2] for the case of f*
*inite
groups) that the homomorphism ff: R(G) ! K0(BG) given by sending the virtual
representation æ to the corresponding map BG ! Z x BU is continuous with
respect to the I-adic topology on R(G), where I is the augmentation ideal, and *
*the
filtration topology on K0(BG) given by
Fm K0(BG) = ker(K0(BG) ! K0(BGm-1 )) ,
where BGm-1 is the m - 1 skeleton of BG. Moreover, K0(BG) is a complete ring
and ff is an isomorphism after I-adic completion of R(G).
Note that the degree m cohomology of BG is determined by its (m + 1)-skeleton
BGm+1 , and so in particular the rth Chern class cr(g) of a map g :BG ! BU is
determined by its restriction to BG2r+1! BU. It follows that cr(g) is zero for *
*all
1 6 r 6 n - 1 if g 2 F2nK0(BG).
Now consider a map f :BG ! BU in FG. As an element of K0(BG) it is
represented by a Cauchy sequence æ1, æ2, . .i.n R(G). By the Whitney sum formul*
*a,
the total Chern class of f satisfies c(f) = c(æi)c(f - Bæi). Hence if f - Bæi 2
F2nK0(BG), then cr(f) = cr(æi) for 1 6 r 6 n - 1. Thus, for each r > 0, the
sequence cr(æ1), cr(æ2), . .i.n H2r(BG) is eventually constant at cr(f) = f*(cr*
*).
As any homogeneous x 2 H*(BU) can involve only finitely many of the cr,
each element f*(x) 2 H*(BG) can be realised in the form Bæ*(x) for some virtual_
representation æ 2 R(G). This completes the proof. |__|
Remark 1.7. These definitions of E-type characteristic classes can also be made
in the integral or p-local rings H*(BG; Z) and H*(BG; Z(p)). If H*(E2*; Z) is
torsion free (as is the case when E is Landweber exact, [8]), then the (mod-p)
6 D. J. GREEN, J. R. HUNTON, AND B. SCHUSTER
E-type characteristic classes are just the mod-p reductions of the integral E-t*
*ype
characteristic classes. In general however this need not be the case. As H*(BU;*
* Z)
is torsion free, the integral analogue of (1.6)holds.
Remark 1.8. The reader may wonder as to the restriction in Definition 1.4 to
even graded spaces E2rin the -spectrum for E. In the case of complex K-theory
and finite groups G this restriction is vacuous: there are no non-trivial maps *
*from
BG to the odd graded spaces [2]. For more general compact groups Proposition 1.6
would however fail without this hypothesis and so (1.4)would not have been the
correct extension of the concept of Chern subring. Below we shall restrict to f*
*inite
G and Landweber exact E; in light of Kriz's examples such as presented in [15],
it is conceivable that even here relaxing Definition 1.4 to include all spaces *
*could
enlarge the subring it defines, but the effect remains unclear. The theorems be*
*low
are proved considering the even spaces alone.
Remark 1.9. There is of course no a priori reason why the subring ChE (G) de-
fined by (1.4)should lie in Heven(BG), since there may well be odd dimensional
elements of H*(E2r) which pass to non-trivial elements of H*(BG). However, we
shall need from the main result of the next section, Theorem 2.7, and onwards t*
*hat
the spectrum E considered is Landweber exact. We can thus appeal to the work
of [13] from which it can easily be deduced that whenever E is Landweber exact
H*(E2r) has no odd dimensional elements.
2. Representations by categories of elementary abelians
Quillen proved in [18] that the variety of H*(BG) can be built from the eleme*
*n-
tary abelian p-subgroups using conjugacy information. This result was extended
in [9] to cover the varieties of certain large subrings of H*(BG) as well. The *
*object
of this section is to show that for a wide range of theories E, the varieties o*
*f the
E-Chern subrings ChE (G) have a similar description. We start by recalling the
definitions and the main result of [9], in a slightly sharpened form.
Definition 2.1.For a compact Lie group G and homogeneously generated subring
R of H*(BG), define the category C(R) as that with objects the elementary abeli*
*an
p-subgroups V of G, and morphisms from W to V to be the set of injective group
homomorphisms f :W ! V satisfying
f* ResV(x) = ResW (x) (2.1)
for every homogeneous x 2 R.
Remark 2.2. This is a variation on the definition given in [9] where the condi*
*tion
on morphisms f :W ! V was weakened to satisfying equation (2.1)modulo the
nilradical. However, for subrings R that are closed under the action of the Ste*
*enrod
algebra, the two definitions are equivalent. To see this, suppose that f is a m*
*orphism
in the öm dulo nilradical" version of C(R) with the property that, for some x 2*
* R,
the class w = f* ResV(x) - ResW(x) is a non-zero nilpotent in H*(BW ). For any
non-zero homogeneous element of H*(BW ) there is an operation ` in the mod p
Steenrod algebra such that `(w) is non-nilpotent: this is immediate for p = 2; *
*for
p odd, see Lemma 2.6.5 of [21]. Naturality of ` implies that
f* ResV(`x) - ResW(`x) = `(w) .
As R is closed under the Steenrod algebra, `(x) lies in R, contradicting f bein*
*g a
morphism in C(R).
CHROMATIC CHARACTERISTIC CLASSES 7
Definition 2.3.Let G be a compact Lie group. A virtual representation æ of G is
called p-regular if it has positive virtual dimension and restricts to every el*
*ementary
abelian p-subgroup as a direct sum of copies of the regular representation. A s*
*ubring
R of H*(BG) is called large if it contains the Chern classes of some p-regular
representation.
Theorem 2.4. [9, 6.1] Let G be a compact Lie group and R a homogenously gener-
ated subring of H*(BG) which is both large and closed under the Steenrod algebr*
*a.
Then the natural map R ! limV 2C(R)H*(BV ) induces a homeomorphism
colimvar(H*(BV )) ! var(R) . |___|
V 2C(R)
The problem, however, is to identify the category C(R) for a given R.
Example 2.5. The Quillen category A = A(G) is the category with objects the
elementary abelian p-subgroups of G, and morphisms generated by inclusion and
conjugation. It is shown in [9, 9.2] that A is the category C(H*(BG)), thus rec*
*ov-
ering one of the main results of [18]:
colimVv2Aar(H*(BV )) ~=var(H*(BG)) .
Example 2.6. It is shown in [9] that the category A(1)of elementary abelian p-
subgroups of G and morphisms the injective homomorphisms f :W ! V which take
each element w 2 W to a conjugate of itself is the category C(Ch(G)) associated
to the Chern subring.
We come to the main result of this section. Recall that the class of Landweber
exact spectra [16] includes examples such as BP -theory, complex cobordism, the
Johnson-Wilson theories E(n) and their In-adic completions [E(n)as well as Mora*
*va
E-theory, complex K-theory, various forms of elliptic cohomology [17] and their
completions (in particular the completions (Ell)cö f Baker [4]).
Theorem 2.7. Let E be a Landweber exact spectrum and let G be a compact Lie
group. Then ChE (G) is closed under the Steenrod algebra and is large in the se*
*nse
of (2.3). Hence there is a category C(Ch E(G)), which we shall write as AE (G) *
*or
just AE , such that
var(Ch E(G)) ~=colimVv2Aar(H*(BV )) .
E
Proof.Closure of ChE (G) under the Steenrod algebra is immediate from its defi-
nition and it suffices to show that it is also large. This follows from the nex*
*t_result
and the fact that the Chern subring Ch(G) is large [9]. |_*
*_|
Proposition 2.8.For E Landweber exact the homomorphism (cE1)*: H*(E02) !
H*(BU) is surjective for all n > 1. Hence for such E there is an inclusion of
subrings Ch(G) ChE (G) for all compact Lie groups G and in particular ChE (G)
contains all Chern classes of complex representations of G.
Here cE1is the first E-theory Chern class: recall that
E*(BU) = E*[[cE1, cE2, . .].] with cEi2 E2i(BU) ;
thus cE1:BU ! E02is an element of eE2(BU) where, as before, E02means the base
point component of E2.
8 D. J. GREEN, J. R. HUNTON, AND B. SCHUSTER
We need to introduce further notation. Recall that any Landweber exact spec-
trum E is certainly complex oriented. For any complex oriented spectrum, denote
by xE :CP 1 ! E02in eE2(CP 1) the orientation map. The definition of cE1is not
independent of this and in fact cE1:BU ! E02restricts to the complex orientation
xE on CP 1 [1, Part II, 4.3(ii)]. We assume that the orientation xK has been
chosen compatibly so that the diagram
cE1 0
BU ___________-E2
I@@ `
xK @ xE
CP 1
commutes.
Take basis elements fin, n 2 N, of H*(CP 1) with fin 2 H2n(CP 1), dual to
the basis of H*(CP 1) given by the powers of the orientation xH . For a complex
oriented theory E define the elements bEn2 Hn(E02) by bEn= (xE )*(fin).
Finally, as Er(X) is an abelian group for any space X, the space Er has an
H-space product Er x Er ! Er which leads, in mod p homology, to a product
*: H*(Er) H*(Er) ! H*(Er).
Lemma 2.9. Let E be any complex oriented spectrum and let K denote complex
K-theory. Then (cE1)* (bK1)*r= (bE1)*r for all r.
Proof.The diagram above, in homology, says (cE1)*(bK1) = bE1. The lemma follows
by noting that cE12 E2(BU) is a primitive element in the Hopf algebra E*(BU),
and hence represents an (unstable) additive cohomology operation K0(-) ! E2(-).
Thus the map cE1commutes with *-products and
K *r E K*r E *r
(cE1)* (b1 ) = (c1 )*(b1 )= (b1 )
for all r. |___|
Proof of Proposition 2.8.By the work of Bendersky and the second author [8],
H*(E2r) is polynomial (under the *-product) for any Landweber exact E and bE16=
0. It follows that (cE1)*(bK1*r) = (bE1)*r is never zero. But cKr2 H2r(BU) is d*
*ual
to (bK1)*r 2 H2r(BU), with respect to the basis (bK1)*r1* (bK2)*r2* . .*.(bKm)**
*rm.
Hence there is an jr 2 H2r(E02) such that (cE1)*(jr) cKr mod (cK1, . .,.cKr-1*
*).
This proves (cE1)* is surjective.
The Chern subring Ch(G) can be generated by elements of the form Bæ*(x) for
some representations æ: G ! U and homogeneous elements x 2 H*(BU). By the
first part of the proposition for any such x there is a homogeneous y 2 H*(E02)*
*_with
(cE1)*(y) = x. Hence Bæ*(x) = (cE1Bæ)*(y) lies in ChE (G). |_*
*_|
Remark 2.10. For the theory [E(n), in terms of which our main theorem is state*
*d,
we could have appealed to [11, 3.11] instead of [8].
3.The category AE
In this section, under the restriction to finite groups G, we identify the ca*
*tegory
AE for theories E satisfying certain conditions. The identification of AE will *
*depend
on a positive natural number n associated to E, and thus we begin by defining a
family of categories of elementary abelian subgroups of G, indexed by such n.
CHROMATIC CHARACTERISTIC CLASSES 9
Definition 3.1.For 0 6 n 6 1, define A(n)= A(n)(G) to be the category with
objects the elementary abelian p-subgroups of G and morphisms the injective gro*
*up
homomorphisms f :W ! V such that
8w1, . .,.wn 2 W 9g 2 G 81 6 i 6 n f(wi) = gwig-1 .
In particular, if t is the p-rank of G, there are equivalences of categories
A(t)= A(t+1)= . .=.A(1)
and this common category is the Quillen category A. As noted in Example 2.6,
it is proved in [9, 7.1] that A(1)is the category C(Ch (G)) of elementary abeli*
*ans
associated to the Chern subring. Moreover, by [9, 9.2], for each 2 6 n < 1 ther*
*e is
a subring R which satisfies the conditions of Theorem 2.4 and has category C(R)*
* =
A(n). The current paper arose from the desire to find a topological constructio*
*n of
such a ring R.
One way to explain why A(1)is the right category for Ch K(G) is via group
characters. On the one hand, K0(BG) is a completed ring of (virtual) characters.
On the other hand, the morphisms in A(1)are the group homomorphisms which
preserve the values of characters for G. That is, f :W ! V lies in A(1)if and o*
*nly
if it satisfies the equation
f* ResV(Ø) = ResW (Ø)
for every character Ø of G.
Switching attention to A(n)we recall the work of Hopkins, Kuhn and Ravenel
[10]. The morphism f :W ! V lies in A(n)if and only if it satisfies the analago*
*us
equation for every generalised character (class function) Ø of G, i.e., for eve*
*ry
function on the set Gn,pof commuting n-tuples of elements of G having p-power
order which is constant on conjugacy classes and takes values in a certain E*-a*
*lgebra
L(E*), which is, roughly speaking, the smallest E*-algebra which contains all r*
*oots
of each equation of the form [pk](x) = 0: let E*cont(BZnp) = colimrE*(B(Z=pr)n)
and S the multiplicatively closed subset generated by Euler classes of continuo*
*us
homomorphisms Znp! S1, then L(E*) = S-1E*cont(BZnp).
For suitable complex oriented E, an element x 2 E*(BG) gives rise to such a
class function in essentially the following way: a commuting n-tuple (g1, . .,.*
*gn)
in G as above can be thought of as a homomorphism ff: Znp! G. The value of
a generalised character*(class function) afforded by x on ff is then given by t*
*he
composite E*(BG) (Bff)----!E*cont(BZnp) -! L(E*). Theorem C of [10] asserts that
the character map ØG associating to each x 2 E*(BG) the character it affords
induces an isomorphism
L(E*) E* E*(BG) ØG--!Cln,p(G; L(E*)) .
Thus E*(BG) is related to the ring of generalised class functions in the same w*
*ay
as ordinary characters to the K-theory of BG.
Here a suitable theory means a complex oriented ring spectrum E, whose coef-
ficients E* are a complete, graded, local ring with maximal ideal m, and residue
characteristic p > 0, such that the mod m reduction of its formal group law has
height n and p-1E* is non-zero. These, together with Landweber exactness condi-
tion needed to apply Theorem 2.7, constitute some of our requirements on E.
We need however E to satisfy one further property. Given a space X and any
cohomology theory E*(-) represented by an -spectrum E*, we have a `Hurewicz'
10 D. J. GREEN, J. R. HUNTON, AND B. SCHUSTER
construction
HE (X): E*(X) -! Hom (H*(E*), H*(X))
where Hom denotes, say, morphisms of graded Fp algebras. The map HE (X)
is given by sending an element ff 2 Er(X), thought of as a homotopy class of
maps ff: X ! Er, to its corresponding cohomology homomorphism ff*: H*(Er) !
H*(X).
Property 3.2. Say the theory E satisfies Property 3.2(n) if
(a) E is Landweber exact and satisfies the hypotheses of [10], namely: the coe*
*ffi-
cients E* are a complete, graded, local ring with maximal ideal m and resi*
*due
characteristic p > 0, such that p-1E* is nonzero and the mod m reduction of
the formal group law has height n;
(b) the Hurewicz map HE (X) is injective when X = BV , the classifying space of
an elementary abelian group of finite rank.
Remark 3.3. In the next section we prove that the Baker-Würgler theory [E(n)
satisfies Property 3.2(n); it appears that a similar proof works for Morava E t*
*heory.
It may well be that the property (b) follows from (a); we know of no examples
satisfying (a) which do not also satisfy (b). On the other hand, the Johnson-
Wilson (incomplete) theories E(n) are examples of Landweber exact spectra for
which property (b) holds, but not (a).
Property (b) is similar to a result of Lannes and Zarati (see, for example, [*
*21,
8.1]) who prove such an injectivity property under the additional assumption th*
*at
the infinite loop space Er concerned has finite type cohomology. However, any
theory E satisfying part (a) will be far from having finite type cohomology and*
* we
use very different methods in the next section to establish the result for the *
*[E(n).
Theorem 3.4. Let G be a finite group and suppose E satisfies Property 3.2(n).
Then there is a homeomorphism of varieties
var(ChE (G)) ! colimvar(H*(BV )) .
A(n)
Theorem 0.1 will of course follow from this and Theorem 4.1.
To prove Theorem 3.4 we introduce a further category CE = CE (G) of elementary
abelians, defined in terms of the theory E.
Definition 3.5.For a finite group G, let CE = CE (G) be the category with ob-
jects the elementary abelian p-subgroups of G and morphisms the injective group
homomorphisms f :W ! V such that f* ResV = ResW holds in E-cohomology:
f* *
E*(BW ) oe______E (BV )
I@@ `
ResW @ ResV
E*(BG)
We prove Theorem 3.4 in two stages, identifying respectively A(n)with CE , and
CE with AE . The former uses part (a) of (3.2); the latter needs part (b).
Proposition 3.6.Suppose the theory E satisfies part (a) of Property 3.2(n). Then
CE = A(n)for all finite groups.
CHROMATIC CHARACTERISTIC CLASSES 11
Proof.Firstly recall from Theorem C of [10] that the kernel of the character map
consists entirely of p-torsion, as L(E*) is faithfully flat over p-1E*. So the *
*character
map is injective for elementary abelian groups.
Suppose f :W ! V is in A(n)but not in CE . Then there is an x 2 E*(BG) such
that y := f* ResV(x) - ResW(x) 6= 0. The injectivity of the character map impli*
*es
that W has a rank (at most) n subgroup S such that ResS(y) 6= 0: just take S
to be the subgroup generated by any commuting n-tuple on which the generalised
class function associated to y does not vanish. Now let T = f(S) and h = f|S.
Then h: S ! T is an isomorphism, and is induced by conjugation by some g 2 G.
Since conjugation by an element of G leaves x fixed, we arrive at
ResS(y) = h*ResT (x) - ResS(x) = ResS(g*x - x) = 0 ,
a contradiction.
Now suppose that f :W ! V is in CE but not in A(n). Then there is an
elementary abelian S W of rank at most n, such that h = f|S :S ! T = f(S)
is an isomorphism not induced by conjugation in G. From the definition of CE it
is clear that h lies in CE . Let g1, . .,.gm be a (minimal) generating set for *
*S, and
set gm+1 = . .=.gn = 1 if necessary. Then (g1, . .,.gn) and (h(g1), . .,.h(gn))
are two non-conjugate n-tuples in G, and hence are separated by a generalised
class function. Surjectivity of the character map gives us a class x 2 E*(BG) w*
*ith_
ResS(x) - h*ResT (x) 6= 0, i.e. ResV(x) and f* ResW(x) are distinct. |*
*__|
Remark 3.7. (1) In the definition of CE , the condition that morphisms be mono-
morphisms is redundant: suppose f :W ! V has kernel K. Then ResK is trivial
on E*(BG), but this cannot happen unless K = 1, as the character isomorphism
gives a nontrivial class in the image of restriction.
(2) Instead of CE it might seem more appropriate to consider a category C0Econ-
sisting of all abelian subgroups and group homomorphisms inducing commutative
triangles as in (3.5). The respective variant of Proposition 3.6 would still ho*
*ld, by
essentially the same arguments. We have refrained from doing so since our con-
struction ultimately ends up in mod p cohomology, where the difference cannot
be seen. This other approach would be relevant were we using p-local or integral
cohomology; compare the final remarks in section 17 of Quillen's paper [18].
(3) By construction, every morphism in CE is also in AE . Thus combining Propo-
sition 3.6 with Theorem 2.7 yields the chain of inclusions A A(n) AE A(1).
Proposition 3.8.Suppose the theory E is Landweber exact and satisfies part (b)
of Property 3.2(n). Then CE = AE for all finite groups.
Proof.As just noted, it is immediate that CE AE and it is the reverse inclusi*
*on
we must show. Equivalently, we need to show that the category CE does not
change upon passing to the subrings of mod p cohomology generated by E-type
characteristic classes.
12 D. J. GREEN, J. R. HUNTON, AND B. SCHUSTER
So, assume f :W ! V is in AE but not in CE . Consider the diagram
f* *
E*(BW ) oe_____________________E (BV )
\| kQQ j3 \|
|| Q Q jj ||
| Q Q j j j |
H || E*(BG) |H|
| | |
| | |
| H | |
| | |
|? | f* ?|
Hom (H*(E*), H*(BW )) oe_ |____Hom (H*(E*), H*(BV ))
kQQ || j3
Q Q || jj
Q Q | j j j
?|
Hom (H*(E*), H*(BG))
Injectivity of the two outside vertical maps is the assumption that E satisfies*
* part
(b) of Property 3.2(n). That f lies in AE implies the commutativity of the image
of the top triangle in the bottom one; that f does not lie in CE means that the*
* top
triangle does not commute. Commutativity of the `sides' of the prism follows fr*
*om
the naturality of the HE (X) construction with respect to maps of the space X.
A contradiction now follows by chasing round an element x 2 E*(BG) for which_
ResW (x) - f* ResV(x) 6= 0. |__|
4. Injectivity of HEd(n)(BV )
The goal of this section is to prove that the In-adically complete theory [E(*
*n)[6]
satisfies Property 3.2(n). Satisfaction of part (a) is well established (we see*
* that it
is Landweber exact in [6] and it is noted in [10] that the other properties lis*
*ted in
part (a) also hold). Thus we must demonstrate that the maps
H[E(n)(BV ): [E(n)*(BV ) ! Hom (H*(E[(n)*), H*(BV ))
are injective for all finite rank elementary abelian p-groups V .
In fact, we shall prove the following, equivalent result in homology.
Theorem 4.1. Let V be a finite rank elementary abelian p-group. Then
H[E(n)(BV ): [E(n)*(BV ) ! Hom (H*(BV ), H*(E[(n)*))
defined by ff: BV ! [E(n)r 7! ff*: H*(BV ) ! H*(E[(n)r) is injective. Here
Hom denotes the morphisms in the category of graded cocommutative Fp coalgebras.
We start by noting that the map HE (X) (in either homology or cohomology,
but from now we shall work with the homology variant) satisfies good algebraic
properties.
Proposition 4.2.Suppose that E is a ring spectrum and X is any space. Then
Hom (H*(X), H*(E*)) has a natural E*-algebra structure and
HE (X): E*(X) -! Hom (H*(X), H*(E*))
is a graded E*-algebra homomorphism.
CHROMATIC CHARACTERISTIC CLASSES 13
Proof.This is essentially formal, but it is a good opportunity to introduce the
notation and operations from coalgebraic algebra that are needed in the main pr*
*oof
of Theorem 4.1 together with explicit formulæ. For basic references to the alge*
*braic
properties and rules of manipulation in coalgebraic algebras (Hopf rings with f*
*urther
structure), see [12, 19].
As each Er is an H-space, each H*(Er) is a Hopf algebra with product * as
introduced in section 2 and coproduct
_ :H*(Er) ! H*(Er) H*(Er) .
However, as E is a ring spectrum the graded product in E*(X) is represented by
maps Er x Es ! Er+s giving a further product
O: Hm (Er) Hn(Es) ! Hm+n (Er+s) .
An element e 2 E-r = Er = ß0(Er), thought of as a map from a point into
Er, gives rise to an element [e] 2 H0(Er) as the image of 1 2 H0(point). Such an
element is grouplike and satisfies [d]*[e] = [d+e] (where defined) and [d]O[e] *
*= [de];
the subobject of all such elements, written Fp[E*], forms a sub-coalgebraic rin*
*g and
H*(E*) is a coalgebraic algebra over this coalgebraic ring. In fact, H0(E*) = F*
*p[E*]
and this should be thought of as the classical group-ring construction, endowed*
* with
extra structure. Note that [0] = b0 and is the * unit (and is distinct from 0) *
*and
[1] is the O unit (and is distinct from 1, which, however, is identical to [0])*
*. Note
also that a O [0] is [0] if a 2 H0(E*) but is 0 otherwise.
It is of course entirely formal that the set of coalgebra maps from H*(X), an*
* Fp
coalgebra, to H*(E*), an algebra in the category of Fp coalgebras, carries itse*
*lf an
algebra structure. However, it will be useful to identify the operations explic*
*itly.
Addition of say f, g 2 Hom (H*(X), H*(E*)) is given by the composite
H*(X) -*!H*(X) H*(X) f-g!H*(E*) H*(E*) -*!H*(E*)
and the product is described similarly using the O product. The zero element is
given by the composite
H*(X) ! H*(point) ! H*(E*)
where the second map is that which in H0(-) sends 1 to [0]; the unit is similar*
*, using
the map representing [1]. Finally, the E* action is given as follows: if e 2 E**
* = E-*
and f :H*(X) ! H*(E*), then ef is the map H*(X) ! H*(E*) sending x 2 H*(X)
to [e] O f(x). It is left to the reader to check that, with these operations,_H*
*E (X) is
an E*-algebra homomorphism. |__|
Remark 4.3. It will be useful to note that the construction HE (X) is not only
natural in the space X (as used in the previous section), but is also natural in
the spectrum E. The strategy of the proof of Theorem 4.1 will be to prove the
analagous result for HK(n)(BZ=p), i.e. in Morava K-theory for the rank 1 case,
and then deduce (4.1)from naturality in the spectrum via the Baker-Würgler tower
[7] linking K(n) and [E(n), and the application of an appropriate Künneth theor*
*em.
Recall that the group monomorphism Z=p ! S1 induces a homomorphism
E*(CP 1) ! E*(BZ=p). For a complex oriented theory E we shall just write
x 2 E2(BZ=p) for the image of the complex orientation xE .
14 D. J. GREEN, J. R. HUNTON, AND B. SCHUSTER
Proposition 4.4.[7, 10, 20] FornE = K(n) or [E(n), E*(BZ=p) is the free E*
module on basis {1, x, . .,.xp -1}. Moreover, for both these theories there is*
* a
Künneth isomorphism
(r) *
E*(B(Z=p)r) = E*(BZ=p) . . .E (BZ=p).
E* E*
|___|
As the homomorphism H*(BZ=p) ! H*(CP 1) is an isomorphism in even de-
grees, we shall extend the notation of section 2 and write fir 2 H2r(BZ=p) for *
*the
corresponding elements. NotePthat the coproduct _ :H*(BZ=p) ! H*(BZ=p)
H*(BZ=p) acts by _(fir) = i+j=rfii fij. Note also that fi0 = 1.
Proposition 4.5.(a) Suppose r > 0. The following formulæ describe some of the
action of the map HE (BZ=p) on the class xr 2 E2r(BZ=p) for a complex oriented
theory E.
HE (BZ=p)(xr): fir 7! bOr1,
fit 7! 0, if 0 < t < r,
fi0 = 1 7! [0] = 1.
(b) (Action on x0.) Suppose e 2 E*. Then
HE (BZ=p)(e): fit 7! 0, if0 < t,
fi0 = 1 7! [e].
Proof.Given the description of the map HE (X) in the proof of Proposition 4.2,
these are essentially straightforward calculations in the `Hopf ring calculus' *
*of [19].
For example, HE (BZ=p)(x)(fit) = bt, by definition of bt. Then HE (BZ=p)(xr)(fi*
*t)
is computed by the composite
X X X
fit7! fij1 . . .fijr7! bj1 . . .bjr7! bj1O. .O.bjr.
j1+...+jr=t j1+...+jr=t j1+...+jr=t
If t = 0 there is just one term in the sum - all ji = 0 and [0] O . .O.[0] = [0*
*]. If
0 < t < r then, in every term in the sum, at least one ji= 0 and at least one j*
*k > 0;
thus each summand contains the element bjiO bjk= [0] O bjk= 0 and so the whole
sum is zero. If t = r then the sum has one term in which all the ji= 1, giving *
*the
bOr1of the proposition, and all other summands are zero, as in the previous cas*
*e._
Part (b) follows immediately from the definitions. |__|
Proposition 4.6.The morphism
HK(n)(BZ=p): K(n)*(BZ=p) -! Hom (H*(BZ=p), H*(K(n) *))
is injective.
P pn-1
Proof.By (4.4)a typical element of K(n)*(BZ=p) is of the form i=0 cixiwhere
ci 2 K(n)* = Fp[vn, v-1n]. In fact, it suffices to consider homogeneous elemen*
*ts,
given the construction of H. We shall suppress the unit vn (and corresponding
[vn] 2 H*(K(n) *), as in [23]) for simplicity of notation and so shall assume c*
*i2 Fp.
Then generally homogeneous elements of degree 2i are of the simple form cixi,
ci2 Fp and 0 6 i < pn; this is not quitentrue if i 0 mod 2(pn - 1) in which c*
*ase
the general element is c0 + cpn-1xp -1.
So, for i 6 0 mod 2(pn - 1), it suffices to show that, for ci6= 0,
i
HK(n)(BZ=p) cix
CHROMATIC CHARACTERISTIC CLASSES 15
acts non-trivially on some fit. By Proposition 4.5 it sends fii to [ci] O bOi1*
*; as b1
is primitive and i > 0, this is just cibOi1. From [23] we know that all the bO*
*r12
H2r(K(n) 2r) are non-trivial if r < pn. n
For i 0 mod 2(pn - 1) we must consider the general element c0 + cpn-1xp -1.
If cpn-1 = 0 then (4.5)(b) shows HK(n)(BZ=p)(c0) is non-zero on fi0. Otherwise,
assuming cpn-1 6= 0,
n-1
HK(n)(BZ=p)(c0 + cpn-1xpn-1): fipn-17![c0] * ([cpn-1] OnbOp1 )
= [c0] * (cpn-1bOp1-1) .
As [c0] is a * unit (with * inverse [-c0]), this last expression is non-zero_an*
*d the
proof is complete. |__|
n-1
It is interesting to note that in H*(K(n) *) although bOp1 6= 0, one more O
power of b1 (or even one more suspension) kills this element. In this sense the
above proof only `just' works.
We now recall the tower of spectra defined in [7] (and implicitly in [6]). Th*
*is is
a tower
. .-.! E(n)=Ik+1n-! E(n)=Ikn-! . .-.! K(n) = E(n)=I1n
where the spectrum E(n)=Iknhas homotopy E(n)*=Ikn. The homotopy limit of this
tower is the Baker-Würgler spectrum [E(n). Unstably, (i.e., passing to spectr*
*a),
this corresponds to a tower of fibrations of the relevant spaces; the fibre of *
*the map
(E(n)=Ik+1n)*-! (E(n)=Ikn)*
is a product of copies of K(n)* indexed by a basis of Ik+1n=Ikn, i.e., by monom*
*ials
in the vl, 0 6 l < n (using the convention of putting v0 = p) of degree k + 1.
This tower of spectra gives rise to Baker and Würgler's K(n) Bockstein spectr*
*al
sequence [7]. An example of this sequence is that for the space BZ=p in which t*
*he
E2-page is just
E[(n)* K(n)*(BZ=p).
K(n)*
This is entirely in even dimensions and the sequence, converging to (E[(n))*(BZ*
*=p),
collapses; cf. Proposition 4.4. We shall prove the rank 1 case of Theorem 4.1 by
examining the Hurewicz image of this spectral sequence.
Theorem 4.7. The morphism
H[E(n)(BZ=p): [E(n)*(BZ=p) -! Hom (H*(BZ=p), H*(E[(n)*))
is injective.
Proof.Let 0 6= ff 2 (E[(n))s(BZ=p) and consider it as a map BZ=p ! [E(n)s.
Either there is some integer k > 1 for which composition of ff with the maps in*
* the
Baker-Würgler tower gives an essential map ~ff:BZ=p ! (E(n)=Ik+1n)sbut a null
map to (E(n)=Ikn)s, or else ff maps to a non-zero element of K(n)s(BZ=p) (at the
bottom of the tower).
In the former case, the map ~fflifts to an essential map to the fibre of the *
*map
(E(n)=Ik+1n)s-! (E(n)=Ikn)s, a product of spaces from the spectrum for K(n).
Thus in both this or the second case, ff gives rise to a non-trivial map
Y
a: BZ=p -! K(n)ri
i
16 D. J. GREEN, J. R. HUNTON, AND B. SCHUSTER
where the product is finite, indexed by the monomials, wi say, in the vl of deg*
*ree
k. (Note that the ri will generally differ from the original s, their value dep*
*ending
on the dimension of the wi concerned.)
Consider the commutative diagram
H*(BZ=p) _____________-H*(E[(n)s)
| |
a* || ||
Y ?| ?|
H*( K(n) ri)______-H*((E(n)=Ik+1n)s) .
i
The top map is H[E(n)(BZ=p)(ff), the map we wish to show to be non-trivial.
This will follow by finding an element fir 2 H*(BZ=p) which passes to something
non-zero in H*((E(n)=Ik+1n)s) and we do this by examining the action of a* (via
4.6)Qand the bottom horizontal map which is given by the inclusion of the fibre
': iK(n)ri! (E(n)=Ik+1n)s. Q
First examine the map a: BZ=p ! iK(n)ri. Keeping track of the compo-
nents and the monomials wi they correspond to, we can write this map as a tuple
(. .,.wicixti, . .).where, as before, we assume ci2 Fp and 0 6 ti< pn (and, str*
*ictly
speaking, in the dimension congruentnto 0 mod 2(pn - 1), components may be of
the form wi(c0,i+ cpn-1,ixp -1)). The composite with the inclusion, in homology,
Y
H*(BZ=p) -a*!H*( K(n) ri) -! H*((E(n)=Ik+1n)s)
sends an element fir to
X
*i[wi] O '*HK(n)(BZ=p) cixti (fi0i)
where the sum is overPall the terms in the iterated coproduct of fir, and where
we write _(fir)n= . . .fi0i . ... (Again, reading the longer expression c0,i+
cpn-1,ixp -1 in the displayed formula where necessary.)
If all the powers of x in this expression are zero, so that we are just deali*
*ng
with a `constant' term, then the resultPis easy - taking fi0 in the top left ha*
*nd of
the commutative square we map to i[ciwi] 6= 0 2 H*((E(n)=Ik+1n)*), where the
ci2 Fp (not all zero) are the coefficients of x0 in each factor in the above ex*
*pression
for a.
So suppose r is the smallest positive power of x which appears in the express*
*ion
for a above. By (4.6), the image of fir in the bottom right of the commutative
square is then (up to a * multiple of * invertible elements [c0,j])
_ !
X X
[wi] O cibOr1= ci[wi] O bOr1
i i
where the sum is now over only some of the indexing elements i (namely those for
which ti= r). P
It suffices now to show that expressions of the form ( ici[wi])ObOr1are non-*
*zero
in H*((E(n)=Ik+1n)*). This follows from the following lemma, thus completing_the
present proof. |__|
CHROMATIC CHARACTERISTIC CLASSES 17
Lemma 4.8. Let wj be a set of monomials in the vl, 0 < l < n, of degree k
andPof some fixed homotopy dimension and suppose cj 2 Fp, not all zero. Then
( icj[wj])O bOq1are non-zero in H*((E(n)=Ik+1n)*) for all 0 6 q < pn.
Proof.As O product with b1Prepresents (double) suspension, one way to prove this
would be to observe that ( icj[wj])was a non-zeroPelement of H0((E(n)=Ik+1n)*)
and that this suspendsPto a non-zero element ( icjwj)of H*(E(n)=Ik+1n). Then
every intermediateP( icj[wj])O bOq1must be non-zero as well. Although the stat*
*e-
ment about ( icj[wj])2 H0((E(n)=Ik+1n)*) is true, it is not true that this sus-
pends to a non-zero element of the stable gadget as, indeed, H*(E(n)=Ik+1n) = 0.
However, this proof would work if H was replaced with K(n) and the strategy
of proof will be to deduce the H result from that for K(n) by arguing with the
Atiyah-Hirzebruch spectral sequence.
First check thePstatements for K(n). It is a basic fact on the homology of
spectra that ( icj[wj])6= 0 2 K(n)0((E(n)=Ik+1n)*) as this is just the group-r*
*ing
[19]. To see the stable result it suffices to check that the right unit
E(n)*=Ik+1n-! K(n)*(E(n)=Ik+1n)
is an inclusion on sums of monomials of degree k. However, the cofibration of
spectra `
oK(n) -! E(n)=Ik+1n-! E(n)=Ikn
gives rise to a long exact sequence in Morava K-theory. The elementsWin question
map to 0 in K(n)*(E(n)=Ikn) but lift non-trivially in K(n)*( oK(n)).
Recall that in H*(K(n) *) the element b1 suspends by iterated O product with
itselfnto the elements bOr16= 0 2 H2r(K(n)n2r). These are non-zero until we get
to bOp1-1 (non-zero) suspending to bOp1-1O e1 = 0 2 H2pn-1(K(n) 2pn-1). Here
we write e1 2 H1(K(n) 1) for the single suspensionnelement following the notati*
*on
of [19]. However, in K(n)*(K(n) *) we have bOp1-1O e1 = vne1 and suspension
continues indefinitely, ultimately reaching the stable element given by the ima*
*ge of
1 under the right unit in K(n)*(K(n)).
Now consider the map of Atiyah-Hirzebruch spectral sequences
Y Q
K(n)* H*( K(n) r) =) K(n)*( K(n) r)
Fp ? i ? i
?y ?y
K(n)* H*((E(n)=Ik+1n)s) =) K(n)*((E(n)=Ik+1n)s).
Fp
iP j Q
Take an element j cj[wj] O bOq12 H*( K(n) ri) with 0 6 q < pn. We
know it is a permanent (and non-trivial) cycleQin the top spectral sequence [23]
and that regarded as an element of K(n)*( K(n) ri) it maps non-trivially to
K(n)*((E(n)=Ik+1n)*). The only way the version of this element in H*(-) could
map to zero in H*((E(n)=Ik+1n)*) would be if the K(n)*(-) version dropped Atiya*
*h-
Hirzebruch filtration in mapping to K(n)*((E(n)=Ik+1n)*). As dim([v] O bOq1)_6_
2pn - 2 this cannot happen, for dimensional reasons. |__|
Theorem 4.1 now follows from (4.4)and (4.7)together with the observation that_
H*(B(Z=p)r) also satisfies a Künneth isomorphism. |__|
18 D. J. GREEN, J. R. HUNTON, AND B. SCHUSTER
Remark 4.9. The method of proof used for Theorem 4.1 can be adapted to es-
tablish analagous results for other theories. One of the key elements of our pr*
*oof is
the use of the Baker-Würgler tower, and the recent work of Baker and Lazarev [5]
now allows such towers to be built in quite general circumstances. In particula*
*r, it
would seem that height n Morava E-theory also satisfies Property 3.2(n).
Baker's study [4] of the homotopy type of elliptic spectra shows that if " is
any suitable prime ideal of Ell*, the complete spectrum (Ell)c" splits as a wed*
*ge
of suspensions of [E(2)(see [4] for precise details of the spectra Ell and idea*
*ls "
considered). Theorem 4.1 thus shows that all these complete spectra also satisfy
Property 3.2(2) and thus, in some sense, identifies the subring of `elliptic ch*
*arac-
teristic classes' in the mod p cohomology of a finite group (even though we are*
* still
not sure what an `elliptic object' actually is).
5.Examples
We finish by sketching some of the calculations for the chromatic subrings
Ch[E(n)(G) for the simplest non-trivial example, that of G the alternating group
A4. We take the prime p to be 2.
As the 2-rank of A4 is two, the category A(2)is the Quillen category A. The
skeleton of A may be represented as
Automorphism group 1 1 C3
________________tt
Rank of elem. abelian1 2
Here, the nodes represent isomorphism classes of elementary abelians, labelled *
*by
2-rank and automorphism group. The edges represent equivalence classes of mor-
phisms under conjugacy, the label denoting the stabilizer in the automorphism
group of the target group. Thus here there are 3 = |C3 : 1| morphisms from a
rank 1 to a rank 2 elementary abelian.
The skeleton of the category A(1)is
1 C2 S3
________________tt
1 2
Thus A is strictly contained in A(1)and so, by Theorem 0.1, the subring Ch[E(1)*
*(A4)
is strictly contained in Ch[E(2)(A4). The following calculations demonstrate an
element in the latter not in the former as well as showing explicitly the equiv*
*alence
var(Ch[E(2)(A4)) = var(H*(BA4)).
Let V be the Sylow 2-subgroup of A4. The Weyl group of V in A4 is the
cyclic group C3, permuting the non-trivial elements transitively. As V is abeli*
*an,
H*(BA4) is the ring of NA4(V )-invariants in H*(BV ). Let x, y be the basis for*
* V *
dual to the basis (1 2)(3 4), (1 3)(2 4) for V . Then we have
H*(BA4) ~=F2[x, y]C3 |x| = |y| = 1
where the C3-action is x 7! y 7! x + y. Over F4 we can diagonalize this action *
*and
so calculate the invariants: H*(BA4) is generated by D1, D0 and j, where
D1 = x2 + xy + y2 D0 = x2y + xy2 j = x3 + x2y + y3 .
CHROMATIC CHARACTERISTIC CLASSES 19
Observe that D1 and D0 are Dickson invariants, and that j is the orbit sum of x*
*2y.
The natural permutation representation ß of A4 has Chern classes c2(ß) = D21and
c3(ß) = D20. These generate the Chern subring Ch(A4) = Ch[E(1)(A4), and are
[E(2)-type classes as well by Proposition 2.8.
Now [E(2)*(BV ) ~= [E(2)*[[w, z]]=([2]F (w), [2]F (z)) where [2]F (-) denotes*
* the
2-series of the formal group law for [E(2). Set ` = TrBA4BV(w2z). By the Mack*
*ey
formula, we have
ResBV` = w2z + z2(w +F z) + (w +F z)2w :BV ! [E(2)6.
Write fi(s) for the generatingPfunction of the fii 2 H2i(BZ=2), that is fi(s) i*
*s the
formal power series i>0fiisi 2 H*(BZ=2)[[s]], and similarly regard fi(s) fi*
*(t)
as the corresponding generating function in H*(BV ) = H*(BZ=2) H*(BZ=2).
F2
Likewise, write b(s) for the generating function of the biin H*([E(2)2). The st*
*andard
Ravenel-Wilson relation [19, 3.6] in H*(E*) for any complex oriented theory E n*
*ow
allows us to compute
(ResBV `)*(fi(s) fi(t))
O2 O2 O2
= b(s) O b(t)* b(t) O b(s + t)* b(s + t) O b(s).
This reduces to
bO30+ ~b(s)O2O ~b(t) + ~b(t)O2O ~b(s + t) + ~b(s + t)O2O ~b(s) mod indecompo*
*sables
P
where ~b(t) is defined as b(t) - b0 = r>1brtr (recall that br O b0 = 0 for r *
*> 0 and
that b0 is the * unit).
The (s, t)-degree 3 part of this expression is
bO31(s2t + t2(s + t) + (s + t)2s) = bO31(s3 + s2t + t3) .
Note that bO312 H6([E(2)6) represents a non-zero indecomposable: for example,
it maps to the corresponding element in H6(K(2) 6) through the Baker-Würgler
tower, and this represents a non-zero indecomposable here [23]. By duality, the*
*re
is a class fl 2 H6([E(2)6) which kills all decomposables in H6 and sends bO31to*
* 1.
Then the [E(2)-type characteristic class `*(fl) restricts to H*(BV ) as j2. He*
*nce
Ch[E(2)(A4) contains Ch[E(1)(A4) as a subring, and the inclusion of Ch[E(2)(A4)*
* in
H*(BA4) is an inseparable isogeny.
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Department of Mathematics, University of Wuppertal, Gaußstr. 20, D-42097 Wup-
pertal, Germany
E-mail address: green@math.uni-wuppertal.de, schuster@math.uni-wuppertal.de
Department of Mathematics and Computer Science, University Road, Leicester, L*
*E1
7RH, England
E-mail address: J.Hunton@mcs.le.ac.uk