Subalgebras of group cohomology
defined by infinite loop spaces
John Robert Hunton & Björn Schuster
30th November 2001
Abstract
We study natural subalgebras ChE (G) of group cohomology defined
in terms of infinite loop spaces E and give representation theoretic de-
scriptions of those based on QS0 and the Johnson-Wilson theories E(n).
We describe the subalgebras arising from the Brown-Peterson spectra
BP and as a result give a simple reproof of Yagita's theorem that the
image of BP *(BG) in H*(BG; Fp) is F -isomorphic to the whole coho-
mology ring; the same result is shown to hold with BP replaced by any
complex oriented theory E with a map of ring spectra E ! HFp which
is non-trivial in homotopy. We also extend the constructions to define
subalgebras of H*(X; Fp) for any space X; when X is finite we show that
the subalgebras ChE(n)(X) give a natural unstable chromatic filtration
of H*(X; Fp).
1 Introduction
The main aim of this paper is to study certain natural subalgebras ChE (G)
of H*(G), the cohomology of a finite group G with coefficients in Fp, which
are defined using infinite loop spaces associated to a spectrum E. Though not
originally defined in this manner, perhaps the first example of such a subal-
gebra is that of the Chern subring, Ch(G), defined and studied in detail by
Thomas [29 ] and others. The Chern subring is the subring of H*(G) gener-
ated by the Chern classes of its irreducible complex representations; it is in
some sense that part of group cohomology most accessible using the tools of
representation theory.
1
In [12 ] the construction ChE (G) was introduced for a group G and a spec-
trum E and was shown to give a subalgebra of H*(G), natural in G and closed
under the action of the Steenrod algebra. We review this construction and de-
velop further some of its basic properties in Section 2. Taking E to be the
spectrum representing complex K-theory, ChK (G) was shown in [12 ] to coin-
cide with Thomas' Chern subring Ch(G). The main goal of [12 ] however was
to study the subalgebras Ch[E(n)(G) based on Baker and Würgler's spectra
[E(n), n = 1, 2, . .,.see [4, 5], and to give these subalgebras a representation
theoretic description as well. The main tools used were a `Hopf ring' analysis
of the spaces in the -spectra for [E(n) [18 , 20, 33], the generalised charact*
*er
theory describing (E[(n))*(BG) of [17 ] and the work of Green and Leary [11 ] on
the varieties associated to subrings of H*(G). The reinterpretation of Ch(G)
as ChK (G) was similar, though easier, using the well known descriptions of
the cohomology of BU and Atiyah's theorem on K*(BG) [2].
Despite the work of [12 ], a number of basic aspects of the subalgebras
ChE (G) remained mysterious. In particular, it was not clear exactly how
the subalgebras were related when the spectrum E was changed, even to a
closely related spectrum, nor was it clear what subalgebras were produced
when some very basic examples of spectra were taken, such as the Brown-
Peterson spectra BP , the (uncomplete) Johnson-Wilson theories E(n) [21 ] or
the universal example QS0.
When a spectrum E has an associated theorem describing E*(BG) as a
functor from groups to algebras, just as Atiyah's theorem [2] identifies K*(BG)
with a certain completion R[(G) of the complex representation ring, it is rea-
sonable to suppose there is a related theorem describing ChE (G), provided
a good enough understanding of the spaces in the E-theory -spectrum can
be established. This was the case for the [E(n) spectra in [12 ] using [17 ] and
[18 ]. In a similar way, we show below that a description of ChQS0 (G) can
be made using the Segal conjecture [9] and classical work on the homology of
QS0 [25 ]. Perhaps of more interest though are the cases of subalgebras based
on BP or E(n) as these spectra have at present no associated character the-
ory or analogue of Atiyah's, Carlsson's or Hopkins-Kuhn-Ravenel's theorems
[2, 9, 17].
2
The case of BP -theory turns out to yield results through one further prop-
erty of the ChE (G) construction, not considered in [12 ]. By its nature, ChE (*
*G)
is essentially a construction in unstable homotopy, and as such there is an ass*
*o-
ciated notion of stabilisation. Using this idea we show that ChBP (G) is in fact
F -isomorphic to H*(BG). Moreover, the same is true for ChE (G) whenever
E is a complex oriented spectrum with a map of ring spectra :E ! HFp
which is onto in homotopy. A corollary of this is a very simple reproof of a
theorem of Yagita [34 ] that says that the map *: BP *(BG) ! H*(BG; Fp) is
an F -epimorphism. This work is described in Section 3. Our results also show
that the more interesting subalgebras ChE (G) are going to be those arising
from spectra E which are either non-complex oriented (such as QS0) or else
are complex oriented but periodic, such as K-theory or the Johnson-Wilson
theories E(n).
The case of QS0, already indicated, is detailed in Section 4. In Section 5 we
examine how ChE (-) varies as a functor when the spectrum E is changed, con-
centrating in particular on changing from E to an exact E-module spectrum,
though even this simple case is more complicated than might be thought. Suf-
ficient is proved to show that, for finite groups G, the subalgebras ChE(n)(G)
and Ch[E(n)(G) are equal, and so the representation theoretic description of
Ch[E(n)(G) in [12 ] applies also to the subalgebras ChE(n)(G). It should be
noted that the generalised character theory of [17 ] applies only to the com-
pleted spectra; this suggests that the results of [12 ] ought perhaps be indepe*
*n-
dent of the work of Hopkins, Kuhn and Ravenel [17 ].
Much of the work of this paper applies to define subalgebras ChE (X) of
H*(X) for an arbitrary space X, not just for spaces of the form X = BG
for a finite group G. It is useful and in fact necessary in some sections to
work in such generality: we do so as far as possible throughout. In the final
section we concentrate on the case where X has the homotopy type of a finite
CW complex. Here the spectra E(n) allow the definition of a new `unstable'
chromatic filtration of H*(X), compatible with the standard stable chromatic
filtration of Ravenel and others (see, for example, [27 , Def. 7.5.3]) defined *
*in
terms of Bousfield localisation. For X the classifying space of a finite group,
a residue of this result exists at the level of varieties.
3
Notation conventions Throughout p will denote a fixed prime and H*(G) or
H*(X) denotes the group or singular cohomology of G or X respectively with
constant coefficients in the field Fp of p elements. When E is a p-local spec-
trum, such as BP , E(n) and so on, the prime considered will be assumed to
be the same one. We identify H*(G) (group cohomology of G) with H*(BG)
(singular cohomology of its classifying space), generally using the former nota-
tion except when desiring to stress the topological nature of the construction.
By the variety of an Fp-algebra R, denoted var(R), we mean the set of algebra
homomorphisms from R to an algebraically closed field k of characteristic p,
topologised with the Zariski topology. In general R will be graded, and as
discussed in [11 ] and [12 ] var(R) will be homeomorphic to var(Reven) (even
when p = 2).
Acknowledgements Both authors are pleased to acknowledge the support of
the London Mathematical Society who, through a Scheme 4 grant, enabled the
second author to visit the University of Leicester where much of this research
was carried out. The first author also gratefully acknowledges the support of
a Leverhulme Research Fellowship.
2 The basic construction
We shall suppose throughout that E denotes a spectrum [1] representing a
generalised cohomology theory E*(-) and for convenience shall suppose the
coefficients E* = E*(point) are concentrated in even degrees. Moreover, we
shall follow [20 , 28 ] and write E_r for the rth space in the spectrum for
E, so E_r = 1-r E and E_r represents E-cohomology in degree r, that is
Er(X) = [X, E_r] for any space X. For reasons discussed further in [12 ] it is
convenient to restrict attention to the even graded spaces E_2r, and we shall do
so throughout. (In Section 4 we restrict further to the single space E_0.) Note
that each E_ris an H-space with product E_rx E_r! E_rrepresenting addition
in E-cohomology and arising from the loop space structures E_r+1 = E_r;
moreover, each E_r is an infinite loop space.
4
Definition 2.1 For a spectrum E and space X, define ChE (X) to be the
subalgebra of H*(X) = H*(X; Fp) generated, as an Fp-algebra, by elements of
the form f*(x) where f 2 E2r(X) (so f may be considered as a map f :X !
E_2r) and x a homogeneous element in H*(E_2r).
Remark 2.2 An element of Hk(X) may of course be represented by a (ho-
motopy class of a) map X ! H_k where H_k denotes an Eilenberg-Mac Lane
space of type K(Fp, k). It is thus equivalent to declare that ChE (X) is the
subalgebra of H*(X) generated by maps X ! H_k which have a factorisation
X -f! E_2r -x! H_k
for some r.
Remark 2.3 It is immediate that this construction behaves well with respect
` Q
to disjoint unions of spaces and ChE ( iXi) = iChE (Xi). With this obser-
vation we shall largely restrict ourselves to connected spaces X.
Remark 2.4 With the aid of the Cartan formula and the fact that ChE (X)
by definition is closed under ring operations, it is not hard to check that the
subalgebra ChE (X) is closed under the action of the Steenrod algebra.
Example 2.5 ([12 ] Prop. 1.6) Let G be a compact Lie group and write
ChE (G) for ChE (X) with X = BG. Write K for the complex K-theory
spectrum. Then ChK (G) is equivalent to the classical Chern subring [29 ] of
H*(G) generated by Chern classes of irreducible representations.
Corollary 2.6 For G a finite group and E a Landweber exact theory [23 ],
ChE (G) is a finitely generated Fp-algebra and H*(G) is a finitely generated
ChE (G) module.
Proof We see from [12 , Prop. 2.8] that if E is Landweber exact and X is
any space, there is an inclusion ChK (X) ChE (X). As we can identify (2.5)
ChK (G) with the Chern subring, over which we know [29 ] H*(G) is finitely
generated as a module, the inclusion ChK (X) ChE (X) immediately tells us
5
that H*(G) is a finitely generated module over ChE (G). The finite generation
of ChE (G) as an Fp-algebra now follows from basic commutative algebra [3,
Prop. 7.8] and the finite generation of H*(G) as an Fp-algebra [10 , 26, 30, 31*
*].
In general however, there seem at first sight to be potentially a lot of maps
f :X ! E_2rthat need to be considered in the construction of ChE (X). The
following technical lemma usefully cuts down on othe quantity of the maps
needed. For a CW complex X we write X(m) for its m-skeleton and recall
Milnor's short exact sequence
0 -! lim1E2r-1(X(m)) -! E2r(X) -! limE2r(X(m)) -! 0 .
Recall also that the skeleta {X(m)} define a topology (which we shall refer
to as the skeletal topology on the group lim E2r(X(m)) where the open neigh-
bourhoods of zero are given by
i j
Fm E2r(X) = ker E2r(X) -! E2r(X(m)) .
Lemma 2.7 Suppose X is a CW complex. Then ChE (X) is generated by
elements f*(x) where f :X ! E_2rrun over a set of topological generators of
the E*-module Eeven(X) modulo phantom maps.
Proof First note that we can disregard any phantom map f :X ! E_2r.
Suppose such a map gave rise to an element f*(x) 2 Hk(X) via some element
x 2 Hk(E_2r). As Hk(X) ~=Hk(X(m)) for any m > k+1 with the isomorphism
induced by restriction æ :X(m) ! X, the element f*(x) is non-zero if and only
if the composite
X(m) -fj!E_2r -x! H_k
is non-zero. However, by definition of f being a phantom map, fæ is trivial
and we are just picking up the zero element. We can thus restrict the elements
f we need to those having non-zero image in lim Eeven(X(m)).
6
Next observe that is is sufficient to consider maps f :X ! E_2r which
are E*-module generators of Eeven(X). The E*-action on E*(X) may be
represented by maps
~ :E2t x E_2r! E_2(r+t)
where the group E2t is regarded as a space with the discrete topology. Now
consider an element of ChE (X) given by (fff)*(x) where f 2 E2r(X) and
ff 2 E2t. The map fff is represented by the composite
X -! X x X ffxf-!E2t x E_2r-~! E_2(r+t)
but this is equivalent to a composite
X -g! {ff} x E_2r~=E_2r -~! E_2(r+t)
where we write ~ also for the restriction of ~ to this specific component. Thus
we have the equality
(fff)*(x) = g*(x~) .
Meanwhile, addition in E2r(X) is represented by a map
oe :E_2rx E_2r-! E_2r.
Consider an element of ChE (X) given by (f1 + f2)*(x) with fi 2 E_2r and
x 2 Ht(X). Then this element can be written as the composite
X -! X x X f1xf2-!E_2rx E_2r-ff!E_2r -x! H_t.
P
If we write oe*(x) as x0 x002 H*(E_2r) H*(E_2r) this shows that (f1 +
P
f2)*(x) = f*1(x0)f*2(x00), an element in the ring generated using just f1 and
f2.
Finally, we show that we need only consider topological generators of
Eeven(X) mod phantom maps. Once again consider a generator f*(x) 2
ChE (X) and let x 2 Hk(E_2r). Suppose f 2 E2r(X) is represented by a
sequence of elements fm 2 E2r(X), m = 1, 2, . .,.where f - fm 2 Fm E2r(X)
and fm is in the (algebraic) E*-span of a given set of topological generators.
As Hk(X) ~= Hk(X(m)) for all m > k + 1, the elements f*(x) and (fm )*(x)
are identical for all m > k + 1. This concludes the proof.
7
Remark 2.8 If E is a ring spectrum and so E*(X) is an E*-algebra, a similar
argument to the one above for sums shows that we can reduce the maps
f 2 E*(X) further to include only topological ring generators of E*(X) mod
phantoms.
Warning 2.9 The result (2.7) does not imply that the ChE (X) may be com-
puted for a CW complex X in terms of ChE (X(m)). While an element of
ChE (X) certainly gives rise, by restriction, to an element of lim ChE (X(m)),
it is possible to have a tower of elements in lim ChE (X(m)) which do not lift
to ChE (X).
We conclude this section with some remarks about the relation of the
subalgebra ChE (X) to the Bousfield E-localisation of X. Recall [7] defines for
a generalised homology theory E*(-) a localisation functor LE with a natural
transformation jX : X 7! LE X satisfying certain universal properties.
Proposition 2.10 For any space X, the subalgebra ChE (X) H*(X) is
contained in the image of j*X: H*(LE X) ! H*(X).
Proof The spaces E_2r are themselves E-local. Hence any map f :X ! E_2r
factors through the localisation jX : X 7! LE X. Thus any generator of
ChE (X) lies in j*X. As j*X is an algebra homomorphism it is closed under
sums and products and the result follows.
Remark 2.11 The inclusion ChE (X) Im (j*X) of (2.10) will usually be
strict. For example, if E is any Landweber exact spectrum, H*(E_2r) lies
entirely in even dimensions [6] and hence ChE (X) Heven(X); there are many
examples where Im (j*X) contains odd dimensional elements, for example if we
take X = BZ=p and E = BP .
However, even the inclusion ChE (X) Im (jevenX) for such spectra will
also likely be strict. For an example here, let X be the classifying space of an
elementary abelian group of rank 3 then (as X is then a nilpotent space) [7]
tells us that LBP X is just the p-localisation of X and so Im (jevenX) is the w*
*hole
of Heven(X). We shall see by Corollary 3.5 that ChBP (X) can be identified
with the image of BP *(X) under the Thom map, and for this particular X
this image is not the whole of Heven(X).
8
3 Stabilisation and BP -theory
There is no reason to suppose that the maps x :E_2r! H_k used in the defini-
tion of ChE (X) commute with the loop space structure on these spaces. This
observation allows us to make the following definition.
Definition 3.1 Define the subalgebra dChE (X) of ChE (X) H*(X) to be
that generated by elements f*(x) where x :E_2r ! H_k is a d-fold loop map,
i.e., there is map y :E_2r+d = -d E_2r! H_k+d = -d H_k with x = dy.
Likewise, define ChsE(X), the stable E-Chern subalgebra, to be the intersection
of all dChE (X), that is the subalgebra generated by elements f*(x) where
x :E_2r! H_k is a map of infinite loop spaces.
Remark 3.2 The process of stabilisation may yield only the trivial subalge-
bra. For example, the K-theory subalgebra ChK (X) is in general non-trivial
(for example, taking X to be BZ=p, we have ChK (BZ=p) = Heven(BZ=p)).
However, there are no non-trivial stable maps K ! HFp and thus ChsK(X) =
0 for all X. Similar remarks apply with K replaced by any of the E(n) spectra.
Now suppose the spectrum E is a ring spectrum and has a Thom map,
by which for the moment we just mean a distinguished map of ring spectra
:E ! HFp which is onto in homotopy. Our principal examples are E = BP
or MU, but we also include spectra such as k(n), P (n) and BP ; the periodic
spectra such as K(n) and E(n) are obviously not included. We write also
for the corresponding individual infinite loop maps E_2r! H_2r.
Definition 3.3 Define the subalgebra ChE (X) of H*(X) to be that generated
by elements of the form f*( ) where f 2 E2r(X) and is the appropriate
infinite loop map, as just given. Of course this is just the image of E*(X) in
H*(X) under the ring homomorphism *. In this situation we have natural
inclusions
ChE (X) ChsE(X) ChE (X) H*(X).
9
Before the next result let us remark again that until otherwised mentioned,
all spectra such as the E and T following are supposed to have coefficients
concentrated in even degrees. We recall the notion of Landweber exact spectra
[23 ]. This class of spectra include complex cobordism MU and the Brown-
Peterson theories BP , the Johnson-Wilson spectra E(n) and their various
completions such as [E(n) and Morava E-theory, it also contains complex K-
theory and many examples of elliptic spectra. The topology of the spaces in the
spectrum for a Landweber exact theory is well understood [6, 16, 18, 20, 28];
in particular, H*(E_2r; Z) is torsion free and concentrated in even dimensions.
Proposition 3.4 Let E be any Landweber exact spectrum and suppose T to be
a ring spectrum with Thom map. Then for any space X there is an inclusion
of subalgebras
ChE (X) ChT (X) .
Proof The Thom map :T ! HFp induces a map of Atiyah-Hirzebruch
spectral sequences for each space E_2rwhich on E2-pages is a surjection
H*(E_2r; T *) -! H*(E_2r; Fp) .
By the properties of H*(E_2r; Z) for a Landweber exact spectrum E mentioned
above, both these spectral sequences collapse and we conclude that any map
x :E_2r! H_k can be lifted through the Thom map E_2r-!l T_k -! H_k. (Note
that only even values of k arise as H*(E_2r) is concentrated in even dimensions*
*.)
Thus any element f*(x) 2 ChE (X) with f :X ! E_2rand x 2 Hk(E_2r) may
be written as (lf)*( ) 2 ChT (X).
The spectra MU and BP each satisfy the hypotheses on both E and T in
Proposition 3.4 (necessarily assuming the prime p of the underlying field Fp is
the same as that for the version of BP considered). As ChT (X) ChT (X),
by choosing both E and T to be BP (and likewise choosing both to be MU),
(3.4) gives us
10
Corollary 3.5 For any space X we have equalities
ChBP (X) = ChsBP(X) = ChBP (X)
ChMU (X) = ChsMU (X) = ChMU (X) .
Remark 3.6 This result should be contrasted with Remark 3.2 which, with
Corollary 2.6 observes that for E any of the periodic Landweber exact spectra
listed above, the subalgebras ChE (X) and ChsE(X) are quite different - the
former, for X = BG, containing the whole Chern subring, the latter being
trivial.
Choosing one of E or T to be BP and the other to be MU, Proposition 3.4
tells us
Corollary 3.7 For X any space, ChBP (X) = ChMU (X).
Theorem 3.8 Let T be a complex oriented ring spectrum with a Thom map
and suppose G to be a finite group. Then the inclusions ChT (G) H*(G)
and ChT (G) H*(G) are F -isomorphisms. Hence, by the first observation
in (3.3), for such T the map *: E*(BG) ! H*(G) is an F -epimorphism.
We offer two proofs of this result, both resting on the stabilisation results j*
*ust
proved. The first proof uses also the main theorem of [12 ], while the second
uses instead work of Carlson [8] or of Green and Leary [11 ] on corestriction of
Chern classes.
First Proof By Proposition 3.4, for every n = 1, 2, . .,.there are inclusions
Ch[E(n)(G) ChT (G) ChT (G) H*(G) .
However, by [12 , Theorem 0.2] the inclusion Ch[E(n)(G) H*(G) is an F -
isomorphism if n is not less than the p-rank of G.
Second Proof By Proposition 3.4 there are inclusions
ChK (G) ChT (G) ChT (G) H*(G) .
11
______
We follow [11 ] and write R(G) for the closure of a functorial subring R(G)
H*(G) under corestriction of elements from R(H) for all subgroups H of G.
_________
By [11 , x8] or [8] (see [11 ] for detailed discussion), ChK (G) is F -isomorp*
*hic
_________
to H*(G), implying ChT (G) is also F -isomorphic to H*(G). However, as
_________
corestriction is a stable construction, ChT (G) = ChT (G) and so ChT (G) and
hence ChT (G) is F -isomorphic to H*(G).
Remark 3.9 If we take T = BP or k(n) in Theorem 3.8 we recover Yagita's
results [34 , (4.2), (4.3)]. As noted at the end of the previous section, the
inclusions ChBP (G) Heven(G) can however be strict, for example when G
is elementary abelian of rank 3. In the next section we shall show that the
result (3.8) can fail when T is not complex oriented.
Corollary 3.10 Suppose T is a complex oriented ring spectrum with a Thom
map and G is a finite group. Let A(G) be the category of elementary abelian
subgroups of G with morphisms those inclusions V ! W given by conjugation
by an element of G. Then there is a homeomorphism of varieties
var(ChT (G)) ~= colim var(H*(V )) .
V 2A(G)
Proof This follows from Theorem 3.8 which identifies var (ChT (G)) with
var(H*(G)) and Quillen's theorem [26 ] which describes var(H*(G)) as the
colimit of the H*(V ) over the category indicated.
4 Subalgebras defined by QS0
Suppose throughout this section that G is a finite group. The main result
below identifies the subalgebra ChQS0 (G) of H*(G) by which we mean the
subalgebra of H*(G) generated by the images in cohomology of maps BG !
QS0. Recall that QS0 = nlim!1 n nS0 and is the infinite loop space that
represents stable cohomotopy in degree zero, i.e., ß0s(X) = [X, QS0]. We
restrict attention here to the zero space QS0 as, by the Segal conjecture [9],
there are no non-trivial maps BG ! QSn for n > 0. The cases n < 0 present
formidable difficulties.
12
We begin by recalling the algebra Sh = Sh(G) H*(G) of [13 , Def. 1.2].
For a finite G-set X of cardinality n, a choice of bijection between X and
the set {1, . .,.n} induces a homomorphism æX : G ! n where n is the
symmetric group on n letters. It thus induces an algebra homomorphism
æ*X: H*( n) ! H*(G). For a fixed space X two choices of æX differ only by an
inner automorphism of n and so the homomorphism æ*X: H*( n) ! H*(G)
depends only on X and not on the choice of bijection. Then Sh = Sh(G) is
defined as the subalgebra of H*(G) generated by elements of the form æ*X(x)
as X runs over all finite G sets and where the x are homogeneous elements of
H*( n).
Theorem 4.1 For G finite there is an algebra isomorphism ChQS0 (G) ~=
Sh(G).
Proof First we show that Sh(G) is a subalgebra of ChQS0 (G). By definition,
Sh(G) is generated by elements of the form f*(x) where f is drawn from some
particular class of maps BG ! B n and x 2 H*( n). Given such an element,
consider the composite
BG -f! B n -in!B 1 -D! QS0
where in is induced by the incusion of n in the infinite symmetric group, and
D is the Dyer-Lashof map. By [25 ] D induces an isomorphism in cohomology,
while by Nakaoka [24 , x7] the induced map i*n:H*(B 1 ) ! H*(B n) is a
surjection. Thus there is an element y 2 H*(QS0) with x = (Din)*(y). Hence
f*(x) = (Dinf)*(y) 2 ChQS0 (G).
To show the converse we use the Segal conjecture [9]. Recall that this
identifies the homotopy classes of maps BG ! QS0 with the elements of
the Burnside ring, completed at the augmentation ideal. Specifically, to an
element of the Burnside ring A(G) we can associate a finite G-set X; the Segal
conjecture tells us that the map which sends this element to the resulting
composite
BG -f! B n -in!B 1 -D! QS0
13
(where again, n is the cardinality of X) is continuous (with respect to the
topologies induced by powers of the augmentation ideal in A(G) and the skeleta
in ß0s(BG)) and an isomorphism after completion; the ring ß0s(BG) is already
complete under the skeletal topology. Thus we can take topological generators
of ß0s(BG) from the ring A(G) itself and any element g*(y) 2 ChQS0 (G) rep-
resented by such an element g 2 ß0s(BG) and y 2 H*(QS0) can be realised as
an element f*(x) 2 Sh(G) where x = (Din)*(y). Lemma 2.7 now shows that
this is sufficient to prove that ChQS0 (G) is contained in Sh(G).
Remark 4.2 A representation theoretic description of the variety associated
to Sh(G) is given in [13 ]. The work there combined with Theorem 4.1 shows
there is a natural homeomorphism
var(ChQS0 (G)) ~= colim var(H*(V ))
V 2Ah(G)
where Ah(G) denotes the category whose objects are elementary abelian sub-
groups of G and whose morphisms are those injective group homomorphisms
f :V ! W for which f(U) is conjugate in G to U for every subgroup U of V .
Remark 4.3 It is also noted in [13 ] that in general neither Sh(G) nor Ch(G),
the classical Chern subring, are contained in the other. See [13 ] for worked
examples.
5 Subalgebras defined by exact spectra
In this section we examine the relation between ChE (X) and ChF (X) for
spectra E and F that are themselves related; specifically, we are interested in
the case where F is exact over E, by which we mean that E is a ring spectrum,
F an E-module spectrum and the corresponding homology theories are related
by a natural equivalence
F*(X) ~=F* E*(X)
E*
for any space X.
14
It is of course not true that if F is exact over E then ChE (X) and ChF (X)
are identical: for example, complex K-theory is exact over BP -theory, but
the inclusion ChK (X) ChBP (X) is certainly strict in many cases. However,
exactness of F over E is sufficient, under suitable finiteness conditions, to g*
*ive
in general an inclusion ChF (X) ChE (X); see the results (5.2), (5.3), (5.4)
and (5.8) below.
Our main goal here though, Theorem 5.7, is to show that the complete
theories [E(n) used in [12 ] are not the only ones that can give the topological
interpretation of the subgroup categories A(n) of [11 , 12], and that the more
classical Johnson-Wilson theories E(n) [21 ] satisfy a similar theorem. The
Baker-Würgler completions [E(n), or equivalently the Morava E-theories, were
necessary in [12 ] so as to apply the generalised character theory of Hopkins,
Kuhn and Ravenel [17 ] and the completeness assumptions seem integral to
that character theory. Thus Theorem 5.7 below has apparently by-passed
the subtleties of [17 ] and suggests it ought to be possible to find a proof of
Theorem 5.7 without reference to [17 ]; it would be interesting to see such a
direct argument.
Throughout this section we shall not only continue the convention that any
subalgebra ChE (X) or ChF (X) is defined using just the even graded spaces in
the E or F -spectra, but we shall further restrict these subalgebras to those
lying in even dimensional cohomology, that is, by ChE (X) we shall really mean
ChE (X) \ Heven(X), etc. This convention is made for notational simplicity
and could be avoided if one wanted. However, as the main examples we wish
to discuss here are of Landweber exact spectra, we see by [6] or by [20 , 28] t*
*hat
this is not actually a restriction at all: there are no odd dimensional elements
in the original subalgebras ChE (X) for such E.
We begin with the following general observation about the subalgebras
ChE (X) and ChF (X) for arbitrary spectra E and F (not necessarily exact
or even one a module spectrum over the other) arrived at by considering the
universal example of X = F_2*.
Proposition 5.1 Suppose the cohomology of each space Heven(F_2r) is of fi-
nite type and suppose ChE (F_2r) = Heven(F_2r) for each r. Then ChF (X)
ChE (X) for every space X.
15
Proof Consider a generator f*(x) of ChF (X), where f :X ! F_2r and x 2
H2t(F_2r). As ChE (F_2r) = Heven(F_2r), the finiteness assumption means that
we can write x as a finite sum of products of elements g*i(yi) 2 ChE (F_2r), wi*
*th
gi: F_2r! E_2siand yi 2 H*(E_2si). The element f*(x) may thus be written
as the composite
Y Q gi Y Y
X -f! F_2r -! F_2r- ! E_2si-! H_2ti-! H_2t
i i i
where the last arrow denotes the summing and multiplication operations. The
Q
composite of the final two arrows denotes an element of H2t( iE_2si) and as
such we can write, using the Künneth theorem for mod p cohomology, in the
P
form y(1) . . .y(q)(where the sum is potentially infinite). If we write fi
Q
for the projection of the composite X -f! F_2r-! iE_2sito the ith factor,
we see that
X
f*(x) = f*1(y(1)) . .f.*q(y(q)) ,
an element of ChE (X): note that this sum will contain only a finite number
of non-zero terms by virtue of the finite type assumption on Heven(F_2r), even
P Q
if the first sum y(1) . . .y(q)2 H*( iE_2si) is infinite.
Proposition 5.2 Suppose F is an exact E-module spectrum and that the
space X is of finite type. Then ChF (X) ChE (X).
Proof Assume X is connected, for else we can argue by connected compo-
nents. If X had the homotopy type of a finite CW complex then F *(X) =
F * E*(X); if X is only of finite type then this statement is only true after
E*
completion of the tensor product. The finite type hypothesis however tells us
that the completion is with respect to the skeletal topology (as in Section 2):
` '
F *(X) = F *c E*(X) = lim F * E*(X(m)) .
E* m!1 E*
Thus topological generators of F *(X) may be taken from elements of the
(uncompleted) tensor product F * E*(X). Such an element, considered as
E*
a map X ! F_2r, may thus be lifted as a finite sum of maps through spaces
16
F *x E_2si, where F *is regarded as a discrete space, and thus we can represent
f as a composite
Y
X -! (F *x E_2si)- ! F_2r.
i
By the connectedness assumption, such a map actually lifts through a finite
product of spaces E_2si.
Now consider a generator f*(x) 2 ChF (X) as usual. Factoring f as above
Q
as a map X -! iE_2si-! F_2rtogether with the finite type assumption on
X now shows f*(x) to be in the subalgebra ChE (X).
Corollary 5.3 If F is an exact E-module spectrum and the spaces F_2r rep-
resenting F are of finite type, then ChF (X) ChE (X) for every space X.
Proof We apply Proposition 5.2 with X = F_2rand deduce that ChF (F_2r)
ChE (F_2r) H*(F_2r). But ChF (F_2r) = H*(F_2r) by definition of ChF (-) and
hence ChE (F_2r) = H*(F_2r). The result now follows from Proposition 5.1.
Remark 5.4 There are a number of examples to which this corollary may
be applied. First of all, we could take E = MU and F = K and we recover
the inclusion ChK (X) ChMU (-) of Propositions 2.5 and 3.4. New results
however include the inclusions
ChKO (X) ChMSp (X) ChKO (X) ChMSpin (X) ChKO (X) ChEl(X)
ChK (X) ChMSpinc (X) ChEl(X) ChMSpin (X)
where MSpinc is the self-conjugate spin cobordism of [14 ] and El is the (non-
complex oriented) integral elliptic theory of Kreck and Stolz [22 ]; see [15 ].
With care, the arguments above may be extended to prove also the inclusion
ChK(n)(X) ChP(n)(X)
using Yagita's In version of the Landweber exact functor theorem [35 ].
Lemma 5.5 Suppose E and F are spectra equipped with a map of spectra
OE :E ! F inducing an inclusion in mod p homology of the corresponding
-spectra, H*(E_*) ,! H*(F_*). Then ChE (X) ChF (X) for any space X.
17
Proof If OE induces an injection in homology, it induces a surjection H*(F_*) i
H*(E_*) in cohomology. Thus, given a generator f*(x) of ChE (X) with f :X !
E_2rand x 2 H2t(E_2r), the element x may be lifted as x = OE*(y) for some
y 2 H2t(F_2r) and f*(x) can be represented as a composite
X -f! E_2r -ffi!F_2r-! H_2t,
an element of ChF (X).
This result will give us a converse inclusion to that offered by the previous
results. Of course there is no reason to suppose that even if F is exact over E
there should be an inclusion of the homology of theory -spectra. For example,
K is an exact BP -module spectrum but the map H*(BP__*) ! H*(K_*) is not
an inclusion. Thus we cannot conclude from (5.4) the equality of ChBP (-)
and ChK (-), but then, as noted above, they are not equivalent.
We can however apply these results to the case of E = E(n) and F = [E(n):
we know [4, 5] that [E(n) is an exact spectrum over E(n) and it is shown in
[18 ] that the map H*(E(n)_*) ! H*(E[(n)_*) induced by the completion is an
inclusion. We thus obtain by (5.2) and (5.5)
Corollary 5.6 For X with the homotopy type of a finite type CW complex,
there is an equivalence
ChE(n)(X) = Ch[E(n)(X) .
Recall the categories A(n) = A(n)(G) of elementary abelian subgroups of a
group G defined in [11 ]: the objects of A(n)(G) are the elementary abelian
subgroups of G, denoted W , V , etc., and the morphisms from W to V are
those injective group homomorphisms f :W ! V with the property that if
U is any subgroup of W of rank at most n, the restriction of f to U is given
by conjugation by some element of G. Corollary 5.6, together with the main
result of [12 ], now gives
18
Theorem 5.7 For a finite group G there is a homeomorphism of varieties
var(ChE(n)(G)) ~= colim var(H*(V )) .
V 2A(n)(G)
We can now also explain the relationship between ChE(1)(X) and ChK (X).
For X the classifying space of a finite group it was shown in [12 ] that Ch[E(1*
*)(X)
and ChK (X) are F -isomorphic, the latter being the classical Chern subring.
The former, by (5.6), is equivalent to ChE(1)(X).
Proposition 5.8 For any space X the subalgebras ChE(1)(X) and ChK (X)
of H*(X) are equal.
Proof Write KZ(p)for p-local complex K-theory; the localisation map Z !
Z(p)induces a map of ring spectra K -!l KZ(p)and in fact KZ(p)is exact over
K via this map. Moreover, a routine calculation shows that the induced map
H*(K_*) -l*!H*(KZ(p)_*) is an inclusion (for example, the calculations of the
form of [19 , x3] show that in positive dimensions this is an isomorphism). Thus
it follows from Corollary 5.3 and Lemma 5.5 that ChK (X) = ChKZ(p)(X) for
all X. However, KZ(p) splits into a sum of suspensions of copies of E(1)
and in fact is exact over E(1); moreover, the splitting gives the necessary
inclusion of homology to allow us to use (5.3) and (5.5) again to prove that
ChE(1)(X) = ChKZ(p)(X).
6 Unstable chromatic filtrations
In [12 ] it was shown for a finite group G that the set of subalgebras Ch[E(n)(*
*G)
(which by Corollary 5.6 we can now identify with ChE(n)(G)) formed an "F -
filtrationö f H*(G), in the sense that their associated varieties formed a se-
quence of quotient spaces
var(H*(G)) i . .i.var(Ch E"(n+1)(G)) i var(Ch [E(n)(G)) i . .i.k .
19
The exact relation between the subalgebras ChE(n)(G) and ChE(n+1)(G) is
however uncertain: there is no reason to suppose that there is an actual in-
clusion ChE(n)(G) ChE(n+1)(G).
In this section we examine the subalgebras ChE(n)(X) for a space X with
the homotopy type of a finite CW complex. Here we have a stronger result
which shows the subalgebras ChE(n)(X) form an actual filtration of the algebra
Heven(X); by the work of Section 2 this is as subalgebras over the Steenrod
algebra and compatible with the (unstable) chromatic filtration of the space X
(cf. [27 , Def. 7.5.3] and see Remark 6.2 below). Our main result is as follows.
Theorem 6.1 Let X be a finite CW complex. Then for every n = 1, 2, . . .
there is an inclusion ChE(n)(X) ChE(n+1)(X) as subalgebras of Heven(X).
Proof We use the associated spectra BP and Wilson's Splitting Theorem
of [32 ]. Specifically, the spectrum BP has homotopy Z(p)[v1, . .,.vn] with
vi of dimension |vi| = 2(pi- 1), and for r < (pn + . .+.p + 1)
BP__2r' BP__2rx BP__2r+|vn+1|
as H-spaces [32 , Corollary 5.5]. Recall also that we can define the spectrum
E(n) as v-1nBP and the space E(n)_2rcan be constructed as the homotopy
colimit of
BP__2r-vn!BP__2r-|vn|vn-!BP__2r-2|vn|-! . . .
where the maps, as indicated, represent in homotopy multiplication by vn.
Now consider a generator of ChE(n)(X) which we take to be represented by
f*(x) for a map f :X ! E(n)_2rand an element x 2 H*(E(n)_2r). It suffices to
show that f*(x) 2 ChE(n+1)(X). Viewing E(n)_2r as the homotopy colimit as
mentioned, compactness of X means that the map f factors through some in-
termediate space BP__2r-k|vn|for k sufficiently large. Perhaps taking k to be
even larger, view this space as a factor in BP__2r-k|vn|= BP__2r-k|vn*
*|x
BP__2r-k|vn|+|vn+1|by the Wilson splitting theorem. Then we have a fac-
torisation of f as
X -f1!BP__2r-k|vn|-f2!E(n)_2r
20
where f2 is projection on the first factor followed by localisation with respect
to powers of vn. Note that f*2(x) will be an element of the form y 1 in
H*(BP__2r-k|vn|) = H*(BP__2r-k|vn|) H*(BP__2r-k|vn|+|vn+1|) .
Now consider the localisation of BP__2r-k|vn|with respect to powers
of vn+1 . For s sufficiently small, the map
BP__s = BP__s x BP__s+|vn+1|
??
?yvn+1
BP__s-|vn+1| = BP__s-|vn+1|x BP__sx BP__s+|vn+1|
is the inclusion as the last two factors. In particular (again assuming s suffi-
ciently small), the vn+1 -localisation map BP__s- ! E(n_+_1)_s gives an
epimorphism in cohomology. We thus have a diagram
X -f1!BP__2r-k|vn|-f2!E(n)_2r,
??
?yg
E(n_+_1)_2r-k|vn|
where g is the vn+1 localisation map, and an element z 2 H*(E(n_+_1)_2r-k|vn|)
with g*(z) = f*2(x). Then f*(x) = (gf1)*(z) 2 ChE(n+1)(X), showing ChE(n)(X)
to be contained in ChE(n+1)(X) as claimed.
Remark 6.2 Recall the ä lgebraic" chromatic filtration of a finite spectrum
X (for example, [27 , Def. 7.5.3]). This is given by a tower of p-local spectra
L0X - L1X - L2X - . . .- X
and is defined as the corresponding filtration of ß*(X) by the kernels of the h*
*o-
momorphisms (jX )*: ß*(X) ! ß*(LnX). Here Ln is localisation with respect
to the spectra E(n). The corresponding tower in the category of spaces gives
the filtration of H*(X) by images of the j*X. Proposition 2.10 shows the set
21
of subalgebras ChE(n)(X) to be compatible with this filtration, but what
is not immediate for general spaces X is that the subalgebras ChE(n)(X) are
nested. Theorem 6.1 shows that, under suitable finiteness conditions, there is
a filtration
ChE(1)(X) . . .ChE(n)(X) ChE(n+1)(X) . . .H*(X)
with a levelwise inclusion in the unstable algebraic chromatic filtration.
Remark 6.3 In the light of the results of this section, the equivalences pro*
*ved
in (5.6) and (5.8), the inclusion of (3.4) and the non-equivalences in general *
*of
ChE(n)(-) and ChE(n+1)(-) noted in [12 ], we are tempted to wonder whether,
for suitable E and X, the subalgebra ChE (X) in fact only depends on the
Bousfield equivalence class of E. In the first instance, `suitable' might inclu*
*de
complex oriented E and spaces X which are either finite CW complexes or
else are models of BG for a finite group G
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The Department of Mathematics and Computer Science, University of Leicester, Un*
*i-
versity Road, Leicester, LE1 7RH, England.
Email: J.Hunton@mcs.le.ac.uk
Department of Mathematics, University of Wuppertal, Gaußstraße 20, D-42097 Wup-
pertal, Germany.
Email: schuster@math.uni-wuppertal.de
25