ON THE COHOMOLOGY OF GENERALIZED HOMOGENEOUS
SPACES
J.P. MAY AND F. NEUMANN
Abstract.We observe that work of Gugenheim and May on the cohomology
of classical homogeneous spaces G=H of Lie groups applies verbatim to the
calculation of the cohomology of generalized homogeneous spaces G=H, whe*
*re
G is a finite loop space or a p-compact group and H is a "subgroup" in t*
*he
homotopical sense.
We are interested in the cohomology H*(G=H; R) of a generalized homogeneous
space G=H with coefficients in a commutative Noetherian ring R. Here G is a
"finite loop space" and H is a "subgroup". More precisely, G and H are homotopy
equivalent to BG and BH for path connected spaces BG and BH, and G=H
is the homotopy fiber of a based map f : BH -! BG. We always assume this
much, and we add further hypotheses as needed. Such a framework of generalized
homogeneous spaces was first introduced by Rector [10], and a more recent frame-
work of p-compact groups has been introduced and studied extensively by Dwyer
and Wilkerson [4] and others.
We ask the following question: How similar is the calculation of H*(G=H; R) to
the calculation of the cohomology of classical homogeneous spaces of compact Lie
groups? When R = Fp and H is of maximal rank in G, in the sense that H*(H; Q)
and H*(G; Q) are exterior algebras on the same number of generators, the second
author has studied the question in [8, 9]. There, the fact that H*(BG; R) need *
*not
be a polynomial algebra is confronted and results similar to the classical theo*
*rems
of Borel and Bott [2, 3] are nevertheless proven. The purpose of this note is *
*to
begin to answer the general question without the maximal rank hypothesis, but
under the hypothesis that H*(BG; R) and H*(BH; R) are polynomial algebras.
In fact, we shall not do any new mathematics. Rather, we shall merely point
out that work of the first author [7] that was done before the general context
was introduced goes far towards answering the question. Essentially the followi*
*ng
theorem was announced in [7] and proven in [5]. We give a brief sketch of its p*
*roof
and then return to a discussion of its applicability to the question on hand. L*
*et
BT nbe a classifying space of an n-torus T n.
Theorem 1. Assume the following hypotheses.
(i)ss1(BG) acts trivially on H*(G=H; R).
(ii)R is a PID and H*(BG; R) is of finite type over R.
(iii)H*(BG; R) is a polynomial algebra.
(iv)There is a map e : BT n-! BH such that H*(BT n; R) is a free H*(BH; R)-
module via e*.
____________
Date: April 20, 2000.
1991 Mathematics Subject Classification. Primary 55P35, 57T35; Secondary 57T*
*15.
The first author was partially supported by the NSF.
1
2 J.P. MAY AND F. NEUMANN
Then H*(G=H; R) is isomorphic as an R-module to TorH*(BG;R)(R; H*(BH; R)),
regraded by total degree. Moreover, there is a filtration on H*(G=H; R) such th*
*at
its associated bigraded R-algebra is isomorphic to TorH*(BG;R)(R; H*(BH; R)).
Proof.The first two hypotheses ensure that H*(G=H; R) is isomorphic to the diff*
*er-
ential torsion product TorC*(BG;R)(R; C*(BH; R)). See, for example, [5, p. 21-2*
*5].
The second hypothesis allows Lemma 3.2 there to be applied with Z replaced by R,
thus allowing the finite type over Z hypothesis assumed there to be replaced by*
* the
finite type over R hypothesis assumed here. Therefore there is an Eilenberg-Moo*
*re
spectral sequence that converges from TorH*(BG;R)(R; H*(BH; R)) to H*(G=H; R).
The conclusion of the theorem is that this spectral sequence collapses at E2 wi*
*th
trivial additive extensions, but not necessarily trivial multiplicative extensi*
*ons. The
last hypothesis and a comparison of spectral sequences argument essentially due*
* to
Baum [1] shows that the conclusion holds in general if it holds when BH = BT n.
See [5, p. 37-38]. Here the strange result [5, 4.1] gives that there is a morph*
*ism
g : C*(BT n; R) -! H*(BT n; R)
of differential algebras such that g induces the identity map on cohomology and
annihilates all [1-products.
Now the general theory of differential torsion products of [5] kicks in. In m*
*odern
language, implicit in the discussion of [6, p. 70], there is a model category *
*struc-
ture on the category of A-modules for any DGA A over R such that every right
A-module M admits a cofibrant approximation of a very precise sort. Namely, for
any HA-free resolution X R HA -! HM of HM, there is a cofibrant approxi-
mation P = X R A -! M. Grading is made precise in the cited sources. The
essential point is that P is not a bicomplex but rather has differential with m*
*any
components. When HA is a polynomial algebra and M = R, we can take X to be
an exterior algebra with one generator for each polynomial generator of HA. Her*
*e,
asssuming that A has a [1-product that satisfies the Hirsch formula ([1 is a gr*
*aded
derivation), [5, 2.2] specifies the required differential explicitly in terms o*
*f [1-
products. Using g to replace C*(BT n; R) by H*(BT n; R) in our differential tor*
*sion
product, we see that the differential torsion product TorC*(BG;R)(R; H*(BT n; R*
*))
is computed by exactly the same chain complex as the ordinary torsion product *
*__
TorH*(BG;R)(R; H*(BT n; R)). See [5, 2.3]. The conclusion follows. |*
*__|
Hypotheses (i) and (ii) in the theorem are reasonable and not very restrictiv*
*e.
Hypothesis (iii) is intrinsic to the method at hand. Note that H*(BG; R) can
have infinitely many polynomial generators, so that G need not be finite. The k*
*ey
hypothesis is (iv). Here the following homotopical version of a theorem of Bore*
*l is
relevant. It was first noticed by Rector [10, 2.2] that Baum's proof [1] of Bor*
*el's
theorem is purely homotopical. A generalized variant of Baum's proof is given i*
*n [5,
p. 40-42]. That proof applies directly to give the following theorem. We state *
*it for
H and G as in the first paragraph. However, we are interested in its applicabil*
*ity
to T nand H in Theorem 1, and we restate it as a corollary in that special case.
Theorem 2. Let R be a field and assume the following hypotheses.
(i)ss1(BG) acts trivially on H*(G=H; R).
(ii)H*(BH; R) and H*(BG; R) are polynomial algebras on the same finite num-
ber of generators.
(iii)H*(G=H; R) is a finite dimensional R-module.
ON THE COHOMOLOGY OF GENERALIZED HOMOGENEOUS SPACES 3
Then H*(G=H; R) ~=R H*(BG;R)H*(BH; R) as an algebra and
H*(BH; R) ~=H*(BG; R) R H*(G=H; R)
as a left H*(BG; R)-module. In particular, H*(BH; R) is H*(BG; R)-free.
Corollary 3.Let R be a field and assume given a map e : BT n- ! BH that
satisfies the following properties, where H=T nis the fiber of e.
(i)ss1(BH) acts trivially on H*(H=T n; R).
(ii)H*(BH; R) is a polynomial algebra on n generators.
(iii)H*(H=T n; R) is a finite dimensional R-module.
Then H*(H=T n; R) ~=R H*(BH;R)H*(BT n; R) as an algebra and
H*(BT n; R) ~=H*(BH; R) R H*(H=T n; R)
as a left H*(BH; R)-module. In particular, H*(BT n; R) is H*(BH; R)-free.
When Corollary 3 applies, its conclusion gives hypothesis (iv) of Theorem 1.
We comment briefly on applications to the integral and p-compact settings for t*
*he
study of generalized homogeneous spaces.
Remark 4. A counterexample of Rector [10] shows that not all finite loop spaces*
* H
have (integral) maximal tori. When H does have a maximal torus, hypothesis (iii)
of the Corollary holds by definition. Assuming that H is simply connected, [9, *
*3.11]
describes for which primes p H*(BH; Z) is p-torsion free, so that H*(BH; Fp) is*
* a
polynomial algebra. If R is the localization of Z at the primes p for which H*(*
*H; Z)
is p-torsion free, then H*(BH; R) is also a polynomial algebra, and H*(BT ; R) *
*is
a free H*(BH; R)-module. That is, hypothesis (iv) of Theorem 1 holds for the
localization of Z away from the finitely many "bad primes" for which H*(BH; Fp)
is not a polynomial algebra on n generators.
Remark 5. In the p-compact setting, taking R = Fp, Dwyer and Wilkerson [4,
8.13, 9.7] prove that if H is connected, BH is Fp-complete, H*(H; Fp) is finite
dimensional, and H*(H; Zp) Zp Q is an exterior algebra on n generators, then
there is a map e : BT n-! BH such that H*(H=T n; Fp) is finite dimensional. Here
Corollary 3 applies whenever H*(BH; Fp) is a polynomial algebra on n generators.
References
[1]P.F. Baum. On the cohomology of homogeneous spaces. Topology 7(1968), 15-38.
[2]A. Borel. Sur la cohomologie desespaces fibre principaux et desespaces homo*
*genes de groupes
de Lie compact. Ann Math. 57(1953), 115-207.
[3]R. Bott. An application of Morse theory to the topology of Lie groups. Bull*
*. Soc. Math.
France 84(1956), 251-281.
[4]W.G. Dwyer and C.W. Wilkerson. Homotopy fixed point methods for Lie groups *
*and finite
loop spaces. Ann. Math 139(1994), 395-442.
[5]V.K.A.M. Gugenheim and J.P. May. On the theory and applications of differen*
*tial torsion
products. Memoirs Amer. Math. Soc. No. 142, 1974.
[6]I. Kriz and J.P. May. Operads, algebras, modules, and motives. Asterisque. *
*No. 233. 1995.
[7]J.P. May The cohomology of principal bundles, homogeneous spaces, and two-s*
*tage Postnikov
systems. Bull. Amer. Math. Soc. 74(1968), 334-339.
[8]F. Neumann. On the cohomology of homogeneous spaces of finite loop spaces a*
*nd the
Eilenberg-Moore spectral sequence. J. Pure and Applied Algebra 140(1999), 26*
*1-287.
[9]F. Neumann. Torsion in the cohomology of finite loop spaces and the Eilenbe*
*rg-Moore spectral
sequence. Topology and its Applications 100(2000), 133-150.
[10]D. Rector. Subgroups of finite dimensioanl topological groups. J. Pure Appl*
*. Algebra 1(1971),
253-273.
4 J.P. MAY AND F. NEUMANN
Department of Mathematics, The University of Chicago, Chicago, IL 60637
E-mail address: may@math.uchicago.edu
Mathematisches Institut der Georg-August-Universit"at, G"ottingen, Germany
E-mail address: neumann@uni-math.gwdg.de