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, where G is a nite loop space or a p-compact group and H is a subgroup in the homotopical sense.

We are interested in the cohomology H (G/H; R) of a generalized homogeneous space G/H with coecients in a commutative Noetherian ring R. Here G is a nite 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 ber 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 rst introduced by Rector [10], and a more recent framework 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 theorems 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 rst author [7] that was done before the general context was introduced goes far towards answering the question. Essentially the following theorem was announced in [7] and proven in [5]. We give a brief sketch of its proof and then return to a discussion of its applicability to the question on hand. Let BT n be a classifying space of an n-torus T n . Theorem 1. Assume the following hypotheses. (i) 1 (BG) acts trivially on H (G/H; R). (ii) R is a PID and H (BG; R) is of nite 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 Classication. Primary 55P35, 57T35; Secondary 57T15. The rst author was partially supported by the NSF.
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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 ltration on H (G/H; R) such that its associated bigraded R-algebra is isomorphic to TorH (BG;R) (R, H (BH; R)). Proof. The rst two hypotheses ensure that H (G/H; R) is isomorphic to the dierential torsion product TorC (BG;R) (R, C (BH; R)). See, for example, [5, p. 21-25]. The second hypothesis allows Lemma 3.2 there to be applied with Z replaced by R, thus allowing the nite type over Z hypothesis assumed there to be replaced by the nite type over R hypothesis assumed here. Therefore there is an Eilenberg-Moore 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 with trivial additive extensions, but not necessarily trivial multiplicative extensions. 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 morphism g : C (BT n ; R) H (BT n ; R) of dierential algebras such that g induces the identity map on cohomology and annihilates all 1 -products. Now the general theory of dierential torsion products of [5] kicks in. In modern language, implicit in the discussion of [6, p. 70], there is a model category structure on the category of A-modules for any DGA A over R such that every right A-module M admits a cobrant approximation of a very precise sort. Namely, for any HA-free resolution X R HA HM of HM , there is a cobrant approximation 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 dierential with many 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. Here, asssuming that A has a 1 -product that satises the Hirsch formula (1 is a graded derivation), [5, 2.2] species the required dierential explicitly in terms of 1 products. Using g to replace C (BT n ; R) by H (BT n ; R) in our dierential torsion product, we see that the dierential 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 restrictive. Hypothesis (iii) is intrinsic to the method at hand. Note that H (BG; R) can have innitely many polynomial generators, so that G need not be nite. The key hypothesis is (iv). Here the following homotopical version of a theorem of Borel is relevant. It was rst noticed by Rector [10, 2.2] that Baums proof [1] of Borels theorem is purely homotopical. A generalized variant of Baums proof is given in [5, p. 40-42]. That proof applies directly to give the following theorem. We state it for H and G as in the rst paragraph. However, we are interested in its applicability to T n and H in Theorem 1, and we restate it as a corollary in that special case. Theorem 2. Let R be a eld and assume the following hypotheses. (i) 1 (BG) acts trivially on H (G/H; R). (ii) H (BH; R) and H (BG; R) are polynomial algebras on the same nite number of generators. (iii) H (G/H; R) is a nite dimensional R-module.

ON THE COHOMOLOGY OF GENERALIZED HOMOGENEOUS SPACES

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 eld and assume given a map e : BT n BH that satises the following properties, where H/T n is the ber of e. (i) 1 (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 nite 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 briey on applications to the integral and p-compact settings for the study of generalized homogeneous spaces. Remark 4. A counterexample of Rector [10] shows that not all nite loop spaces H have (integral) maximal tori. When H does have a maximal torus, hypothesis (iii) of the Corollary holds by denition. 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 nitely 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 nite 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 nite 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), 1538. [2] A. Borel. Sur la cohomologie des spaces br principaux et des spaces homog`nes de groupes e e e e de Lie compact. Ann Math. 57(1953), 115207. [3] R. Bott. An application of Morse theory to the topology of Lie groups. Bull. Soc. Math. France 84(1956), 251281. [4] W.G. Dwyer and C.W. Wilkerson. Homotopy xed point methods for Lie groups and nite loop spaces. Ann. Math 139(1994), 395442. [5] V.K.A.M. Gugenheim and J.P. May. On the theory and applications of dierential torsion products. Memoirs Amer. Math. Soc. No. 142, 1974. [6] I. Kriz and J.P. May. Operads, algebras, modules, and motives. Astrisque. No. 233. 1995. e [7] J.P. May The cohomology of principal bundles, homogeneous spaces, and two-stage Postnikov systems. Bull. Amer. Math. Soc. 74(1968), 334339. [8] F. Neumann. On the cohomology of homogeneous spaces of nite loop spaces and the Eilenberg-Moore spectral sequence. J. Pure and Applied Algebra 140(1999), 261-287. [9] F. Neumann. Torsion in the cohomology of nite loop spaces and the Eilenberg-Moore spectral sequence. Topology and its Applications 100(2000), 133150. [10] D. Rector. Subgroups of nite dimensioanl topological groups. J. Pure Appl. Algebra 1(1971), 253273.

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-Universitat, Gottingen, Germany E-mail address: neumann@uni-math.gwdg.de