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ON THE COHOMOLOGY OF GENERALIZED HOMOGENEOUS SPACES

By J. P. May, F. Neumann and Communicated Ralph Cohen

Abstract

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 finite 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 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 ΩBG and ΩBH for path connected spaces BG and BH, andG/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 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) andH ∗ (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 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

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Year: 2001
OAI identifier: oai:CiteSeerX.psu:10.1.1.352.6142
Provided by: CiteSeerX
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