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


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
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