Homotopical Uniqueness of Classifying Spaces

If G is a connected compact Lie group, then for almost all prime numbers p the mod p cohomology ring of the classifying space BG is a finitely generated polynomial algebra. In 1961, N. Steenrod [24] asked in general for a determination of all spaces X such that H∗(X,Fp) is a finitely generated polynomial algebra (i.e., such that X has a polynomial cohomology ring); at that time, the only examples known were spaces of the form X = BG. There has been a lot of subsequent progress on this problem. On one hand, the topological constructions of Sullivan [25, p. 4.28], as exploited by Clark-Ewing [8] and Wilkerson [27], have led to the discovery of exotic spaces X with polynomial cohomology rings. On the other hand, the algebraic arguments of Wilkerson [26] and Adams-Wilkerson [1] have shown in some generality that if X is any space with a polynomial cohomology ring then H∗(X,Fp) must be one of the polynomial algebras listed in [8]. However, there is still a gap here between topology and algebra; in this paper we act to narrow the gap and in some cases to close it. Suppose that p is an odd prime. Let K be the category of unstable algebras [18] over the mod p Steenrod algebra Ap and Kpoly the full subcategory of K consisting of objects which as rings are finitely generated polynomial algebras. If X is a p-complete space with H(X,Fp) ∈ Kpoly we extend the ideas of [1] by associating to X a finite p-adic linear group WX generated by pseudoreflections (see 1.1); given any such finite group W such that p does not divide the order of W, we then show (1.2) that there is up to homotopy exactly one p-complete space X with H∗(X,Fp) ∈ Kpoly and WX = W. In a variety of particular situations (1.3, 1.4) this gives a bijective correspondence between finite group data and homotopy types of p-complete spaces with polynomial cohomology rings.