On the $p$-Regularlty of Stiefel Manifolds

Abstract

A topological space which has the homotopy type of a finite CW-complex is called p-regular if its localization at a prime p is homotopy equivalent to a product of a certain number of spheres localized at p. We note that, if X is pregular, the rational cohomology of X determines the number and dimensions of spheres which appear in the decomposition into the product. J.-P. Serre gave, in one of his celebrated papers ([19]), a necessary and sufficient condition for the p-regularity of classical Lie groups. For example, U(ri) (resp. Sp(n)) is p-regular if and only if n ;g p (resp. 2n ^ p). The purpose of this paper is to generalize the result of Serre to complex and quaternionic Stiefel manifolds. The p-regularity of Stiefel manifolds was also studied by Y. Hemmi ([6]) from the H-space theoretical point of view. He showed that, for an odd prime p, there are infinitely many complex and quaternionic Stiefel manifolds which are p-regular (see [6] Theorem 3.3 for details). Our result is somewhat stronger than his in the sense that, for an odd prime p, we determine whether a complex (quaternionic) Stiefel manifold is p-regular or not except for finitely many undecided cases, and we also deal with the case p = 2. In particular, we settle the p-regularity problem on complex Stiefel manifolds for p = 2 and 3, and there are still two (resp. six) undecided cases for p = 5 (resp. p = 7) (see Example 4.10). Following James ([8]), we denote by Fnfe, Wntk and Xn^k the real, complex and quaternionic Stiefel manifolds O(ri)/0(n — k), U(n)/U(n — k) and Sp(ri)/Sp(n — k) respectively. For an odd prime p, by examining the action of the Steenrod operation P on H*(Wn+ktk; ¥p) and H*(Xn+ktk; Fp), we easily derive a necessary condition that the p-regularity of F^+fcfc(resp. Xn+k^k) implies / c r g p — 1, or k = p and p\n (resp. k g (p l)/2, or k = (p + l)/2 and p\2n + 1). The main results of this paper are as follows.