p-Primary homotopy decompositions of looped stiefel manifolds and their exponents

Abstract

Let p be an odd prime, and fix integers m and n such that 0 < m < n ≤ (p-1)(p-2). We give a p-local homotopy decomposition for the loop space of the complex Stiefel manifold Wn,m. Similar decompositions are given for the loop space of the real and symplectic Stiefel manifolds. As an application of these decompositions, we compute upper bounds for the p-exponent of Wn;m. Upper bounds for p-exponents in the stable range 2m < n and 0 < m ≤ (p-1)(p-2) are computed as well.