$p$–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 $0<m<n\le \left(p-1\right)\left(p-2\right)$. We give a $p$–local homotopy decomposition for the loop space of the complex Stiefel manifold $W_{n,m}$. 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 $W_{n,m}$. Upper bounds for $p$–exponents in the stable range $2m<n$ and $0<m\le \left(p-1\right)\left(p-2\right)$ are computed as well.