Homotopy Decompositions of Looped Stiefel manifolds, and their Exponents

TL;DR

This paper provides homotopy decompositions of looped Stiefel manifolds over complex, real, and symplectic cases at odd primes, and uses these to estimate their p-exponents.

Contribution

It introduces new p-local homotopy decompositions for looped Stiefel manifolds and derives bounds for their p-exponents, extending understanding of their homotopy properties.

Findings

Decompositions for loop spaces of complex, real, and symplectic Stiefel manifolds.

Upper bounds for p-exponents of these manifolds.

Results applicable in the stable range for p-exponent estimates.

Abstract

Let $p$ be an odd prime, and fix integers $m$ and $n$ such that $0<m<n\leq (p-1)(p-2)$. 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 $0<m\leq (p-1)(p-2)$ are computed as well.