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# Homology, Homotopy and Applications

## Volume 18 (2016)

### Number 2

### Mod-$p$ homotopy decompositions of looped Stiefel manifolds

Pages: 59 – 69

DOI: http://dx.doi.org/10.4310/HHA.2016.v18.n2.a3

#### Authors

#### Abstract

Let $W_{n,k}$ be the Stiefel manifold $U(n) / U(n - k)$. For odd primes $p$ and for $k \leqslant (p - 1)(p - 2)$, we give a homotopy decomposition of the based loop space $\Omega W_{n,k}$ as a product of $p - 1$ factors, each of which is the based loops on a finite $H$-space. Similar decompositions are obtained for $Sp(n) / Sp(n - k)$ and $O(n) / O(n - k)$ and upper bounds on the homotopy exponents are obtained.

#### Keywords

Stiefel manifold, loop space, decomposition, exponent

#### 2010 Mathematics Subject Classification

Primary 55P35. Secondary 55Q52.