$p$-local decompositions of projective Stiefel manifolds

TL;DR

This paper investigates the $p$-local homotopy type of complex projective Stiefel manifolds, revealing conditions under which they decompose into products of spheres and projective spaces, extending Yamaguchi's $p$-regularity results.

Contribution

It demonstrates that many projective Stiefel manifolds are $p$-locally a product of a complex projective space and spheres, and shows the $S^1$-equivariant nature of Yamaguchi's $p$-regularity.

Findings

Many projective Stiefel manifolds decompose into products of spheres and projective spaces.

Yamaguchi's $p$-regularity results extend to an $S^1$-equivariant setting.

Conditions for $p$-local decompositions are explicitly characterized.

Abstract

The main objective of this paper is to analyze the $p$-local homotopy type of the complex projective Stiefel manifolds, and other analogous quotients of Stiefel manifolds. We take the cue from a result of Yamaguchi about the $p$-regularity of the complex Stiefel manifolds which lays down some hypotheses under which the Stiefel manifold is $p$-locally a product of odd dimensional spheres. We show that in many cases, the projective Stiefel manifolds are $p$-locally a product of a complex projective space and some odd dimensional spheres. As an application, we prove that in these cases, the $p$-regularity result of Yamaguchi is also $S^1$-equivariant.