Characteristic rank of vector bundles over Stiefel manifolds

TL;DR

This paper computes the characteristic rank of vector bundles over Stiefel manifolds for real, complex, and quaternionic cases, enhancing understanding of their cohomological properties and characteristic classes.

Contribution

It provides explicit calculations of the characteristic rank for vector bundles over Stiefel manifolds across different fields, a novel contribution to topological and geometric analysis.

Findings

Characteristic rank computed for real, complex, and quaternionic Stiefel manifolds

Results clarify the relationship between cohomology classes and Stiefel-Whitney classes

Advances understanding of vector bundle properties over classical manifolds

Abstract

The characteristic rank of a vector bundle $\xi$ over a finite connected $CW$-complex $X$ is by definition the largest integer $k$, $0\leq k\leq \mathrm{dim}(X)$, such that every cohomology class $x\in H^j(X;\mathbb Z_2)$, $0\leq j\leq k$, is a polynomial in the Stiefel-Whitney classes $w_i(\xi)$. In this note we compute the characteristic rank of vector bundles over the Stiefel manifold $V_k(\mathbb F^n)$, $\mathbb F=\mathbb R,\mathbb C,\mathbb H$.