The Schwarz genus of the Stiefel manifold and counting geometric configurations

TL;DR

This paper calculates the Schwarz genus and Lusternik--Schnirelmann category of Stiefel manifolds under Weyl group actions, applying these results to estimate geometric configurations like orthogonal bases and inscribed parallelotopes.

Contribution

It provides explicit computations of the Schwarz genus and LS-category for Stiefel manifolds with Weyl group actions, linking algebraic topology to geometric configuration counting.

Findings

Computed Schwarz genus of Stiefel manifolds under Weyl group actions.

Determined LS-category of quotient spaces of Stiefel manifolds.

Applied topological invariants to estimate geometric configurations.

Abstract

In this paper we compute: the Schwarz genus of the Stiefel manifold $V_k(\mathbb R^n)$ with respect to the action of the Weyl group $W_k:=(\mathbb Z/2)^{k}\rtimes\Sigma_k$, and the Lusternik--Schnirelmann category of the quotient space $V_k(\mathbb R^n)/W_k$. Furthermore, these results are used in estimating the number of: critically outscribed parallelotopes around the strictly convex body, and Birkhoff--James orthogonal bases of the normed finite dimensional vector space.