The Yang indices of Grassmannians

TL;DR

This paper introduces a combinatorial method to estimate the Yang indices of real Grassmannians and Stiefel manifolds, providing new bounds and exact values for small cases, enhancing understanding of their topological properties.

Contribution

It presents an elementary combinatorial approach to compute lower bounds for Yang indices of Grassmannians and Stiefel manifolds, improving bounds for specific cases.

Findings

Yang index of St(n,k) is at least n - k

For odd n, the bound for G(n,2) improves to n-1

Computed possible Yang indices for small n

Abstract

An elementary combinatorial technique for computing lower bounds for the Yang indices of real Stiefel manifolds and oriented real Grassmannians is described. As a demonstration, it shown that the Yang index of $St(n,k)$, and consequently $G(n,k)$, is at least $n - k$. For odd $n$, the bound for $G(n,2)$ can be improved to $n-1$. These are combined with basic properties of the Yang index and Conner-Floyd index and coindex to compute the possible Yang indices of $St(n,k)$ and $G(n,k)$ for small $n$.