permutation action on cohomology of Stiefel manifolds

Question

Let $V_k(\mathbb{R}^n)$ be Stiefel manifolds.

In the paper The cohomology rings of real Stiefel manifolds with integer coefficients, Martin Čadek, Mamoru Mimura, and Jiří Vanžura, J. Math. Kyoto Univ. Volume 43, Number 2 (2003), 411-428., the cohomology rings $$ H^*(V_k(\mathbb{R}^n);\mathbb{Z}_2), $$ $$ H^*(V_k(\mathbb{R}^n);\mathbb{Z}), $$ are obtained.

Let $\Sigma_k$ be permutation grup of order $k$. For any $\sigma\in \Sigma_k$, let $\sigma$ act on $V_k(\mathbb{R}^n)$ by $$\sigma(v_1,\cdots,v_k)=(v_{\sigma(1)},\cdots,v_{\sigma(k)}).$$

I want to know the induced homomorphism on cohomology ring $$ \sigma^*: H^*(V_k(\mathbb{R}^n);\mathbb{Z}_2)\to H^*(V_k(\mathbb{R}^n);\mathbb{Z}_2), $$ $$ \sigma^*: H^*(V_k(\mathbb{R}^n);\mathbb{Z} )\to H^*(V_k(\mathbb{R}^n);\mathbb{Z} ). $$ Is it possible or difficult? How to solve it?

Could you illustrate the example how to obtain the action of $\mathbb{Z}_2$ on $H^*(V_2(\mathbb{R}^n);\mathbb{Z})$?

Answer 1

The space $V_k(\mathbb{R}^n)$ can be identified with the space $L(\mathbb{R}^k,\mathbb{R}^n)$ of linear isometric inclusions $\mathbb{R}^k\to\mathbb{R}^n$. From this point of view, it is clear that the action of $\Sigma_k$ extends to an action of the orthogonal group $O(k)$. Now $SO(k)$ is connected so it acts by maps that are homotopic to the identity, and so the action in cohomology is trivial. Thus, you only need to calculate the effect of a single transposition to get the full action of $\Sigma_k$. I don't remember how that works out, but it can't be hard.