Borsuk-Ulam type theorem for Stiefel manifolds and orthogonal mass partitions

Abstract

We prove a Borsuk--Ulam-type zero theorem for the Stiefel manifold $V_{n,k}$ equipped with a free action of the hyperoctahedral group $B_k=(\mathbb{Z}/2{)}^k⋊S_k$. The existence of a zero reduces to a nonvanishing condition for explicit polynomials in the truncated ring $R_{n,k}={\mathbb{F}}_2[a_1,\dots ,a_k]/(a_1^n,a_2^{n-1},\dots ,a_k^{n-k+1})$, converting a topological problem into finite algebra over~${\mathbb{F}}_2$. As an application we study equipartitions by mutually orthogonal hyperplanes. We prove that if $(P_{k,n}{)}^m\ne 0$ in the partition ring $R_{d,k}={\mathbb{F}}_2[a_1,\dots ,a_k]/(a_1^{d+1},a_2^d,\dots ,a_k^{d-k+2})$, then for any $m$ finite Borel measures in ${\mathbb{R}}^d$ there exist $k$ mutually orthogonal hyperplanes such that every $n$-element subfamily partitions each measure into $2^n$ equal parts. Let ${\Delta }^{\ast }(m,k,n)$ denote the smallest such dimension~$d$. We prove lower bounds on ${\Delta }^{\ast }(m,k,n)$ for all $m$, $k$, $n$ via a Sard-theoretic dimension argument, and the nonvanishing condition above provides algebraic upper bounds. We establish these bounds in several cases and derive exact values, including ${\Delta }^{\ast }(2^j-1,k,2)=2^{j-1}(k+1)-1$ for all $j\ge 1$, $k\ge 2$. In particular, the MVZ upper bound on $\Delta (m,k)$ \cite{MSZ} is achieved by mutually orthogonal hyperplanes: orthogonality comes for free.

27 pages