Knaster's problem for $(Z_2)^k$-symmetric subsets of the sphere $S^{2^k-1}$

TL;DR

This paper extends Knaster's problem to $(Z_2)^k$-symmetric subsets of high-dimensional spheres, calculating topological obstructions and deriving geometric and measure partition results.

Contribution

It introduces a new topological approach to symmetric group actions on spheres, leading to novel geometric and measure partition theorems.

Findings

Calculated Euler class obstruction for $(Z_2)^k$-orbits

Proved existence of skew crosspolytopes inscribed in hypersurfaces

Established equipartition results for measures using symmetric convex fans

Abstract

We prove a Knaster-type result for orbits of the group $(Z_2)^k$ in $S^{2^k-1}$, calculating the Euler class obstruction. Among the consequences are: a result about inscribing skew crosspolytopes in hypersurfaces in $\mathbb R^{2^k}$, and a result about equipartition of a measures in $\mathbb R^{2^k}$ by $(Z_2)^{k+1}$-symmetric convex fans.