Borsuk--Ulam theorems for elementary abelian 2-groups

TL;DR

This paper refines existing Borsuk--Ulam type theorems for elementary abelian 2-groups and tori, providing sharper estimates for the zero-sets of G-maps under specific group actions.

Contribution

It offers a refined dimension estimate for zero-sets of G-maps using classical Borsuk--Ulam theorems, focusing on free actions and finite isotropy groups.

Findings

Improved bounds on the dimension of zero-sets for elementary abelian 2-group actions.

Enhanced understanding of G-map zero-sets with free or finite isotropy group actions.

Application of classical Borsuk--Ulam theorem to refine existing topological estimates.

Abstract

Let $G$ be a compact Lie group and let $U$ and $V$ be finite-dimensional real $G$-modules with $V^G=0$. A theorem of Marzantowicz, de Mattos and dos Santos estimates the covering dimension of the zero-set of a $G$-map from the unit sphere in $U$ to $V$ when $G$ is an elementary elementary abelian $p$-group for some prime $p$ or a torus. In this note, the classical Borsuk--Ulam theorem will be used to give a refinement of their result estimating the dimension of that part of the zero-set on which an elementary abelian $p$-group $G$ acts freely or a torus $G$ acts with finite isotropy groups.