Borsuk-Ulam theorems for elementary abelian 2-groups - Crabb, Michael C. (Topological Methods in Nonlinear Analysis, 2023)
Let $G$ be a fct Lie group and let $U$ and $V$ be finite-dimensional real $G$-moduleswith $V^G=0$. A theorem of Marzantowicz,de Mattos and dos Santos estimates the covering dimension of thezero-set of a $G$-map from the unit sphere in $U$ to $V$ when$G$ is an elementary abelian $p$-group for someprime $p$ or a torus.In this note, the classical Borsuk-Ulam theorem will be used to givea refinement of their result estimating the dimensionof that part of the zero-set on which an elementary abelian$p$-group $G$ acts freely or a torus $G$ acts with finite isotropygroups. The methods also provide an easy answerto a question raised in \cite{DM}.