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-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 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. The methods also provide an easy answer to a question raised in DM.