Borsuk--Ulam theorems for elementary abelian 2-groups

2022/01/24 by M. C. Crabb, Crabb, M. C.
Mathematics · #55M20 #55M25 #55M35 #55N91 #55R25 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2201.09564

openalex publication_date 2022/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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