On algebraic problems behind the Brouwer degree of equivariant maps

Abstract

Given a finite group G and two unitary G-representations V and W, possible restrictions on topological degrees of equivariant maps between representation spheres S(V) and S(W) are usually expressed in a form of congruences modulo the greatest common divisor of lengths of orbits in S(V) (denoted by alpha(V)). Effective applications of these congruences is limited by answers to the following questions: (i) under which conditions, is alpha(V) > 1? and (ii) does there exist an equivariant map with the degree easy to calculate? In the present paper, we address both questions. We show that alpha(V) > 1 for each irreducible non-trivial C[G]-module if and only if G is solvable. This provides a new solvability criterion for finite groups. For non-solvable groups, we use 2-transitive actions to construct complex representations with non-trivial a-characteristic. Regarding the second question, we suggest a class of Norton algebras without 2-nilpotents giving rise to equivariant quadratic maps, which admit an explicit formula for the degree. (C) 2019 Elsevier Inc. All rights reserved.