On the zero set of G-equivariant maps

TL;DR

This paper investigates the local structure of zero sets of smooth G-equivariant maps between vector spaces, providing conditions under which these sets contain smooth manifolds with specific isotropy properties, with applications to bifurcation theory.

Contribution

It introduces an index based on group representations to determine the structure of zero sets near symmetric points, extending understanding of G-equivariant transversality and stratification.

Findings

Zero set near a symmetric zero contains manifolds with specific isotropy when certain representation conditions hold.

The index s(Σ) predicts the dimension and existence of isotropy strata in the zero set.

Provides a systematic method for analyzing zero sets beyond the main theorem's conditions.

Abstract

Let $G$ be a finite group acting on vector spaces $V$ and $W$ and consider a smooth $G$-equivariant mapping $f:V\to W$. This paper addresses the question of the zero set near a zero $x$ of $f$ with isotropy subgroup $G$. It is known from results of Bierstone and Field on $G$-transversality theory that the zero set in a neighborhood of $x$ is a stratified set. The purpose of this paper is to partially determine the structure of the stratified set near $x$ using only information from the representations $V$ and $W$. We define an index $s(\Sigma)$ for isotropy subgroups $\Sigma$ of $G$ which is the difference of the dimension of the fixed point subspace of $\Sigma$ in $V$ and $W$. Our main result states that if $V$ contains a subspace $G$-isomorphic to $W$, then for every maximal isotropy subgroup $\Sigma$ satisfying $s(\Sigma)>s(G)$, the zero set of $f$ near $x$ contains a smooth manifold of zeros with isotropy subgroup $\Sigma$ of dimension $s(\Sigma)$. We also present a systematic method to study the zero sets for group representations $V$ and $W$ which do not satisfy the conditions of our main theorem. The paper contains many examples and raises several questions concerning the computation of zero sets of equivariant maps. These results have application to the bifurcation theory of $G$-reversible equivariant vector fields.