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On the zero set of G-equivariant maps

Published online by Cambridge University Press:  15 July 2009

P-L. BUONO
Affiliation:
Faculty of Science, University of Ontario Institute of Technology, Oshawa, ONT L1H 7K4, Canada. e-mail: Pietro-Luciano.Buono@uoit.ca
M. HELMER
Affiliation:
Faculty of Science, University of Ontario Institute of Technology, Oshawa, ONT L1H 7K4, Canada. e-mail: Pietro-Luciano.Buono@uoit.ca
J. S. W. LAMB
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK

Abstract

Let G be a finite group acting on vector spaces V and W and consider a smooth G-equivariant mapping f: VW. This paper addresses the question of the zero set of f near a zero x with isotropy subgroup G. It is known from results of Bierstone and Field on G-transversality theory that the zero set in a neighbourhood 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(Σ) for isotropy subgroups Σ of G which is the difference of the dimension of the fixed point subspace of Σ in V and W. Our main result states that if V contains a subspace G-isomorphic to W, then for every maximal isotropy subgroup Σ satisfying s(Σ) > s(G), the zero set of f near x contains a smooth manifold of zeros with isotropy subgroup Σ of dimension s(Σ). We also present partial results in the case of 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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2009

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