The concept of an equivariant map naturally arises in the study of manifolds with actions of a fixed group: equivariant maps are maps that commute with group actions on the source and target. An equivariant automorphism of the source of equivariant maps preserves their equivariance. Therefore, the group of equivariant automorphisms of a manifold acts on the space of equivariant maps of this manifold. The structure of orbits of this action is often complicated: it can include discrete (finite or countable) families of orbits as well as continuous ones. An orbit is called equivariant simple if its sufficiently small neighborhood intersects only a finite number of other orbits. In this paper we study singular multivariate holomorphic function germs that are equivariant simple with respect to a pair of actions of a finite cyclic group on the source and target. We present a necessary existence condition for such germs in terms of dimensions of certain vector spaces defined by group actions. As an application of this result, we describe scalar actions of finite cyclic groups for which there exist no equivariant simple singular function germs.
Equivariant topology, singularity theory, classification of singularities, simple singularities
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Cite this paper
Evgeny Astashov. (2018) Necessary Existence Condition for Equivariant Simple Singularities. International Journal of Mathematical and Computational Methods, 3, 37-42
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