Actions of finite groups and smooth functions on surfaces

Abstract

Abstract. Let f : M → be a Morse function on a smooth closed surface, V be a connected component of some critical level of f, and EV be its atom. Let also S(f) be a stabilizer of the function f under the right action of the group of diffeomorphisms Diff(M) on the space of smooth functions on M, and SV (f) = {h ∈ S(f) |h(V ) = V }. The group SV (f) acts on the set π0∂EV of connected components of the boundary of EV . Therefore we have a homomorphism φ : S(f) → Aut(π0∂EV ). Let also G = φ(S(f)) be the image of S(f) in Aut(π0∂EV ). Suppose that the inclusion ∂EV ⊂ M \V induces a bijection π0∂EV → π0(M \V ). Let H be a subgroup of G. We present a sufficient condition for existence of a section s : H → SV (f) of the homomorphism φ, so, the action of H on ∂EV lifts to the H-action on M by f-preserving diffeomorphisms of M. This result holds for a larger class of smooth functions f : M → having the following property: for each critical point z of f the germ of f at z is smoothly equivalent to a homogeneous polynomial 2 → without multiple linear fact.