Actions of finite groups and smooth functions on surfaces
Abstract
Let f:M R 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 ∂ EVπ0(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 R having the following property: for each critical point z of f the germ of f at z is smoothly equivalent to a homogeneous polynomial R2 R without multiple linear factors.
Turn this paper into a lesson
ArcXiv compiles a structured reading guide from this paper's metadata: plain-English importance, contributions, prerequisite concepts, which sections to read first, flashcards, and a quiz. Grounded in the abstract, never invented.