Homotopic types of right stabilizers and orbits of smooth functions on surfaces - Maksimenko, S. I. (Ukrains’kyi Matematychnyi Zhurnal, 2012)
Let M be a smooth connected compact surface, P be either the real line R or a circle S^1. For a subset X ⊂ M, let D(M, X) denote the group of diffeomorphisms of M fixed on X. In this note, we consider a special class F of smooth maps f : M → P with isolated singularities that contains all Morse maps. For each map f ∈ F, we consider certain submanifolds X ⊂ M that are “adopted” with f in a natural sense, and study the right action of the group D(M, X) on C^∞(M, P). The main result describes the homotopy types of the connected components of the stabilizers S(f) and orbits O(f) for all maps f ∈ F. It extends previous results of the author on this topic.