Homotopic types of right stabilizers and orbits of smooth functions on surfaces

  • S. I. Maksimenko

Abstract

Let $\mathcal{M}$ be a smooth connected compact surface, $P$ be either the real line $\mathbb{R}$ or a circle $S^1$. For a subset $X ⊂ M$, let $\mathcal{D}(M, X)$ 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 $f ∈ \mathcal{F}$, we consider certain submanifolds $X ⊂ M$ that are “adopted” with $f$ in a natural sense, and study the right action of the group $\mathcal{D}(M, X)$ on $C^{∞}(M, P)$. The main result describes the homotopy types of the connected components of the stabilizers $S(f)$ and orbits $\mathcal{O}(f)$ for all maps $f ∈ \mathcal{F}$. It extends previous results of the author on this topic.
Published
25.09.2012
How to Cite
MaksimenkoS. I. “Homotopic Types of Right Stabilizers and Orbits of Smooth Functions on Surfaces”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, no. 9, Sept. 2012, pp. 1165-04, http://umj.imath.kiev.ua/index.php/umj/article/view/2651.
Section
Research articles