2017
Том 69
№ 9

All Issues

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

Maksimenko S. I.

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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.

English version (Springer): Ukrainian Mathematical Journal 64 (2012), no. 9, pp 1350-1369.

Citation Example: Maksimenko S. I. Homotopic types of right stabilizers and orbits of smooth functions on surfaces // Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1165-1204.

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