TY - JOUR
AU - S. I. Maksimenko
PY - 2012/09/25
Y2 - 2023/03/27
TI - Homotopic types of right stabilizers and orbits of smooth functions on surfaces
JF - Ukrains’kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 64
IS - 9
SE - Research articles
DO -
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/2651
AB - 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 smoothmaps $f : M → P$ 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.
ER -