Journal of Mathematical Physics, Analysis, Geometry 2020, vol. 16, No 2, pp. 138-160   https://doi.org/10.15407/mag16.02.138     ( to contents , go back )

### Automorphisms of Cellular Divisions of 2-Sphere Induced by Functions with Isolated Critical Points

Anna Kravchenko

Taras Shevchenko National University of Kyiv, Ukraine
E-mail: annakravchenko1606@gmail.com

Sergiy Maksymenko

Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine
E-mail: maks@imath.kiev.ua

Received November 18, 2019, revised January 29, 2020.

Abstract

Let $f:S^2\to \mathbb{R}$ be a Morse function on the $2$-sphere and $K$ be a connected component of some level set of $f$ containing at least one saddle critical point. Then $K$ is a $1$-dimensional CW-complex cellularly embedded into $S^2$, so the complement $S^2\setminus K$ is a union of open $2$-disks $D_1,\ldots, D_k$. Let $\mathcal{S}_{K}(f)$ be the group of isotopic to the identity diffeomorphisms of $S^2$ leaving invariant $K$ and also each level set $f^{-1}(c)$, $c\in\mathbb{R}$. Then each $h\in \mathcal{S}_{K}(f)$ induces a certain permutation $\sigma_{h}$ of those disks. Denote by $G = \{ \sigma_h \mid h \in \mathcal{S}_{K}(f)\}$ the group of all such permutations. We prove that $G$ is isomorphic to a finite subgroup of $SO(3)$.

Key words: surface, Morse function, diffeomorphisms