Automorphisms of cellular divisions of $2$-sphere induced by functions with isolated critical points

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 $D_1,\dots ,D_k$. 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 $G=\{{\sigma }_h\mid h\in {\mathcal{S}}_K(f)\}$ be the group of all such permutations. We prove that $G$ is isomorphic to a finite subgroup of $SO(3)$.

18 pages, 6 figures

References in corpus (1)

• Smooth functions on 2-torus whose Kronrod-Reeb graph contains a cycle