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

TL;DR

This paper studies the symmetries of cellular decompositions of the 2-sphere induced by Morse functions with isolated critical points, showing that the permutation group of certain disk components is isomorphic to a finite subgroup of SO(3).

Contribution

It establishes that the permutation group arising from isotopic diffeomorphisms preserving a level set component is isomorphic to a finite subgroup of SO(3), linking topology with classical symmetry groups.

Findings

The permutation group is finite.

The group is isomorphic to a subgroup of SO(3).

The result connects cellular decompositions with classical symmetry groups.

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 $G = \{ \sigma_h \mid h \in \mathcal{S}_{K}(f)\}$ be the group of all such permutations. We prove that $G$ is isomorphic to a finite subgroup of $SO(3)$.