# Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere

## TL;DR

This paper characterizes the automorphism groups of Kronrod-Reeb graphs for Morse functions on the 2-sphere, revealing their structure when the fixed subtree contains multiple points, thus advancing understanding of the topological symmetries of such functions.

## Contribution

It explicitly computes the groups of graph automorphisms induced by diffeomorphisms for Morse functions on the 2-sphere with non-trivial fixed subtrees.

## Key findings

- The automorphism groups are fully described for Morse functions with fixed subtrees of multiple points.
- The structure of these groups depends on the configuration of the fixed subtree.
- The results extend previous work on orientable surfaces excluding the 2-sphere and 2-torus.

## Abstract

Let $M$ be a compact two-dimensional manifold and, $f \in C^{\infty}(M,\mathbb{R})$ be a Morse function, and $\Gamma_f$ be its Kronrod-Reeb graph. Denote by $\mathcal{O}_{f}=\{f \circ h \mid h \in \mathcal{D}\}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $\mathcal{D}$ on $C^{\infty}(M,\mathbb{R})$, and by $\mathcal{S}(f)=\{h\in\mathcal{D} \mid f \circ h = f\}$ the corresponding stabilizer of this function. It is easy to show that each $h\in\mathcal{S}(f)$ induces a homeomorphism of $\Gamma_f$. Let also $\mathcal{D}_{\mathrm{id}}(M)$ be the identity path component of $\mathcal{D}(M)$, $\mathcal{S}'(f)= \mathcal{S}(f) \cap \mathcal{D}_{\mathrm{id}}(M)$ be group of diffeomorphisms of $M$ preserving $f$ and isotopic to identity map, and $G_f$ be the group of homeomorphisms of the graph $\Gamma_f$ induced by diffeomorphisms belonging to $\mathcal{S}'(f)$. This group is one of the key ingredients for calculating the homotopy type of the orbit $\mathcal{O}_{f}$. Recently the authors described the structure of groups $G_f$ for Morse functions on all orientable surfaces distinct from $2$-torus $T^2$ and $2$-sphere $S^2$. The present paper is devoted to the case $M=S^{2}$. In this situation $\Gamma_f$ is always a tree, and therefore all elements of the group $G_f$ have a common fixed subtree $\mathrm{Fix}(G_f)$, which may even consist of a unique vertex. Our main result calculates the groups $G_f$ for all Morse functions $f:S^{2}\to\mathbb{R}$ whose fixed subtree $\mathrm{Fix}(G_f)$ consists of more than one point.