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

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 $\mathcal{D}(M)$, ${\mathcal{S}}^{\prime }(f)=\mathcal{S}(f)\cap {\mathcal{D}}_{\mathrm{id}}(M)$ 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 ${\mathcal{S}}^{\prime }(f)$. 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.

arXiv comment5 pages

References in corpus (6)

• Homotopy types of stabilizers and orbits of Morse functions on surfaces
• On the homotopy type of the spaces of Morse functions on surfaces
• Topology of the spaces of Morse functions on surfaces
• Orbits of smooth functions on 2-torus and their homotopy types
• Automorphisms of Kronrod-Reeb graphs of Morse functions on compact surfaces
• Connected components of spaces of Morse functions with fixed critical points