Automorphisms of Kronrod–Reeb graphs of Morse functions on compact surfaces

Abstract

Let $M$ be a connected orientable compact surface, $f:M\to\mathbb{R}$ be a Morse function, and $\mathcal{D}_{\mathrm{id}}(M)$ be the group of difeomorphisms of $M$ isotopic to the identity. Denote by $\mathcal{S}'(f)=\{f\circ h = f\mid h\in\mathcal{D}_{\mathrm{id}}(M)\}$ the subgroup of $\mathcal{D}_{\mathrm{id}}(M)$ consisting of difeomorphisms "preserving" $f$, i.e. the stabilizer of $f$ with respect to the right action of $\mathcal{D}_{\mathrm{id}}(M)$ on the space $\mathcal{C}^{\infty}(M,\mathbb{R})$ of smooth functions on $M$. Let also $\mathbf{G}(f)$ be the group of automorphisms of the Kronrod-Reeb graph of $f$ induced by diffeomorphisms belonging to $\mathcal{S}'(f)$. This group is an important ingredient in determining the homotopy type of the orbit of $f$ with respect to the above action of $\mathcal{D}_{\mathrm{id}}(M)$ and it is trivial if $f$ is "generic", i.e. has at most one critical point at each level set $f^{-1}(c)$, $c\in\mathbb{R}$. For the case when $M$ is distinct from $2$-sphere and $2$-torus we present a precise description of the family $\mathbf{G}(M,\mathbb{R})$ of isomorphism classes of groups $\mathbf{G}(f)$, where $f$ runs over all Morse functions on $M$, and of its subfamily $\mathbf{G}^{smp}(M,\mathbb{R}) \subset \mathbf{G}(M,\mathbb{R})$ consisting of groups corresponding to simple Morse functions, i.e. functions having at most one critical point at each connected component of each level set. In fact, $\mathbf{G}(M,\mathbb{R})$, (resp. $\mathbf{G}^{smp}(M,\mathbb{R})$), coincides with the minimal family of isomorphism classes of groups containing the trivial group and closed with respect to direct products and also with respect to wreath products "from the top" with arbitrary finite cyclic groups, (resp. with group $\mathbb{Z}_2$ only).