Finiteness of the homotopy types of right orbits of Morse functions on surfaces

Sergiy Maksymenko

Published 2014-09-15Version 1

Let $M$ be a connected orientable surface, $P$ be either a real line $\mathbb{R}$ or a circle $S^1$, and $f:M \to P$ be a Morse map. Denote by $\mathcal{D}_{\mathrm{id}}$ the group of diffeomorphisms of $M$ isotopic to the identity. This group acts from the right on the space of smooth maps $C^{\infty}(M,P)$ and one can define the stabilizer $\mathcal{S} = \{h \in \mathcal{D}_{\mathrm{id}} \mid f \circ h = f\}$ and the orbit $\mathcal{O} = \{f \circ h \mid h \in \mathcal{D}_{\mathrm{id}} \}$ of $f$ with respect to that action. Then $\mathcal{S}$ acts on the Kronrod-Reeb graph $\Gamma$ of $f$ and we denote by $G$ the group of all automorphisms of $\Gamma$ induced by elements from $\mathcal{S}$. Previously the author shown that $\mathcal{O}$ has a homotopy type of some CW-complex of dimension $2k-1$, where $k$ is the total number of critical points of $f$. Moreover, if $f$ is generic and $M$ differs from $2$-sphere and projective plane, then $\mathcal{O}$ is homotopy equivalent to some torus. In the present paper we refine these facts by proving the following theorem: if $M$ is distinct from the $2$-sphere and $2$-torus then there exists a free action of $G$ on a $p$-dimensional torus $T^p$ for some $p\geq0$ such that the orbit $\mathcal{O}$ (endowed with $C^{\infty}$ topology) is homotopy equivalent to the factor space $T^p/G$.