The first Betti numbers of orbits of Morse functions on surfaces

Abstract

UDC 515.1

Let $M$ be a connected compact orientable surface and let $P$ be the real line $\mathbb{R}$ or circle $S^1.$
The group $\mathcal{D}$ of diffeomorphisms on $M$ acts in the space of smooth mappings $C^{\infty} (M,P)$ by the rule $(f,h)\longmapsto f\circ h,$ where $h \in \mathcal{D},$ $f\in C^\infty (M,P).$
For $f\in C^{\infty}(M,P),$ let $\mathcal{O}(f)$ denote the orbit of $f$ relative to the specified action.
By $\mathcal{M}(M,P)$ we denote the set of isomorphism classes of the fundamental groups $\pi_1\mathcal{O}(f)$ of orbits of all Morse mappings $f\colon M\to P.$

S. I. Maksymenko and B. G. Feshchenko studied the sets of isomorphism classes $\mathcal{B}$ and $\mathcal{T}$ of groups generated by direct products and certain wreath products.
In this case, they succeeded to prove the inclusions $\mathcal{M}(M,P) \subset \mathcal{B}$ under the condition that $M$ is distinct from the 2-sphere $S^2$ and 2-torus $T^2$ and $\mathcal{M} (T^2, \mathbb{R})\subset \mathcal{T}.$
In the present paper, we show that these inclusions are equalities and describe some subclasses from $\mathcal{M} (M,P)$ under certain restrictions on the behavior of functions on the boundary $\partial M.$

We also prove that for any group $G \in \mathcal{B}$ $(G \in \mathcal{T})$, the center $Z(G)$ and the quotient group by the commutator subgroup $G/[G,G]$ are free Abelian groups of the same rank easily calculated by using the geometric properties of a Morse mapping $f$ such that $\pi_1\mathcal{O}(f)\simeq G.$
In particular, this rank is the first Betti number of the orbit $\mathcal{O}(f)$ of $f.$