Connected components of spaces of Morse functions with fixed critical points

Let $M$ be a smooth closed orientable surface and $F=F_{p,q,r}$ be the space of Morse functions on $M$ having exactly $p$ critical points of local minima, $q\ge1$ saddle critical points, and $r$ critical points of local maxima, moreover all the points are fixed. Let $F_f$ be the connected component of a function $f\in F$ in $F.$ By means of the winding number introduced by Reinhart (1960), we construct a surjection $\pi_0(F)\to\mathbb{Z}^{p+r-1}$, in particular $|\pi_0(F)|=\infty$ and the component $F_f$ is not preserved under the Dehn twist about the boundary of any disk containing exactly two critical points, exactly one of which is a saddle point. Let $\mathscr{D}$ be the group of orientation preserving diffeomorphisms of $M$ leaving fixed the critical points, $\mathscr{D}^0$ be the connected component of $\operatorname{id}_M$ in $\mathscr{D},$ and $\mathscr{D}_f\subset\mathscr{D}$ the set of diffeomorphisms preserving $F_f.$ Let $\mathscr{H}_f$ be the subgroup of $\mathscr{D}_f$ generated by $\mathscr{D}^0$ and all diffeomorphisms $h\in\mathscr{D}$ preserving some functions $f_1\in F_f,$ and let $\mathscr{H}_f^\mathrm{abs}$ be its subgroup generated by $\mathscr{D}^0$ and the Dehn twists about the components of level curves of functions $f_1\in F_f.$ We prove that $\mathscr{H}_f^\mathrm{abs}\subsetneq\mathscr{D}_f$ if $q\ge2,$ and construct an epimorphism $\mathscr{D}_f/\mathscr{H}_f^\mathrm{abs}\to\mathbb{Z}_2^{q-1},$ by means of the winding number. A finite polyhedral complex $K=K_{p,q,r}$ associated to the space $F$ is defined. An epimorphism $\mu\colon\pi_1(K)\to\mathscr{D}_f/\mathscr{H}_f$ and finite generating sets for the groups $\mathscr{D}_f/\mathscr{D}^0$ and $\mathscr{D}_f/\mathscr{H}_f$ in terms of the 2-skeleton of the complex $K$ are constructed.