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Let $M$ be a smooth closed orientable surface and $F=F_{p,q,r}$ be the spaceof 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 afunction $f\in F$ in $F$. By means of the winding number introduced by Reinhart(1960), a surjection $\pi_0(F)\to{\mathbb Z}^{p+r-1}$ is constructed. Inparticular, $|\pi_0(F)|=\infty$, and the Dehn twist about the boundary of anydisk containing exactly two critical points, exactly one of which is a saddlepoint, does not preserve $F_f$. Let $\mathscr D$ be the group of orientationpreserving diffeomorphisms of $M$ leaving fixed the critical points, ${\mathscrD}^0$ be the connected component of ${\rm id}_M$ 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 $h\in{\mathscr D}$ which preserve somefunctions $f_1\in F_f$, and let ${\mathscr H}_f^{\rm abs}$ be its subgroupgenerated ${\mathscr D}^0$ and the Dehn twists about the components of levelcurves of functions $f_1\in F_f$. We prove that ${\mathscr H}_f^{\rmabs}\subsetneq{\mathscr D}_f$ if $q\ge2$, and construct an epimorphism${\mathscr D}_f/{\mathscr H}_f^{\rm abs}\to{\mathbb Z}_2^{q-1}$, by means ofthe winding number. A finite polyhedral complex $K=K_{p,q,r}$ associated to thespace $F$ is defined. An epimorphism $\mu:\pi_1(K)\to{\mathscr D}_f/{\mathscrH}_f$ 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.

Connected components of spaces of Morse functions with fixed critical points