Connected components of spaces of Morse functions with fixed critical points

Abstract

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, $q\ge 1$ 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), a surjection ${\pi }_0(F)\to {\mathbb{Z}}^{p+r-1}$ is constructed. In particular, $|{\pi }_0(F)|=\infty$, and the Dehn twist about the boundary of any disk containing exactly two critical points, exactly one of which is a saddle point, does not preserve $F_f$. Let $\mathcal{D}$ be the group of orientation preserving diffeomorphisms of $M$ leaving fixed the critical points, ${\mathcal{D}}^0$ be the connected component of ${\mathrm{id}}_M$ in $\mathcal{D}$, and ${\mathcal{D}}_f\subset \mathcal{D}$ the set of diffeomorphisms preserving $F_f$. Let ${\mathcal{H}}_f$ be the subgroup of ${\mathcal{D}}_f$ generated by ${\mathcal{D}}^0$ and all diffeomorphisms $h\in \mathcal{D}$ which preserve some functions $f_1\in F_f$, and let ${\mathcal{H}}_f^{\mathrm{abs}}$ be its subgroup generated ${\mathcal{D}}^0$ and the Dehn twists about the components of level curves of functions $f_1\in F_f$. We prove that ${\mathcal{H}}_f^{\mathrm{abs}}⊊{\mathcal{D}}_f$ if $q\ge 2$, and construct an epimorphism ${\mathcal{D}}_f/{\mathcal{H}}_f^{\mathrm{abs}}\to {\mathbb{Z}}_2^{q-1}$, by means of the winding number. A finite polyhedral complex $K=K_{p,q,r}$ associated to the space $F$ is defined. An epimorphism $\mu :{\pi }_1(K)\to {\mathcal{D}}_f/{\mathcal{H}}_f$ and finite generating sets for the groups ${\mathcal{D}}_f/{\mathcal{D}}^0$ and ${\mathcal{D}}_f/{\mathcal{H}}_f$ in terms of the 2-skeleton of the complex $K$ are constructed.

12 pages with 2 figures, in Russian, to be published in Vestnik Moskov. Univ., a typo in theorem 1 is corrected

References in corpus (3)

• Lectures on 0/1-polytopes
• Homotopy types of stabilizers and orbits of Morse functions on surfaces
• Path-components of Morse mappings spaces of surfaces

Cited by in corpus (5)

• On the homotopy type of the spaces of Morse functions on surfaces
• Topology of the spaces of Morse functions on surfaces
• Special framed Morse functions on surfaces
• Orbits of smooth functions on 2-torus and their homotopy types
• Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere