Special framed Morse functions on surfaces

Abstract

Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse functions on $M$, and ${\mathbb{F}}^1$ the space of framed Morse functions, both endowed with $C^{\infty }$-topology. The space ${\mathbb{F}}^0$ of special framed Morse functions is defined. We prove that the inclusion mapping ${\mathbb{F}}^0\hookrightarrow {\mathbb{F}}^1$ is a homotopy equivalence. In the case when at least $\chi (M)+1$ critical points of each function of $F$ are labeled, homotopy equivalences $\tilde{\mathbb{K}}\sim \tilde{\mathcal{M}}$ and $F\sim {\mathbb{F}}^0\sim {\mathcal{D}}^0\times \tilde{\mathbb{K}}$ are proved, where $\tilde{\mathbb{K}}$ is the complex of framed Morse functions, $\tilde{\mathcal{M}}\approx {\mathbb{F}}^1/{\mathcal{D}}^0$ is the universal moduli space of framed Morse functions, ${\mathcal{D}}^0$ is the group of self-diffeomorphisms of $M$ homotopic to the identity.

8 pages, in Russian

References in corpus (2)

• On the homotopy type of the spaces of Morse functions on surfaces
• Connected components of spaces of Morse functions with fixed critical points

Cited by in corpus (8)

• On the homotopy type of the spaces of Morse functions on surfaces
• Automorphisms of Kronrod-Reeb graphs of Morse functions on compact surfaces
• Topology of the spaces of functions with prescribed singularities on surfaces
• Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1
• Topological actions of wreath products
• Deformations of functions on surfaces
• Automorphisms of cellular divisions of $2$-sphere induced by functions with isolated critical points
• Topology of spaces of smooth functions and gradient-like flows with prescribed singularities on surfaces