Topology of the spaces of Morse functions on surfaces

Abstract

Let $M$ be a smooth closed orientable surface, and let $F$ be the space of Morse functions on $M$ such that at least $\chi (M)+1$ critical points of each function of $F$ are labeled by different labels (enumerated). Endow the space $F$ with $C^{\infty }$-topology. We prove the homotopy equivalence $F\sim R\times \tilde{\mathcal{M}}$ where $R$ is one of the manifolds $\mathbb{R}{P}^3$, $S^1\times S^1$ and the point in dependence on the sign of $\chi (M)$, and $\tilde{\mathcal{M}}$ is the universal moduli space of framed Morse functions, which is a smooth stratified manifold. Morse inequalities for the Betti numbers of the space $F$ are obtained.

15 pages, in Russian

References in corpus (4)

• Homotopy types of stabilizers and orbits of Morse functions on surfaces
• On the homotopy type of the spaces of Morse functions on surfaces
• Path-components of Morse mappings spaces of 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
• Diffeomorphisms preserving Morse-Bott functions
• Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1
• Symplectomorphisms of surfaces preserving a smooth function, I
• Topological actions of wreath products
• Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere