On the homotopy type of the spaces of Morse functions on surfaces

Abstract

Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse functions on $M$ having fixed number of critical points of each index, moreover at least $\chi (M)+1$ critical points are labeled by different labels (enumerated). A notion of a skew cylindric-polyhedral complex, which generalizes the notion of a polyhedral complex, is introduced. The skew cylindric-polyhedral complex $\tilde{\mathbb{K}}$ (the "complex of framed Morse functions"), associated with the space $F$, is defined. In the case when $M=S^2$, the polyhedron $\tilde{\mathbb{K}}$ is finite; its Euler characteristic is evaluated and the Morse inequalities for its Betti numbers are obtained. A relation between the homotopy types of the polyhedron $\tilde{\mathbb{K}}$ and the space $F$ of Morse functions, endowed with the $C^{\infty }$-topology, is indicated.

32 pages, in Russian

References in corpus (5)

• Homotopy types of stabilizers and orbits of Morse functions on surfaces
• Topology of the spaces of Morse functions on surfaces
• Special framed 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 (15)

• 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 compact surfaces
• Topology of the spaces of functions with prescribed singularities on surfaces
• Smooth functions on 2-torus whose Kronrod-Reeb graph contains a cycle
• Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1
• Homotopy properties of spaces of smooth functions on 2-torus
• Symplectomorphisms of surfaces preserving a smooth function, I
• Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere
• Topological actions of wreath products
• Automorphisms of cellular divisions of $2$-sphere induced by functions with isolated critical points
• Deformations of circle-valued functions on $2$-torus
• Topology of spaces of smooth functions and gradient-like flows with prescribed singularities on surfaces
• Deformations of functions on surfaces