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

TL;DR

This paper investigates the homotopy type of spaces of Morse functions on surfaces, introducing a new complex structure and analyzing its properties, especially for the sphere, to understand the topology of these function spaces.

Contribution

It introduces the skew cylindric-polyhedral complex to study Morse function spaces and relates its topology to the homotopy type of these spaces on surfaces.

Findings

The complex is finite for the sphere and its Euler characteristic is computed.

Morse inequalities for Betti numbers of the complex are established.

A relation between the homotopy types of the complex and the Morse function space is proposed.

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 $\mathbb{\widetilde K}$ (the "complex of framed Morse functions"), associated with the space $F$, is defined. In the case when $M=S^2$, the polyhedron $\mathbb{\widetilde K}$ 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 $\mathbb{\widetilde K}$ and the space $F$ of Morse functions, endowed with the $C^\infty$-topology, is indicated.