Special framed Morse functions on surfaces

TL;DR

This paper studies the topology of spaces of Morse functions on surfaces, introducing special framed Morse functions and proving homotopy equivalences that relate these spaces to moduli spaces and diffeomorphism groups.

Contribution

It defines the space of special framed Morse functions and establishes homotopy equivalences with universal moduli spaces and diffeomorphism groups, advancing understanding of their topological structure.

Findings

Inclusion of special framed Morse functions into framed Morse functions is a homotopy equivalence.

Homotopy equivalences relate the space of Morse functions to moduli spaces and diffeomorphism groups.

Results hold when critical points are sufficiently labeled, with explicit homotopy relations established.

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 $\mathbb{\widetilde K}\sim\widetilde{\cal M}$ and $F\sim\mathbb{F}^0\sim{\mathscr D}^0\times\mathbb{\widetilde K}$ are proved, where $\mathbb{\widetilde K}$ is the complex of framed Morse functions, $\widetilde{\cal M}\approx\mathbb{F}^1/{\mathscr D}^0$ is the universal moduli space of framed Morse functions, ${\mathscr D}^0$ is the group of self-diffeomorphisms of $M$ homotopic to the identity.