Topology of spaces of smooth functions and gradient-like flows with prescribed singularities on surfaces

TL;DR

This paper studies the topology of spaces of smooth gradient-like flows with prescribed singularities on surfaces, establishing their homotopy equivalence to certain manifolds and describing their orbit decompositions.

Contribution

It proves that these spaces are homotopy equivalent to manifolds and describes their decompositions into orbits via transversal fibrations, extending to non-Morse singularities.

Findings

Spaces of gradient-like flows are homotopy equivalent to a manifold.

Decompositions into orbits are given by transversal fibrations.

Results apply to both topological and smooth equivalence classes.

Abstract

By a gradient-like flow on a closed orientable surface $M$, we mean a closed 1-form $\beta$ defined on $M$ punctured at a finite set of points (sources and sinks of $\beta$) such that there exists a Morse function $f$ on $M$, called an energy function of $\beta$, whose critical points coincide with equilibria of $\beta$, and the pair $(f,\beta)$ has a canonical form near each critical point of $f$. Let $\mathcal{B}=\mathcal{B}(\beta_0)$ be the space of all gradient-like flows on $M$ having the same types of local singularities as a flow $\beta_0$, and $\mathcal{F}=\mathcal{F}(f_0)$ the space of all Morse functions on $M$ having the same types of local singularities as an energy function $f_0$ of $\beta_0$. We prove that the spaces $\mathcal{F}$ and $\mathcal{B}$, equipped with $C^\infty$ topologies, are homotopy equivalent to some manifold $\mathcal{M}_s$, moreover their decompositions into $\mathrm{Diff}^0(M)$-orbits are given by two transversal fibrations on $\mathcal{M}_s$. Similar results are proved for topological equivalence classes on $\mathcal{F}$ and $\mathcal{B}$, and for non-Morse singularities.