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

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.