Topology of spaces of smooth functions and gradient-like flows with prescribed singularities on surfaces
Elena Kudryavtseva · arXiv (Cornell University) · 2021
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.