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## Topology of the spaces of Morse functions on surfaces

### Abstract

Let $M$ be a smooth closed orientable surface, and let $F$ be the space of Morse functions on $M$ such that at least $\chi(M)+1$ critical points of each function of $F$ are labeled by different labels (enumerated). Endow the space $F$ with $C^\infty$-topology. We prove the homotopy equivalence $F\sim R\times{\widetilde{\cal M}}$ where $R$ is one of the manifolds ${\mathbb R}P^3$, $S^1\times S^1$ and the point in dependence on the sign of $\chi(M)$, and ${\widetilde{\cal M}}$ is the universal moduli space of framed Morse functions, which is a smooth stratified manifold. Morse inequalities for the Betti numbers of the space $F$ are obtained.Comment: 15 pages, in Russia

Topics: Mathematics - Geometric Topology, Mathematics - Algebraic Topology, 58E05, 57M50, 58K65, 46M18
Year: 2011
DOI identifier: 10.1134/S0001434612070243
OAI identifier: oai:arXiv.org:1104.4792