Deformations of circle-valued Morse functions on surfaces

Abstract

Let $M$ be a smooth connected orientable compact surface and let $\mathcal{F}_{\text{cov}}(M,S^1)$ be a space of all Morse functions $f: M → S^1$ without critical points on $∂M$ such that, for any connected component $V$ of $∂M$, the restriction $f : V → S^1$ is either a constant map or a covering map. The space $\mathcal{F}_{\text{cov}}(M,S^1)$ is endowed with the $C^{∞}$-topology. We present the classification of connected components of the space $\mathcal{F}_{\text{cov}}(M,S^1)$. This result generalizes the results obtained by Matveev, Sharko, and the author for the case of Morse functions locally constant on $∂M$.