Deformations of circle-valued Morse functions on surfaces

Let M be a smooth connected orientable compact surface and let ${\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right)$ 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 ${\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right)$ is endowed with the C ^∞-topology. We present the classification of connected components of the space ${\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right)$. This result generalizes the results obtained by Matveev, Sharko, and the author for the case of Morse functions locally constant on ∂M.