DEFORMATIONS OF CIRCLE-VALUED MORSE FUNCTIONS ON SURFACES

Let M be a smooth connected orientable compact surface and let F-cov(M, S-1) be a space of all Morse functions f: M -> S-1 without critical points on partial derivative M such that, for any connected component V of partial derivative M, the restriction f: V -> S-1 is either a constant map or a covering map. The space F-cov (M, S-1) is endowed with the C 1 - topology. We present the classification of connected components of the space F-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 partial derivative M.