Topological equivalence of functions on oriented surfaces

Abstract

On closed oriented surfaces of genus g ? 1, we consider functions that possess only one saddle critical point in addition to local maxima and minima. We study the problem of the realization of these functions on surfaces and construct an invariant that distinguishes them. For surfaces of genus $g = \frac{{n - 1}}{2}$, where n is a prime number, we calculate the number of topologically nonequivalent functions with one maximum and one minimum.