First Betti Numbers of the Orbits of Morse Functions on Surfaces

Let M be a connected compact orientable surface and let P either the real line ℝ or circle S1. The group D(M) of diffeomorphisms on acts in space smooth mappings C∞(M,P) by rule (f, h) ⟼ f ° h, where h ∈ C∞(M,P). For C∞(M,P), O(f) denote orbit relative to specified action. By ℳ(M,P) we set isomorphism classes for fundamental groups