First Betti Numbers of the Orbits of Morse Functions on Surfaces

Abstract

Let M be a connected compact orientable surface and let P be either the real line ℝ or a circle S1. The group D(M) of diffeomorphisms on M acts in the space of smooth mappings C∞(M,P) by the rule (f, h) ⟼ f ° h, where h ∈ D(M) and f ∈ C∞(M,P). For f ∈ C∞(M,P), let O(f) denote the orbit of f relative to the specified action. By ℳ(M,P) we denote the set of isomorphism classes for the fundamental groups 𝜋1O(f) of orbits of all Morse mappings f : M → P. S. Maksymenko and B. Feshchenko studied the sets of isomorphism classes ℬ and 𝒯 of groups generated by direct products and certain wreath products. They proved the inclusions ℳ(M,P) ∁ B valid under the condition that M differs from the 2-sphere S2 and 2-torus T2 and ℳ(T2,ℝ) ∁ T . We show that these inclusions are equalities and describe some subclasses of M(M,P) under certain restrictions imposed on the behavior of functions on the boundary 𝜕M. We also prove that, for any group G ∈ ℬ (G ∈ T ), the center Z(G) and the quotient group with respect to the commutator subgroup G/[G,G] are free Abelian groups of the same rank, which can be easily found by using the geometric properties of a Morse mapping f such that 𝜋1O(f) ≃ G. In particular, this rank is the first Betti number of the orbit O(f) of the mapping f.