Homotopy Type of Orbits of Morse Functions on Surfaces

Let be a smooth compact surface, orientable or not, with boundary or without it, either the real line R or the circle , and () the group of diffeomorphisms of acting on C∞(M, P) by the rule , for ∈ () and ∈ (, ).Let : → be a Morse function and () be the orbit of f under this action. We prove that () = for k ≥ 3, and () = 0 except for few cases. In particular, () is aspherical, provided so is . Moreover, () is an extension of a finitely generated free abelian group with a (finite) subgroup of the group of automorphisms of the Reeb graph of