Homotopy type of stabilizers and orbits of Morse functions on surfaces

Let M be a smooth compact surface, orientable or not, with boundary or without it. Let also P be either the real line R¹ or the circle S¹. Then the group Diff(M) of diffeomorphisms of M acts on C^∞(M,P) by the rule h⋅ff ∘h^-1, for h ∈ Diff(M) and f ∈ C^∞(M,P).

Let f : M → P be a Morse function and O_f be the orbit of f under this action. We prove that π_kO_f = π_kM for k ≥ 3, and π₂O_f = 0 except for few cases. In particular, O_f is aspherical, provided M is. Moreover, π₁O_f is an extension of a finitely generated free abelian group with a (finite) subgroup of the group of automorphisms of the Kronrod-Reeb graph of f.