Homotopy types of stabilizers and orbits of Morse functions on surfaces

Let $M$ be a smooth compact surface, orientable or not, with boundary or without it, $P$ either the real line $R^1$ or the circle $S^1$, and $Diff(M)$ the group of diffeomorphisms of $M$ acting on $C^{\mathrm{\infty }}(M,P)$ by the rule $h\cdot f\mapsto f\circ h^{-1}$, where $h\in Diff(M)$ and $f \in C^{\infty}(M,P)$. Let $f:M\to P$ be a Morse function and $O(f)$ be the orbit of $f$ under this action. We prove that ${\pi }_{k}O(f)={\pi }_{k}M$ for $k\ge 3$, and ${\pi }_{2}O(f)=0$ except for few cases. In particular, $O(f)$ is aspherical, provided so is $M$. Moreover, ${\pi }_{1}O(f)$ is an extension of a finitely generated free abelian group with a (finite) subgroup of the group of automorphisms of the Reeb graph of $f$. We also give a complete proof of the fact that the orbit $O(f)$ is tame Frechet submanifold of $C^{\mathrm{\infty }}(M,P)$ of finite codimension, and that the projection $Diff(M)\to O(f)$ is a principal locally trivial $S(f)$-fibration.