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 Of be the orbit of f under this action.
We prove that πkOf = πkM for k ≥ 3, and π2Of = 0 except for few cases. In
particular, Of is aspherical, provided M is. Moreover, π1Of 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.