Homotopy groups of orbits of Morse functions on surfaces
Sergiy Maksymenko · arXiv (Cornell University) · 2003
Abstract. Let M be a smooth compact surface, orientable or not, with boundary or without it. Let also P be either the real line R 1 or the circle S 1. Then the group D(M) of diffeomorphisms of M acts on C ∞ (M, P) by the rule h · f ↦ → f ◦ h, for h ∈ D(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 πkO(f) = πkM for k ≥ 3, π2O(f) = 0 except for few cases. In particular, O(f) is aspherical, provided so is M. Moreover, π1O(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. 1.