Homotopy Types of Stabilizers and Orbits of Morse Functions on Surfaces

Abstract

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 D(M) the group of diffeomorphisms of M acting on C^∞(M, P) by the rule h⋅ f = f ∘ h −1 for h ∊ D(M) and f ∊ C^∞ (M,P). Let f: M → P be an arbitrary Morse mapping, Σ f the set of critical points of f, D(M,Σ f ) the subgroup of D(M) preserving Σ f , and S(f), S (f,Σ f ), O(f), and O(f,Σ f ) the stabilizers and the orbits of f with respect to D(M) and D(M,Σ f ). In fact S(f) = S(f,Σ f ). In this paper we calculate the homotopy types of S(f), O(f) and O(f,Σ f ). It is proved that except for few cases the connected components of S(f) and O(f,Σ f ) are contractible, π k O(f) = π k M for k ≥ 3, π2 O(f) = 0, and π1 O(f) is an extension of π1 D(M) ⊕ Z k (for some k ≥ 0) with a (finite) subgroup of the group of automorphisms of the Kronrod-Reeb graph of f. We also generalize the methods of F. Sergeraert to give conditions for a finite codimension orbit of a tame smooth action of a tame Lie group on a tame Fréchet manifold to be a tame Fréchet manifold itself. In particular, we obtain that O(f) and O(f, Σ f ) are tame Fréchet manifolds.