Homotopy Types of Stabilizers and Orbits of Morse Functions on Surfaces

By S. Maksymenko. 2003

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Let M be a smooth compact surface, orientable or not, with boundary or without it, P either the real line ℝ1 or the circle S1, 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, Σf the set of critical points of f, D(M,Σf) the subgroup of D(M) preserving Σf, and S(f), S (f,Σf), O(f), and O(f,Σf) the stabilizers and the orbits of f with respect to D(M) and D(M,Σf). In fact S(f) = S(f,Σf).In this paper we calculate the homotopy types of S(f), O(f) and O(f,Σf). It is proved that except for few cases the connected components of S(f) and O(f,Σf) are contractible, πkO(f) = πkM for k ≥ 3, π2O(f) = 0, and π1O(f) is an extension of π1D(M) ⊕ Zk (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, Σf) are tame Fréchet manifolds.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3178eb51b72c073cb21af8d32c5d1cd9e1c3ad77