Homotopy types of stabilizers and orbits of Morse functions on surfaces

Details Homotopy types of stabilizers and orbits of Morse functions on surfaces Homotopy types of stabilizers and orbits of Morse functions on surfaces

2003-10-06

arxiv.org/abs/math/0310067v5

Annals of Global Analysis and Geometry 29 (2006) 241-285

Mathematics

Geometric Topology

49 pages, 8 figures. This version includes the proof of the fact that the orbits of a finite codimension of tame action of tam

Scientific paper

10.1007/s10455-005-9012-6

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 $C^{\infty}(M,P)$ 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 $k\geq 3$, 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 $C^{\infty}(M,P)$ of finite codimension, and that the projection $Diff(M) \to O(f)$ is a principal locally trivial $S(f)$-fibration.

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