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 $Diff(M)$ 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 $k\ge 3$, 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.

49 pages, 8 figures. This version includes the proof of the fact that the orbits of a finite codimension of tame action of tame Lie group on tame Frechet manifold is a tame Frechet manifold itself

Cited by in corpus (12)

• On the homotopy type of the spaces of Morse functions on surfaces
• Topology of the spaces of Morse functions on surfaces
• Automorphisms of Kronrod-Reeb graphs of Morse functions on compact surfaces
• Connected components of spaces of Morse functions with fixed critical points
• Diffeomorphisms preserving Morse-Bott functions
• Topology of the spaces of functions with prescribed singularities on surfaces
• Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1
• Symmetries of degenerate center singularities of plane vector fields
• Homotopy types of diffeomorphisms groups of simplest Morse-Bott foliations on lens spaces, 2
• Symplectomorphisms of surfaces preserving a smooth function, I
• Topological actions of wreath products
• Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere