Orbits of smooth functions on 2-torus and their homotopy types

Abstract

Let $f:T^2\to \mathbb{R}$ be a Morse function on $2$-torus $T^2$ such that its Kronrod-Reeb graph $\Gamma (f)$ has exactly one cycle, i.e. it is homotopy equivalent to $S^1$. Under some additional conditions we describe a homotopy type of the orbit of $f$ with respect to the action of the group of diffeomorphism of $T^2$. This result holds for a larger class of smooth functions $f:T^2\to \mathbb{R}$ having the following property: for every critical point $z$ of $f$ the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial ${\mathbb{R}}^2\to \mathbb{R}$ without multiple factors.

arXiv comment16 pages, 3 figures

References in corpus (7)

• On the homotopy type of the spaces of Morse functions on surfaces
• Connected components of spaces of Morse functions with fixed critical points
• Functions on surfaces and incompressible subsurfaces
• Smooth functions on 2-torus whose Kronrod-Reeb graph contains a cycle
• Homotopy properties of spaces of smooth functions on 2-torus
• Deformations of circle-valued Morse functions on surfaces
• Structure of the fundamental groups of orbits of smooth functions on surfaces

Cited by in corpus (5)

• Symplectomorphisms of surfaces preserving a smooth function, I
• Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere
• Automorphisms of cellular divisions of $2$-sphere induced by functions with isolated critical points
• Deformations of functions on surfaces
• Deformations of circle-valued functions on $2$-torus