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

TL;DR

This paper investigates the homotopy types of orbits of certain smooth functions on the 2-torus, specifically Morse functions with a single cycle in their Kronrod-Reeb graph, under diffeomorphism group actions.

Contribution

It characterizes the homotopy type of the orbit of Morse functions on the 2-torus with specific critical point structures, extending to a broader class of functions with polynomial-like critical germs.

Findings

Homotopy type of the orbit described for Morse functions with one cycle in the Kronrod-Reeb graph.

Extension of results to functions with critical points equivalent to homogeneous polynomials.

Provides conditions under which the orbit's homotopy type can be explicitly determined.

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.