Deformations of smooth function on $2$-torus whose KR-graph is a tree

TL;DR

This paper characterizes the fundamental group of the orbit space of Morse functions on the 2-torus with a tree-shaped Kronrod-Reeb graph, extending results to functions with certain homogeneous polynomial germs at critical points.

Contribution

It provides a complete description of the fundamental group of the orbit of Morse functions on the 2-torus with tree-shaped Kronrod-Reeb graphs, including a broader class of functions with specific local properties.

Findings

Full description of  (f) for functions with tree-shaped Kronrod-Reeb graphs.

Extension of results to functions with homogeneous polynomial germs at critical points.

Applicable to a broader class of smooth functions on the 2-torus.

Abstract

Let $f:T^2\to \mathbb{R}$ be Morse function on $2$-torus $T^2,$ and $\mathcal{O}(f)$ be the orbit of $f$ with respect to the right action of the group of diffeomorphisms $\mathcal{D}(T^2)$ on $C^{\infty}(T^2)$. Let also $\mathcal{O}_f(f,X)$ be a connected component of $\mathcal{O}(f,X)$ which contains $f.$ In the case when Kronrod-Reeb graph of $f$ is a tree we obtain the full description of $\pi_1\mathcal{O}_f(f).$ This result also holds for more general class of smooth functions $f:T^2\to \mathbb{R}$ which have the following property: for each critical point $z$ of $f$ the germ $f$ of $z$ is smoothly equivalent to some homogeneous polynomial $\mathbb{R}^2\to \mathbb{R}^2$ without multiple points. Translated from Ukrainian