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

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

8 pages