Smooth functions on 2-torus whose Kronrod-Reeb graph contains a cycle

Abstract

Let $f:M\to \mathbb{R}$ be a Morse function on a connected compact surface $M$, and $\mathcal{S}(f)$ and $\mathcal{O}(f)$ be respectively the stabilizer and the orbit of $f$ with respect to the right action of the group of diffeomorphisms $\mathcal{D}(M)$. In a series of papers the first author described the homotopy types of connected components of $\mathcal{S}(f)$ and $\mathcal{O}(f)$ for the cases when $M$ is either a $2$-disk or a cylinder or $\chi (M)<0$. Moreover, in two recent papers the authors considered special classes of smooth functions on $2$-torus $T^2$ and shown that the computations of ${\pi }_1\mathcal{O}(f)$ for those functions reduces to the cases of $2$-disk and cylinder. In the present paper we consider another class of Morse functions $f:T^2\to \mathbb{R}$ whose KR-graphs have exactly one cycle and prove that for every such function there exists a subsurface $Q\subset T^2$, diffeomorphic with a cylinder, such that ${\pi }_1\mathcal{O}(f)$ is expressed via the fundamental group ${\pi }_1\mathcal{O}(f{|}_Q)$ of the restriction of $f$ to $Q$. 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.

17pages, 8 figures