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

Abstract

Let f : M → R be a Morse function on a connected compact surface M, and S(f) and O(f) be respectively the stabilizer and the orbit of f with respect to the right action of the group of diffeomorphisms D(M). In a series of papers the first author described the homotopy types of connected components of S(f) and O(f) for the cases when M is either a 2-disk or a cylinder or χ(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 π1O(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 → R whose KR-graphs have exactly one cycle and prove that for every such function there exists a subsurface Q ⊂ T 2 , diffeomorphic with a cylinder, such that π1O(f) is expressed via the fundamental group π1O(f|Q) of the restriction of f to Q. This result holds for a larger class of smooth functions f : T 2 → 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 R2 → R without multiple factors.