SMOOTH FUNCTIONS ON 2-TORUS WHOSE KRONROD-REEB GRAPH CONTAINS A CYCLE

摘要

Let f: M → R be a Morse function on a, conriected compact surface M, and S(f) aiid 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 sruooth functions on 2-torus T~2 and shown that the computations of π_1O(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. diffcomorphic with a cylinder, such that π_1O(f) is expressed via the fundamental group π_1O(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 polynornial R~2 →R without multiple factors.