2014/11/25 by Sergiy Maksymenko, Maksymenko, Sergiy, Bohdan Feshchenko +1 · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1411.6863
openalex publication_date 2014/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 T2 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:T2→ℝ whose KR-graphs have exactly one cycle and prove that for every such function there exists a subsurface Q⊂ T2, 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:T2→ 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.