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

2018/04/24 by Bohdan Feshchenko, Feshchenko, Bohdan · 1 citation
Computer Science · Mathematics · #37C05 #57R45 #57S05 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1804.08966

openalex publication_date 2018/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f:T²→ ℝ be Morse function on 2-torus T², and O(f) be the orbit of f with respect to the right action of the group of diffeomorphisms D(T²) on C^∞(T²). Let also O_f(f,X) be a connected component of O(f,X) which contains f. In the case when Kronrod-Reeb graph of f is a tree we obtain the full description of π₁O_f(f). This result also holds for more general class of smooth functions f:T²→ 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 ℝ²→ ℝ² without multiple points. Translated from Ukrainian

Cited by

• Smooth functions that split a Klein bottle into two Möbius bands

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