Automorphisms of cellular divisions of 2-sphere induced by functions\n with isolated critical points

2019/11/25 by Kravchenko, Anna, Sergiy Maksymenko, Maksymenko, Sergiy · 1 citation
Mathematics · Medicine · #20E22 #22F50 #57M60 #Algebraic Topology (math.AT) #Cell Adhesion Molecules Research #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1911.10808

openalex publication_date 2019/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f:S²\→ \ℝ be a Morse function on the 2-sphere and K be a\nconnected component of some level set of f containing at least one saddle\ncritical point. Then K is a 1-dimensional CW-complex cellularly embedded\ninto S², so the complement S²\∖ K is a union of open 2-disks\nD₁,\…, D_k. Let \S_K(f) be the group of isotopic to the\nidentity diffeomorphisms of S² leaving invariant K and also each level set\nf^-1(c), c\∈\ℝ. Then each h\∈ \S_K(f) induces a\ncertain permutation \σ_h of those disks. Denote by G = \σ_h\n\| h \∈ \S_K(f) be the group of all such permutations. We\nprove that G is isomorphic to a finite subgroup of SO(3).\n

Citations

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

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