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

Abstract

Let f : S 2 → R be a Morse function on the 2-sphere and K be a connected component of some level set of f containing at least one saddle critical point. Then K is a 1-dimensional CW-complex cellularly embedded into S 2 , so the complement S 2 \ K is a union of open 2-disks D 1 , . . . , D k . Let S K (f ) be the group of isotopic to the identity diffeomorphisms of S 2 leaving invariant K and also each level set f −1 (c), c ∈ R. Then each h ∈ S K (f ) induces a certain permutation σ h of those disks. Denote by G = {σ h | h ∈ S K (f )} be the group of all such permutations. We prove that G is isomorphic to a finite subgroup of SO(3).

Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere

“…We also mention that an algebraic structure of G(f ) for Morse function on S 2 is partially understood [8], and in general this case is more complicated than the case of Morse functions on compact surfaces of genus ≥ 1. Recently S. Maksymenko and A. Kravchenko [9] described special subgroups of G(f ) for Morse functions on S 2 .…”