Journal of Mathematical Physics, Analysis, Geometry
2020, vol. 16, No 2, pp. 138-160     ( to contents , go back )

Automorphisms of Cellular Divisions of 2-Sphere Induced by Functions with Isolated Critical Points

Anna Kravchenko

Taras Shevchenko National University of Kyiv, Ukraine

Sergiy Maksymenko

Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

Received November 18, 2019, revised January 29, 2020.


Let $f:S^2\to \mathbb{R}$ 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 $S^2\setminus K$ is a union of open $2$-disks $D_1,\ldots, D_k$. Let $\mathcal{S}_{K}(f)$ be the group of isotopic to the identity diffeomorphisms of $S^2$ leaving invariant $K$ and also each level set $f^{-1}(c)$, $c\in\mathbb{R}$. Then each $h\in \mathcal{S}_{K}(f)$ induces a certain permutation $\sigma_{h}$ of those disks. Denote by $G = \{ \sigma_h \mid h \in \mathcal{S}_{K}(f)\}$ the group of all such permutations. We prove that $G$ is isomorphic to a finite subgroup of $SO(3)$.

Key words: surface, Morse function, diffeomorphisms

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