Automorphisms of Cellular Divisions of 2-Sphere Induced by Functions with Isolated Critical Points
DOI:
https://doi.org/10.15407/mag16.02.138Ключові слова:
поверхня, функція Морса, дифеоморфізмиАнотація
Нехай $f:S^2\to \mathbb{R}$ -- функція Морса на $2$-сфері і $K$ -- компонента зв'язності деякої множини рівня функції $f$, що містить хоча б одну сідлову критичну точку. Тоді $K$ - це $1$-вимірний CW-комплекс, клітково вкладений в $S^2$, так що доповнення $S^2\setminus K$ є об'єднанням відкритих $2$-дисків $D_1,\ldots, D_k$. Нехай $\mathcal{S}_{K}(f)$ група дифеоморфізмів $S^2$, які ізотопні до тотожного відображення і залишають інваріантними множину $K$ і кожну множину рівня $f^{-1}(c)$, $c\in\mathbb{R}$. Тоді кожен $h\in \mathcal{S}_{K}(f)$ індукує певну перестановку $\sigma_{h}$ вказаних вище дисків. Позначимо через $G = \{ \sigma_h \mid h \in \mathcal{S}_{K}(f)\}$ групу всіх таких перестановок. Ми доведемо, що $G$ ізоморфна скінченній підгрупі в $SO(3)$.Mathematics Subject Classification: 20E22, 57M60, 22F50
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