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Homotopic types of right stabilizers and orbits of smooth functions on surfaces

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Ukrainian Mathematical Journal Aims and scope

Let M be a connected smooth compact surface and let P be either the number line \( \mathbb{R} \) or a circle S 1. For a subset XM, by \( \mathcal{D} \)(M, X) we denote a group of diffeomorphisms of M fixed on X. We consider a special class \( \mathcal{F} \) of smooth mappings f:MP with isolated singularities containing all Morse mappings. For each mapping f\( \mathcal{F} \), we consider certain submanifolds XM “adapted” to f in a natural way and study the right action of the group \( \mathcal{D} \)(M, X) on C ∞( M, P). The main results of the paper describe the homotopic types of the connected components of stabilizers \( \mathcal{S} \)(f) and the orbits \( \mathcal{O} \)(f) of all mappings f\( \mathcal{F} \) and generalize the results of the author in this field obtained earlier.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 9, pp. 1186–1203, September, 2012.

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Maksimenko, S.I. Homotopic types of right stabilizers and orbits of smooth functions on surfaces. Ukr Math J 64, 1350–1369 (2013). https://doi.org/10.1007/s11253-013-0721-x

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