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 X ⊂ M, 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:M → P with isolated singularities containing all Morse mappings. For each mapping f ∈ \( \mathcal{F} \), we consider certain submanifolds X ⊂ M “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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S. Maksymenko, “Homotopy types of stabilizers and orbits of Morse functions on surfaces,” Ann. Global Anal. Geom., 29, No. 3, 241–285 (2006).
S. Maksymenko, “Homotopy dimension of orbits of Morse functions on surfaces,” Trav. Math., 18, 39–44 (2008).
S. Maksymenko, “Functions with isolated singularities on surfaces,” in: Geometry and Topology of Functions on Manifolds, Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, 7, No. 4 (2010), pp. 7–66.
S. Maksymenko, “Functions on surfaces and incompressible subsurfaces,” Meth. Funct. Anal. Topol., 16, No. 2, 167–182 (2010).
F. Sergeraert, “Un théorème de fonctions implicites sur certains espaces de Fréchet et quelques applications,” Ann. Sci. École Norm. Super., 5, No. 4, 599–660 (1972).
S. Smale, “Diffeomorphisms of the 2-sphere,” Proc. Amer. Math. Soc., 10, 621–626 (1969).
J. S. Birman, “Mapping class groups and their relationship to braid groups,” Commun. Pure Appl. Math., 22, 213–238 (1969).
C. J. Earle and J. Eells, “A fibre bundle description of Teichmüller theory,” J. Different. Geom., 3, 19–43 (1969).
C. J. Earle and A. Schatz, “Teichmüller theory for surfaces with boundary,” J. Different. Geom., 4, 169–185 (1970).
A. Gramain, “Le type d’homotopie du groupe des difféomorphismes d’une surface compacte,” Ann. Sci. École Norm. Super., 6, No. 4, 53–66 (1973).
R. T. Seeley, “Extension of C ∞ functions defined in a half space,” Proc. Amer. Math. Soc., 15, 625–626 (1964).
S. Maksymenko, “Smooth shifts along trajectories of flows,” Topol. Appl., 130, No. 2, 183–204 (2003).
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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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DOI: https://doi.org/10.1007/s11253-013-0721-x