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Smooth functions on 2-torus whose Kronrod-Reeb graph contains a cycle. (English) Zbl 1340.57009

The authors consider a class of Morse functions \(f:\;\mathbb T^2\to \mathbb R\) whose Kronrod-Reeb graphs contain exactly one cycle. It is proved that in this case there exists a subsurface \(Q\subset \mathbb T^2\) diffeomorphic to a cylinder, such that the fundamental group of the orbit \(O(f)\) with respect to the group of diffeomorphisms is expressed via \(\pi_1 (O(f|_Q))\). A similar result is obtained for a certain wider classes of functions.
The paper continues a series of publications devoted to the calculations of the homotopy types of \(O(f)\) and the stabilizer \(S(f)\); see, for example, S. I. Maksymenko and B. G. Feshchenko [Ukr. Math. J. 66, No. 9, 1346–1353 (2015; Zbl 1354.57036); translation from Ukr. Mat. Zh. 66, No. 9, 1205–1212 (2014)]; Elena A. Kudryavtseva [Sb. Math. 204, No. 1, 75–113 (2013; Zbl 1277.58007)].

MSC:

57R45 Singularities of differentiable mappings in differential topology
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
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