For citation:
Shubin D. D. Topology of ambient manifolds of non-singular Morse – Smale flows with three periodic orbits. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 6, pp. 863-868. DOI: 10.18500/0869-6632-2021-29-6-863-868
Topology of ambient manifolds of non-singular Morse – Smale flows with three periodic orbits
The purpose of this study is to establish the topological properties of three-dimensional manifolds which admit Morse – Smale flows without fixed points (non-singular or NMS-flows) and give examples of such manifolds that are not lens spaces. Despite the fact that it is known that any such manifold is a union of circular handles, their topology can be investigated additionally and refined in the case of a small number of orbits. For example, in the case of a flow with two non-twisted (having a tubular neighborhood homeomorphic to a solid torus) orbits, the topology of such manifolds is established exactly: any ambient manifold of an NMS-flow with two orbits is a lens space. Previously, it was believed that all prime manifolds admitting NMS-flows with at most three non-twisted orbits have the same topology. Methods. In this paper, we consider suspensions over Morse – Smale diffeomorphisms with three periodic orbits. These suspensions, in turn, are NMS-flows with three periodic trajectories. Universal coverings of the ambient manifolds of these flows and lens spaces are considered. Results. In this paper, we present a countable set of pairwise distinct simple 3-manifolds admitting NMS-flows with exactly three non-twisted orbits. Conclusion. From the results of this paper it follows that there is a countable set of pairwise distinct three-dimensional manifolds other than lens spaces, which refutes the previously published result that any simple orientable manifold admitting an NMS-flow with at most three orbits is lens space.
- Asimov D. Round handles and non-singular Morse – Smale flows. Annals of Mathematics. 1975; 102(1):41–54. DOI: 10.2307/1970972.
- Campos B, Cordero A, Mart´ inez-Alfaro J, Vindel P. NMS flows on three-dimensional manifolds with one saddle periodic orbit. Acta Mathematica Sinica. 2004;20(1):47–56. DOI: 10.1007/s10114-003-0305-z.
- Pochinka O, Shubin D. Nonsingular Morse – Smale flows of n-manifolds with attractor-repeller dynamics [Electronic resource]. arXiv: 2105.13110. arXiv Preprint; 2021. 17 p. Available from: https://arxiv.org/abs/2105.13110.
- Medvedev T, Nozhdrinova E, Pochinka O. On periodic data of diffeomorphisms with one saddle orbit. Topology Proceedings. 2019;54:49–68.
- Hatcher A. Notes on Basic 3-Manifold Topology [Electronic resource]. Ithaca, NY: Cornell University; 2007. 61 p. Available from: https://pi.math.cornell.edu/ hatcher/3M/3M.pdf.
- Kosniowski C. A First Course in Algebraic Topology. Cambridge: Cambridge University Press; 1980. 269 p. DOI: 10.1017/CBO9780511569296.
- Grines VZ, Medvedev TV, Pochinka OV. Dynamical Systems on 2- and 3-Manifolds. Vol. 46 of Developments in Mathematics. Switzerland: Springer International Publishing; 2016. 295 p. DOI: 10.1007/978-3-319-44847-3.
- Matveev SV, Fomenko AT. Algorithmic and Computer Methods for Three-Manifolds. Netherlands: Springer; 1997. 337 p. DOI: 10.1007/978-94-017-0699-5.
- Friedl S. Algebraic Topology [Electronic resource]. Regensburg: University of Regensburg; 2019. 326 p. Available from: http://www.mathematik.uni-regensburg.de/friedl/papers/2016_algebraic-top....
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