ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


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

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
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Language: 
Russian
Article type: 
Short communication
UDC: 
530.182

Topology of ambient manifolds of non-singular Morse – Smale flows with three periodic orbits

Autors: 
Shubin Danila Denisovich, National Research University "Higher School of Economics"
Abstract: 

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.

Acknowledgments: 
The results were prepared in the course of the study (No. 21-04-004) within the framework of the Program “Scientific Foundation of the National Research University Higher School of Economics (HSE)” in 2021 – 2022. The author thanks O. V. Pochinka for setting the task and fruitful discussions
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Received: 
29.05.2021
Accepted: 
27.07.2021
Published: 
30.11.2021