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

Danila Shubin

doi:10.18500/0869-6632-2021-29-6-863-868

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.

Highlights

Topology of ambient manifolds of non-singular Morse – Smale flows with three periodic orbits
Nonsingular Morse – Smale flows of n-manifolds with attractor-repeller dynamics [Electronic resource] // arXiv: 2105.13110. arXiv Preprint, 2021. 17 p
Notes on Basic 3-Manifold Topology [Electronic resource]

Summary

Introduction

Topology of ambient manifolds of non-singular Morse – Smale flows with three periodic orbits. Что в трубчатой окрестности периодических орбит НМС-поток топологически сопряжён с надстройкой над диффеоморфизмом a : R2 → R2 вида a(x1, x2) ↦→ (λ1x1, λ2x2); в случае, когда λ1 > 0, λ2 > 0, соответствующая орбита получится притягивающей, когда λ1 > 0, λ2 < 0 – седловой, λ1 < 0, λ2 < 0 – отталкивающей. В силу того, что несущее многообразие любого НМС-потока является Согласно [2], НМС-потоки с нескрученными орбитами, не более, чем одна из которых является седловой, тоже допускают только линзовые пространства

Objectives

Results

Conclusion