Link as a complete invariant of Morse-Smale 3-diffeomorphisms

A. A. Nozdrinov ; A. I. Pochinka

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 25 (2023) no. 1, pp. 531-541

Résumé

In this paper we consider gradient-like Morse-Smale diffeomorphisms defined on the three-dimensional sphere $\mathbb S^3$. For such diffeomorphisms, a complete invariant of topological conjugacy was obtained in the works of C. Bonatti, V. Grines, V. Medvedev, E. Pecu. It is an equivalence class of a set of homotopically non-trivially embedded tori and Klein bottles embedded in some closed 3-manifold whose fundamental group admits an epimorphism to the group $\mathbb Z$. Such an invariant is called the scheme of the gradient-like diffeomorphism $f:\mathbb S^3\to\mathbb S^3$. We single out a class $G$ of diffeomorphisms whose complete invariant is a topologically simpler object, namely, the link of essential knots in the manifold $\mathbb S^2\times\mathbb S^1$. The diffeomorphisms under consideration are determined by the fact that their non-wandering set contains a unique source, and the closures of stable saddle point manifolds bound three-dimensional balls with pairwise disjoint interiors. We prove that, in addition to the closure of these balls, a diffeomorphism of the class $G$ contains exactly one nonwandering point, which is a fixed sink. It is established that the total invariant of topological conjugacy of class $G$ diffeomorphisms is the space of orbits of unstable saddle separatrices in the basin of this sink. It is shown that the space of orbits is a link of non-contractible knots in the manifold $\mathbb{S}^2 \times \mathbb{S}^1$ and that the equivalence of links is tantamount to the equivalence of schemes. We also provide a realization of diffeomorphisms of the considered class along an arbitrary link consisting of essential nodes in the manifold $\mathbb{S}^2 \times \mathbb{S}^1$.

Détail

Keywords: Morse-Smale diffeomorphism, knot, link, topological conjugacy
Mots-clés : invariant.