ISSN 2079-6900 (Print) 
ISSN 2587-7496 (Online)

Middle Volga Mathematical Society Journal

DOI 10.15507/2079-6900.25.202301.531-541

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 37D15

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

A. A. Nozdrinov, A. I. Pochinka

Higher School of Economics (Nizhny Novgorod, Russian Federation)

Abstract. 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 $\mathbb{S}^2 \times \mathbb{S}^1$.

Key Words: Morse-Smale diffeomorphism, knot, link, topological conjugacy, invariant

For citation: A. A. Nozdrinov, A. I. Pochinka. Link as a complete invariant of Morse-Smale 3-diffeomorphisms. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 25:1(2023), 531–541. DOI:

Submitted: 05.12.2022; Revised: 10.01.2023; Accepted: 15.02.2023

Information about the authors:

Alexey A. Nozdrinov, Post-graduate student, Department of Fundamental Mathematics, National Research University «Higher School of Economics» (25/12 Bolshaya Pecherskaya St., Nizhny Novgorod 603155, Russia), ORCID:,

Arseniy I. Pochinka, Student of the Faculty of Informatics, Mathematics and Computer Science, National Research University «Higher School of Economics» (25/12 Bolshaya Pecherskaya St., Nizhny Novgorod 603155, Russia),

All authors have read and approved the final manuscript.

Conflict of interest: The authors declare no conflict of interest.

Creative Commons Attribution 4.0 International License This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.