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

Abstract

In this paper we consider gradient-like Morse-Smale diffeomorphisms defined on the three-dimensional sphere S3 . 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 Z . Such an invariant is called the scheme of the gradient-like diffeomorphism f:S3→S3 . 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 S2×S1 . 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 S2×S1 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 S2×S1 .