Topological classification of Morse–Smale diffeomorphisms on 3-manifolds

Abstract

The topological classification of even the simplest Morse–Smale diffeomorphisms on $3$-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies primarily in the possibility of “wild” behavior of separatrices of saddle points. Another difference between Morse–Smale diffeomorphisms in dimension $3$ and their surface analogues lies in the variety of heteroclinic intersections: a connected component of such an intersection may not be only a point, as in the $2$-dimensional case, but also a curve, compact or noncompact. The problem of topological classification of Morse–Smale cascades on $3$-manifolds either without heteroclinic points (gradient-like cascades) or without heteroclinic curves was solved in a series of papers from 2000 to 2016 by C. Bonatti, V. Grines, F. Laudenbach, V. Medvedev, E. Pecou, and O. Pochinka. The present article is devoted to completing the topological classification of the set $MS\left(M^3\right)$ of orientation-preserving Morse–Smale diffeomorphisms $f$ on a smooth closed orientable $3$-manifold $M^3$. The complete topological invariant for a diffeomorphism $f\in MS\left(M^3\right)$ is the equivalence class of its scheme $S_f$ which contains information on the periodic data and the topology of embedding of $2$-dimensional invariant manifolds of the saddle periodic points of $f$ into the ambient manifold.