Topological classification of Morse-Smale diffeomorphisms on 3-manifolds

TL;DR

This paper provides a comprehensive topological classification of orientation-preserving Morse-Smale diffeomorphisms on 3-manifolds, addressing complex behaviors of separatrices and heteroclinic intersections that differ from lower-dimensional cases.

Contribution

It introduces a complete topological invariant, the scheme class, for Morse-Smale diffeomorphisms on 3-manifolds, extending previous classifications to include complex saddle behaviors.

Findings

Classified Morse-Smale diffeomorphisms using scheme classes.

Identified the role of heteroclinic curves and points in classification.

Extended classification to all orientation-preserving cases on 3-manifolds.

Abstract

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 possible "wild" behaviour of separatrices of saddle points. Another difference between Morse-Smale diffeomorphisms in dimension 3 from their surface analogues lies in the variety of heteroclinic intersections: a connected component of such an intersection may be not only a point as in the two-dimensional case, but also a curve, compact or non-compact. The problem of a 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 2006 by Ch. Bonatti, V. Grines, F. Laudenbach, V. Medvedev, E. Pecou, O. Pochinka. The present paper is devoted to a complete topological classification of the set $MS(M^3)$ of orientation preserving Morse-Smale diffeomorphisms $f$ given on smooth closed orientable 3-manifolds $M^3$. A complete topological invariant for a diffeomorphism $f\in MS(M^3)$ is an equivalent class of its scheme $S_f$, which contains an information on a periodic date and a topology of embedding of two-dimensional invariant manifolds of the saddle periodic points of $f$ into the ambient manifold.