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15 September 2019 Topological classification of Morse–Smale diffeomorphisms on 3-manifolds
C. Bonatti, V. Grines, O. Pochinka
Duke Math. J. 168(13): 2507-2558 (15 September 2019). DOI: 10.1215/00127094-2019-0019


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(M3) of orientation-preserving Morse–Smale diffeomorphisms f on a smooth closed orientable 3-manifold M3. The complete topological invariant for a diffeomorphism fMS(M3) is the equivalence class of its scheme Sf 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.


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C. Bonatti. V. Grines. O. Pochinka. "Topological classification of Morse–Smale diffeomorphisms on 3-manifolds." Duke Math. J. 168 (13) 2507 - 2558, 15 September 2019.


Received: 18 October 2017; Revised: 24 February 2019; Published: 15 September 2019
First available in Project Euclid: 7 September 2019

zbMATH: 07131293
MathSciNet: MR4007599
Digital Object Identifier: 10.1215/00127094-2019-0019

Primary: 37B25
Secondary: 37D15 , 57M30

Keywords: Morse–Smale diffeomorphism , topological classification

Rights: Copyright © 2019 Duke University Press


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Vol.168 • No. 13 • 15 September 2019
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