[Paper Review] Morse-Smale 3-diffeomorphisms with saddles of the same unstable manifold dimension

Evgenii Mikhailovich Osenkov, O. V. Pochinka|arXiv (Cornell University)|Oct 12, 2023

Mathematical Dynamics and FractalsMathematics3 citations

TL;DR

This paper proves that any closed, connected 3-manifold admitting a Morse-Smale diffeomorphism in which all saddle points have the same unstable manifold dimension must be homeomorphic to the 3-sphere. The proof uses induction on the number of saddle points, leveraging topological gluing techniques and structural properties of unstable manifolds to show that such systems cannot exist on non-spherical 3-manifolds, generalizing Reeb’s classical result for source-sink systems.

ABSTRACT

In this paper, we consider a class of Morse-Smale diffeomorphisms defined on a closed 3-manifold (non-necessarily orientable) under the assumption that all their saddle points have the same dimension of the unstable manifolds. The simplest example of such diffeomorphisms is the well-known ``source-sink'' or ``north pole - south pole'' diffeomorphism, whose non-wandering set consists of exactly one source and one sink. Such systems, as Reeb showed back in 1946, can be realized only on the sphere. We generalize his result, namely, we show that diffeomorphisms from the considered class also can be defined only on the 3-sphere.

Motivation & Objective

• To generalize Reeb’s 1946 theorem on source-sink diffeomorphisms on the 3-sphere to a broader class of Morse-Smale diffeomorphisms.
• To investigate the topological constraints imposed by uniform unstable manifold dimension across all saddle points in 3-dimensional diffeomorphisms.
• To determine whether such diffeomorphisms can exist on 3-manifolds other than the 3-sphere.
• To establish that the only closed, connected 3-manifold supporting such a system is the 3-sphere.

Proposed method

• Induction on the number of saddle points in the non-wandering set of the diffeomorphism.
• Identification of a saddle point with no heteroclinic intersections using an ordering argument.
• Construction of a new manifold via gluing open subsets of 3-manifolds using diffeomorphisms lifted from a model map on the 3-sphere.
• Decomposition of the original manifold into a connected sum of two 3-manifolds via a topological gluing process.
• Application of the pasting lemma to ensure the resulting map is a smooth diffeomorphism on the connected sum.
• Use of topological embeddings and tameness properties to preserve manifold structure and ensure smoothness after gluing.

Figure 2: The space $\mathcal{L}_{\sigma}.$

Experimental results

Research questions

• RQ1 Can Morse-Smale diffeomorphisms with all saddle points having the same unstable manifold dimension exist on 3-manifolds other than the 3-sphere?
• RQ2 What topological restrictions does uniform unstable manifold dimension across saddles impose on the ambient 3-manifold?
• RQ3 Does the existence of such a diffeomorphism force the ambient manifold to be homeomorphic to the 3-sphere?
• RQ4 How does the structure of the non-wandering set constrain the global topology of the 3-manifold in this class?
• RQ5 Can the connected sum decomposition of the manifold be used to inductively classify such systems?

Key findings

• Any closed, connected 3-manifold admitting a Morse-Smale diffeomorphism where all saddle points have the same unstable manifold dimension is homeomorphic to the 3-sphere.
• The proof proceeds by induction on the number of saddle points, with the base case being the classical source-sink system on the 3-sphere.
• The existence of a saddle point with no heteroclinic intersections allows the construction of a connected sum decomposition of the manifold.
• Each component of the connected sum supports a diffeomorphism from the same class with fewer saddle points.
• By induction, each component must be homeomorphic to the 3-sphere, so the original manifold is a connected sum of 3-spheres, hence itself a 3-sphere.
• The result generalizes Reeb’s 1946 theorem, which applied only to systems with one source and one sink.

Figure 3: The orbits space of the sink basin