What question did this study set out to answer?

The aim is to classify gradient-like flows on closed four-dimensional manifolds based on their topological properties.

June 20, 2026

Classification of gradient-like flows without heteroclinic intersections on four-dimensional manifolds

Puntos clave

The aim is to classify gradient-like flows on closed four-dimensional manifolds based on their topological properties.
Defined gradient-like flows lacking heteroclinic intersections on four-dimensional manifolds.
Reduced classification to the problem of distinguishing isomorphic framed graphs.
Developed an algorithm for generating canonical representatives of topological equivalence classes.
Demonstrated that topological equivalence of flows corresponds to isomorphism of their framed graphs.
Identified combinatorial invariants that facilitate classification of flow behavior.
Described the topology of the ambient manifold through these graph relationships.

Resumen

We consider a class of gradient-like flows without heteroclinic intersections defined on closed manifolds of dimension four. We show that, for such flows, the problem of complete topological classification can be reduced to the combinatorial problem of distinguishing special framed graphs describing the mutual arrangement of invariant manifolds and the action of the flow on a wandering set. Namely, flows are topologically equivalent if and only if their framed graphs are isomorphic. Using the obtained combinatorial invariant, we provide an algorithm for constructing a canonical representative of each class of topological equivalence and describe the topology of the ambient manifold.

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