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

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.