Classification of nonsingular $4$-flows without heteroclinic intersections

A regular topological flow on a closed $n$-manifold is a flow whose chain-recurrent set consists of finitely many topologically hyperbolic fixed points and periodic orbits. Such a flow is said to be nonsingular if its chain-recurrent set contains no fixed points. The topological equivalence of low-dimensional nonsingular flows was considered in a number of papers under assumptions of various generality. Classification results for dimension higher than 3 are few. However, it is known that there are four-dimensional nonsingular flows with wildly embedded invariant saddle manifolds. In the paper, a class of nonsingular flows without heteroclinic intersections on closed orientable 4-manifolds is considered. It is shown that a complete invariant for such flows is a scheme consisting of two-dimensional tori and Klein bottles embedded in a closed 3-manifold. A class of admissible schemes is defined, and for every admissible scheme, a standard representative in the class of flows under consideration is constructed.