A criterion to detect a nontrivial homology of an invariant set of a flow in R³

Consider a flow in R³ and let K be the biggest invariant subset of some compact region of interest N ⊆ R³. The set K is often not computable, but the way the flow crosses the boundary of N can provide indirect information about it. For example, classical tools such as Wa\.{z}ewski's principle or the Poincar\'e-Hopf theorem can be used to detect whether K is nonempty or contains rest points, respectively. We present a criterion that can establish whether K has a nontrivial homology by looking at the subset of the boundary of N along which the flow is tangent to N. We prove that the criterion is as sharp as possible with the information it uses as an input. We also show that it is algorithmically checkable.