Minimizing the number of heteroclinic curves of a 3-diffeomorphism with fixed points with pairwise different Morse indices

We consider Morse–Smale $3$-diffeomorphisms whose nonwandering set consists of exactly four fixed points with pairwise distinct Morse indices. The question of which closed $3$-manifolds admit such diffeomorphisms remains open. The set of these manifolds is known to contain all lens spaces. Moreover, on all manifolds except $\mathbb{S}^2\times\mathbb{S}^1$, such diffeomorphisms have heteroclinic curves. We prove that the number of heteroclinic diffeomorphism curves on a given manifold can be minimized by reducing to finitely many noncompact heteroclinic curves that are orientable intersections of invariant saddle manifolds. This result paves the way to an exhaustive description of closed $3$-manifolds that the diffeomorphisms in question.