Convexity of Morse Stratifications and Spines of 3-manifolds

The gradient fields v of nonsingular functions f on compact 3folds X with boundary are used to generate their spines K(f, v). We study the transformations of K(f, v) that are induced by deformations of the data (f, v). We link the Matveev complexity of c(X) with counting the trajectories of the v-flow that are tangent to the boundary ∂X at a pair of distinct points (call them double-tangent trajectories). Let MC(X) be the minimum number of such trajectories, minimum being taken over all nonsingular v’s. We call MC(X) the Morse complexity of X. Next, we prove that there are only finitely many irreducible and boundary irreducible X with no essential annuli of bounded Morse complexity. In particular, there exists only finitely many hyperbolic manifolds X with bounded MC(X). For such X, their normalized hyperbolic volume gives an upper bound of MC(X). Also, if an irreducible and boundary irreducible X with no essential annuli admits a nonsingular gradient flow with no double-tangent trajectories, then X is a standard ball. All these and many other results of the paper rely on a careful study of the stratified geometry of ∂X relative to the v-flow. It is characterized by failure of ∂X to be convex with respect to a generic flow v. It turns out, that convexity or its lack have profound influence on the topology of X. This phenomenon is in the focus of the paper.