Using simplicial volume to count multi-tangent trajectories of traversing vector fields

Abstract

For a non-vanishing gradient-like vector field on a compact manifold X[superscript n+1] with boundary, a discrete set of trajectories may be tangent to the boundary with reduced multiplicity n, which is the maximum possible. (Among them are trajectories that are tangent to ∂X exactly n times.) We prove a lower bound on the number of such trajectories in terms of the simplicial volume of X by adapting methods of Gromov, in particular his “amenable reduction lemma”. We apply these bounds to vector fields on hyperbolic manifolds.