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 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.

Counting maximally broken Morse trajectories on aspherical manifolds

“…The following statement is the integral analogue of [1, Lemma 4, Corollary 5] and of [2,Lemma 4]. Proof.…”

“…In this section we prove the integral oriented version of the amenable reduction lemma [1, Lemma 4, Corollary 5], [2,Lemma 4]. It will allow us to estimate the integral oriented norm of cycles in terms of some conditions on the shape of the simplices composing them.…”

Using simplicial volume to count maximally broken Morse trajectories

“…Acknowledgments. The main theorem was conjectured by Gabriel Katz related to our collaboration on the paper [AK15]. I would like to thank him and my advisor Larry Guth for conversations about that project.…”

“…To prove this Lemma 3, we prove the Generalized Localization Lemma (Lemma 6), for which Lemma 3 and the Localization Lemma of [Gro09] are special cases. The proof is very similar to the proof of the Localization Lemma (given with more detail in [AK15]), and it relies on the Amenable Reduction Lemma of [Gro09]. There is one main way in which the proof of the Generalized Localization Lemma goes beyond what is needed for Lemma 3.…”

“…In the papers [Gro09] and [AK15], the stratified simplicial norm of a homology class is the infimal sum of coefficients of cycles that represent the homology class and satisfy the conditions above. In this paper, we only count the simplices for which every vertex is in a different stratum; these are called essential simplices, a term which is defined in a slightly more general context later in the paper.…”