Complexity of Shadows & Traversing Flows in Terms of the Simplicial Volume

TL;DR

This paper investigates the complexity of flow-generated stratifications on manifolds with boundary using Gromov's simplicial volume, providing universal bounds and obstructions related to the fundamental group's homology.

Contribution

It combines Gromov's amenable localization with Poincaré duality to establish bounds on flow stratifications based on the simplicial volume of the doubled manifold.

Findings

Lower bounds on the number of flow strata components

Obstructions to k-convex flows based on homology norms

Connections between simplicial volume and flow complexity

Abstract

We combine Gromov's amenable localization technique with the Poincar\'{e} duality to study the traversally generic vector flows on smooth compact manifolds $X$ with boundary. Such flows generate well-understood stratifications of $X$ by the trajectories that are tangent to the boundary in a particular canonical fashion. Specifically, we get lower estimates of the numbers of connected components of these flow-generated strata of any given codimension. These universal bounds are basically expressed in terms of the normed homology of the fundamental groups $\pi_1(D(X))$, where $D(X)$ denotes the double of $X$. The norm here is the Gromov simplicial semi-norm in homology. It turns out that some close relatives of the normed spaces $H_\ast(D(X); \R)$ form obstructions to the existence of $k$-convex traversally generic vector flows on $X$.