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

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 $X$.