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

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We combine Gromov's amenable localization technique with the Poincaré 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 π₁(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(D(X);) form obstructions to the existence of k-convex traversally generic vector flows on X.