  Let $M$ be a compact smooth Riemannian $n$-manifold with boundary. We combine
Gromov's amenable localization technique with the Poincar\'{e} duality to study
the {\sf traversally generic} geodesic flows on $SM$, the space of the
spherical tangent bundle. Such flows generate stratifications of $SM$, governed
by rich universal combinatorics. The stratification reflects the ways in which
the flow trajectories are tangent to the boundary $\partial(SM)$. Specifically,
we get lower estimates of the numbers of connected components of these
flow-generated strata of any given codimension $k$ in terms of the normed
homology $H_k(M; \mathbf R)$ and $H_k(DM; \mathbf R)$, where $DM =
M\cup_{\partial M} M$ denotes the double of $M$. The norms here are the {\sf
simplicial semi-norms} in homology. The more complex the metric on $M$ is, the
more numerous the strata of $SM$ and $S(DM)$ are. %So one may regard our
estimates as analogues of the Morse inequalities for the geodesics on manifolds
with boundary. It turns out that the normed homology spaces form obstructions
to the existence of globally $k$-{\sf convex} traversally generic metrics on
$M$. We also prove that knowing the geodesic scattering map on $M$ makes it
possible to reconstruct the stratified topological type of the space of
geodesics, as well as the amenably localized Poincar\'{e} duality operators on
$SM$.
