Title: Applying Gromov's Amenable Localization to Geodesic Flows,
Abstract: Let $M$ be a compact connected smooth Riemannian $n$-manifold with boundary.
We combine Gromov's amenable localization technique with the Poincaré
duality to study the 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 geodesic flow trajectories interact with the boundary $\d(SM)$.
Specifically, we get lower estimates of the numbers of connected components of
these flow-generated strata of any given codimension $k$. These universal
bounds are expressed in terms of the normed homology $H_k(M; \R)$ and $H_k(DM;
\R)$, where $DM = M\cup_{\d M} M$ denotes the double of $M$. The norms here are
the Gromov 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 some close relatives of the normed homology spaces form
obstructions to the existence of globally $k$-convex traversally generic
metrics on $M$.