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Gromov’s Amenable Localization and Geodesic Flows

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Abstract

Let M be a compact 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 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 \(H_k(M; \mathbb R)\) and \(H_k(DM; \mathbb R)\), where \(DM = M\cup _{\partial M} M\) denotes the double of M. The norms here are the simplicial semi-norms in homology. The more complex the metric on M is, the more numerous the strata of SM and S(DM) are. It turns out that the normed homology spaces form obstructions to the existence of globally k-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 Poincaré duality operators on SM.

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Notes

  1. For a non-compact X, \(\Vert c\Vert _{\ell _1}\) may be infinite.

  2. In particular, traversally generic metrics are boundary generic and of the gradient type.

  3. Recall that the “quotient norm” of a given vector \(V\) in the quotient space is defined to be the infimum of the norms of all the vectors (in the original space) that represent \(V\).

  4. We have suppressed in (3.5) the dependence of homology and cohomology on the coefficients \({\mathsf {R}}\).

  5. This is the only place where the hypotheses that \(v^g\) is traversally generic (and not only boundary generic and traversing) seems to be important!

  6. See [1, 3] for an accurate definition of the stratified simplicial norm.

References

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Acknowledgements

I am grateful to the referee whose questioning improved the clarity of my arguments.

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Correspondence to Gabriel Katz.

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Katz, G. Gromov’s Amenable Localization and Geodesic Flows. Qual. Theory Dyn. Syst. 20, 19 (2021). https://doi.org/10.1007/s12346-021-00448-y

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