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Using simplicial volume to count multi-tangent trajectories of traversing vector fields

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Abstract

For a non-vanishing gradient-like vector field on a compact manifold \(X^{n+1}\) with boundary, a discrete set of trajectories may be tangent to the boundary with reduced multiplicity n, which is the maximum possible. (Among them are trajectories that are tangent to \(\partial X\) exactly n times.) We prove a lower bound on the number of such trajectories in terms of the simplicial volume of X by adapting methods of Gromov, in particular his “amenable reduction lemma”. We apply these bounds to vector fields on hyperbolic manifolds.

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Acknowledgments

The authors would like to thank Larry Guth (Hannah’s advisor) for initiating the collaboration, actively supervising most of our meetings, and improving the exposition in the paper. The authors also thank the referee for clarifying wording and suggesting future directions.

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Authors and Affiliations

  1. MIT, Cambridge, MA, 02139, USA

    Hannah Alpert & Gabriel Katz

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  1. Hannah Alpert
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  2. Gabriel Katz
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Correspondence to Hannah Alpert.

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Alpert, H., Katz, G. Using simplicial volume to count multi-tangent trajectories of traversing vector fields. Geom Dedicata 180, 323–338 (2016). https://doi.org/10.1007/s10711-015-0104-6

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  • DOI: https://doi.org/10.1007/s10711-015-0104-6

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