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Geometriae Dedicata

, Volume 180, Issue 1, pp 323–338 | Cite as

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

  • Hannah Alpert
  • Gabriel Katz
Original Paper

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.

Keywords

Traversing vector field Simplicial volume Simplicial norm Amenable group 

Mathematics Subject Classification (2010)

53C23 57N80 58K45 

Notes

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.

References

  1. 1.
    Alpert, H.: Using simplicial volume to count maximally broken Morse trajectories (2015). http://arxiv.org/pdf/1506.04789v1
  2. 2.
    Fujiwara, K., Manning, J.: Simplicial volume and fillings of hyperbolic manifolds. Algebr. Geom. Topol. 11(4), 2237–2264 (2011)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Golubitsky, M., Guillemin, V.: Stable mappings and their singularities. In: Halmos, P.R., Moore, C.C. (eds.) Graduate Texts in Mathematics, vol. 14. Springer, New York (1974)Google Scholar
  4. 4.
    Gromov, M.: Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56, 5–99 (1982)MathSciNetMATHGoogle Scholar
  5. 5.
    Gromov, M.: Singularities, expanders and topology of maps. I. Homology versus volume in the spaces of cycles. Geom. Funct. Anal. 19(3), 743–841 (2009)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar
  7. 7.
    Katz, G.: Convexity of Morse stratifications and gradient spines of 3-manifolds. JP J. Geom. Topol. 9(1), 1–119 (2009)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Katz, G.: Traversally generic & versal vector flows: semi-algebraic models of tangency to the boundary. Asian J. Math. (2014). http://arxiv.org/pdf/1407.1345v1

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.MITCambridgeUSA

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