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Local systems on the free loop space and finiteness of the Hofer-Zehnder capacity

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“Local coefficients bring an extra level of complication that one tries to avoid whenever possible.”

— Allen Hatcher, Algebraic Topology

Abstract

In this article we examine under which conditions symplectic homology with local coefficients of a unit disk bundle \(D^*M\) vanishes. For instance this is the case if the Hurewicz map \(\pi _2(M)\rightarrow H_2(M;{\mathbb {Z}})\) is nonzero. As an application we prove finiteness of the \(\pi _1\)-sensitive Hofer-Zehnder capacity of unit disk bundles in these cases. We also prove uniruledness for such cotangent bundles. Moreover, we find an obstruction to the existence of H-space structures on general topological spaces, formulated in terms of local systems.

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Acknowledgments

The third author wishes to thank Nancy Hingston for inspiring comments, and the College of New Jersey for hospitality during the summer 2015.

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Authors and Affiliations

  1. Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstrasse 62, 48149, Münster, Germany

    Peter Albers

  2. Universität Augsburg, Universitätsstrasse 14, 86159, Augsburg, Germany

    Urs Frauenfelder

  3. UPMC Univ Paris 06, UMR 7586, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Universités, Case 247, 4 place Jussieu, 75005, Paris, France

    Alexandru Oancea

Authors
  1. Peter Albers
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  2. Urs Frauenfelder
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  3. Alexandru Oancea
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Corresponding author

Correspondence to Peter Albers.

Additional information

P. Albers partially funded by SFB 878. A. Oancea partially funded by the European Research Council, StG-259118-STEIN.

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Albers, P., Frauenfelder, U. & Oancea, A. Local systems on the free loop space and finiteness of the Hofer-Zehnder capacity. Math. Ann. 367, 1403–1428 (2017). https://doi.org/10.1007/s00208-016-1401-6

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  • DOI: https://doi.org/10.1007/s00208-016-1401-6

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