“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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References
Abbondandolo, A., Schwarz, M.: On the Floer homology of cotangent bundles. Commun. Pure Appl. Math. 59(2), 254–316 (2006)
Abbondandolo, A., Schwarz, M.: Corrigendum: on the Floer homology of cotangent bundles. Commun. Pure Appl. Math. 67(4), 670–691 (2014)
Abouzaid, M.: Symplectic cohomology and Viterbo’s theorem. In: Free Loop Spaces in Geometry and Topology, vol. 24 of IRMA Lectures in Mathematics and Theoretical Physics. EMS Publishing House (2015)
Albers, P., Frauenfelder, U., van Koert, O., Paternain, G.P.: Contact geometry of the restricted three-body problem. Commun. Pure Appl. Math. 65(2), 229–263 (2012)
Bangert, V.: Existence of a complex line in tame almost complex tori. Duke Math. J. 94(1), 29–40 (1998)
Biolley, A.-L.: Cohomologie de Floer, hyperbolicités symplectique et pseudocomplexe [Floer cohomology, symplectic and pseudocomplex hyperbolicities]. PhD thesis, École Polytechnique (2003). Available online at https://tel.archives-ouvertes.fr/pastel-00000702/document
Biolley, A.-L.: Floer homology, symplectic and complex hyperbolicities (2004). arXiv:math/0404551
Bourgeois, F., Eliashberg, Y., Hofer, H., Wysocki, K., Zehnder, E.: Compactness results in symplectic field theory. Geom. Topol. 7, 799–888 (2003)
Bourgeois, F., Oancea, A.: An exact sequence for contact- and symplectic homology. Invent. Math. 175(3), 611–680 (2009)
Cartan, H., Leray, J.: Relations entre anneaux d’homologie et groupes de Poincaré. In: Topologie algébrique, Colloques Internationaux du Centre National de la Recherche Scientifique, no. 12, pp. 83–85. Centre de la Recherche Scientifique, Paris (1949)
Cieliebak, K.: Handle attaching in symplectic homology and the chord conjecture. J. Eur. Math. Soc. (JEMS) 4(2), 115–142 (2002)
Cieliebak, K., Eliashberg, Y.: From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds, vol. 59 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI (2012)
Cieliebak, K., Hofer, H., Latschev, J., Schlenk, F.: Quantitative symplectic geometry. In: Dynamics, Ergodic Theory, and Geometry, vol. 54 of Math. Sci. Res. Inst. Publ., pp. 1–44. Cambridge Univ. Press, Cambridge (2007)
Eilenberg, S., Mac Lane, S.: On the groups of \(H(\Pi, n)\). I. Ann. Math. 2(58), 55–106 (1953)
Greenberg, M.J., Harper, J.R.: Algebraic topology. A first course, vol. 58 of Mathematics Lecture Note Series. Benjamin/Cummings Publishing Co. Inc., Reading, Mass. (1981)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Hofer, H., Zehnder, E.: A New Capacity for Symplectic Manifolds. Analysis. et cetera, pp. 405–427. Academic Press, Boston (1990)
Hofer, H., Zehnder, E.: Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel (1994)
Hopf, H.: Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen. Ann. Math. 2(42), 22–52 (1941)
Hopf, H.: Fundamentalgruppe und zweite Bettische Gruppe. Comment. Math. Helv. 14, 257–309 (1942)
Irie, K.: Hofer-Zehnder capacity of unit disk cotangent bundles and the loop product. J. Eur. Math. Soc. (JEMS) 16(11), 2477–2497 (2014)
Kang, J.: Symplectic homology of displaceable Liouville domains and leafwise intersection points. Geom. Dedicata 170, 135–142 (2014)
Labourie, F.: Lectures on Representations of Surface Groups, vol. 17 of Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich (2013)
Lang, S.: Introduction to Complex Hyperbolic Spaces. Springer-Verlag, New York (1987)
Liu, C., Wang, Q.: Symmetrical symplectic capacity with applications. Discrete Contin. Dyn. Syst. 32(6), 2253–2270 (2012)
Massot, P., Niederkrüger, K., Wendl, C.: Weak and strong fillability of higher dimensional contact manifolds. Invent. Math. 192(2), 287–373 (2013)
McCleary, J.: A User’s Guide to Spectral Sequences, vol. 58 of Cambridge Studies in Advanced Mathematics, 2nd edn. Cambridge University Press, Cambridge (2001)
Ritter, A.: Novikov-symplectic cohomology and exact Lagrangian embeddings. Geom. Topol. 13(2), 943–978 (2009)
Ritter, A.F.: Deformations of symplectic cohomology and exact Lagrangians in ALE spaces. Geom. Funct. Anal. 20(3), 779–816 (2010)
Ritter, A.F.: Topological quantum field theory structure on symplectic cohomology. J. Topol. 6(2), 391–489 (2013)
Salamon, D., Weber, J.: Floer homology and the heat flow. Geom. Funct. Anal. 16(5), 1050–1138 (2006)
Seidel, P.: A remark on the symplectic cohomology of cotangent bundles, after Kragh. Informal note (2010)
Steenrod, N.E.: Homology with local coefficients. Ann. Math. 2(44), 610–627 (1943)
Viterbo, C.: Functors and computations in Floer homology with applications. II. Prépublication Orsay 98-15. Available online at http://www.math.ens.fr/~viterbo/FCFH.II.2003.pdf (1998)
Viterbo, C.: Functors and computations in Floer homology with applications. I. Geom. Funct. Anal. 9(5), 985–1033 (1999)
Weinstein, A.: On the hypotheses of Rabinowitz’ periodic orbit theorems. J. Differ. Equ. 33(3), 353–358 (1979)
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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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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