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Automatic transversality in contact homology I: regularity

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Abstract

This paper helps to clarify the status of cylindrical contact homology, a conjectured contact invariant introduced by Eliashberg, Givental, and Hofer in 2000. We explain how heuristic arguments fail to yield a well-defined homological invariant in the presence of multiply covered curves. We then introduce a large subclass of dynamically convex contact forms in dimension 3, termed dynamically separated, and demonstrate automatic transversality holds, thereby allowing us to define the desired chain complex. The Reeb orbits of dynamically separated contact forms satisfy a uniform growth condition on their Conley–Zehnder index under iteration, typically up to large action; see Definition 1.15. These contact forms arise naturally as perturbations of Morse–Bott contact forms such as those associated to \(S^1\)-bundles. In subsequent work, we give a direct proof of invariance for this subclass and, when further proportionality holds between the index and action, powerful geometric computations in a wide variety of examples.

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Notes

  1. Nondegeneracy of the contact form means that all the Reeb orbits associated to the Reeb vector field \(R_\alpha \) are nondegenerate; see below.

  2. Simple is synonymous with embedded in other literature.

  3. The symplectization of \((M,\alpha )\) is given by the manifold \(\mathbb {R}\times M\) and symplectic form

    $$\begin{aligned} \omega = e^\tau (d\alpha - \alpha \wedge d\tau ) = d (e^\tau \alpha ). \end{aligned}$$

    Here \(\tau \) is the coordinate on \(\mathbb {R}\) and \(\alpha \) is understood to be the 1-form on \(\mathbb {R}\times M\), obtained via pullback under the projection \(\mathbb {R}\times M \rightarrow M\) and \({J}\) is an \(\alpha \) -compatible almost complex structure; see Definition 2.1.

  4. Note that when considering genus 0 domains, we can alternatively fix a standard \(j_0\) on \(\Sigma :=S^2\) and only keep track of the location of the punctures.

  5. Note that \((S^2 {\setminus } \{x\}, j_0)\) is biholomorphic to \((\mathbb {C}, j_0)\), hence the terminology plane.

  6. The different ways of defining differential is rather ambiguous in older literature.

  7. Co-oriented means that there exists a globally defined \(\alpha \in \Omega ^1(M)\) such that \(\xi = \ker \alpha \).

  8. From [60] Theorem 0, regularity achieved in Condition (A) induces an orbifold structure. This is upgraded to a manifold structure in Lemma 4.2(b) of [33] and mentioned in the text after this remark.

  9. We use the convention \(\omega (X_H, \cdot ) = dH.\)

  10. See Definition 4.13 for the precise definition of a building.

  11. In [15], they only require the exclusion of contractible orbits of degree 1 in order to prove \(\partial ^2=0\) in Proposition 1.9.1. The exclusion of the contractible orbits of degree 0 and \(-1\) is necessary to obtain the existence of a chain map and the homotopy of homotopies respectively when proving invariance.

  12. While the result is expected to be true, there appear to be problems with some details of Dragnev’s proof in [14], particularly regarding the claimed existence of certain rather special cut-off functions on page 757. Rather than rely on [14] directly, we appeal to [25, 37, 60]. It is not immediately clear whether or not the proof has a simple fix, however a complete proof should follow by modifying arguments in [2, 60, 61]. A detailed proof of the desired result in the more general setting of stable Hamiltonian structures appears in Wendl’s blog: https://symplecticfieldtheorist.wordpress.com/2014/11/27/generic-transversality-in-symplectizations-part-1/, https://symplecticfieldtheorist.wordpress.com/2014/11/27/generic-transversality-in-symplectizations-part-2/, https://symplecticfieldtheorist.wordpress.com/2014/11/28/an-easy-proof-of-the-pi-du-lemma/.

  13. Otherwise we must keep track of a \(-1\) associated to each building component arising from the compactification \(\overline{\mathcal {M}}^{{J}}(x;z)\), which is annoying.

  14. The smooth one parameter family of diffeomorphic contact manifolds can be obtained via the flow of the Liouville vector field \(u \frac{\partial }{\partial u} + v \frac{\partial }{\partial v}\) on \(\mathbb {C}^2 {\setminus } \{ 0 \}\), which has been appropriately reparametrized so that at time 0 one starts from \(E_-\) and lands on \(E_+\) at time 1.

  15. Isomorphism classes of symplectic vector bundles are in a 1-1 correspondence with complex vector bundles. As a result \((\xi , \bar{J})\) is frequently said to be a complex vector bundle, and one suppresses the ‘almost’ in almost complex structure despite the fact that we do not require elements of \(\mathcal {J}\) to be integrable.

  16. In earlier literature these were sometimes referred to as finite energy parametrized surfaces.

  17. This includes unbranched multiply covered cylinders.

  18. The proof of Theorem 3.7 is modeled on the blog post by Chris Wendl, https://symplecticfieldtheorist.wordpress.com/2014/12/10/somewhere-injective-vs-multiply-covered/. This was in turn modeled on the “shorter” proof of Proposition 2.5.1 [43]. The “longer” proof of Proposition 2.5.1 was adapted to give a proof of Theorem 3.7 for finite energy planes, subsuming the 11-page Appendix of [24]. This longer proof extends to asymptotically cylindrical pseudoholomorphic curves we are considering, but this paper is already long enough.

  19. This theorem holds for almost complex manifolds with noncompact cylindrical ends approaching codimension 1 manifolds \(M_\pm \) equipped with stable Hamiltonian structures, and allows for Morse–Bott orbits.

  20. This is in the case that the orbits are nondegenerate. If the orbits are Morse–Bott, the definition is more complicated; see [60].

  21. Any somewhere injective curve \(u\) of index \(\le 1\) is automatically immersed from Proposition 3.17 thus by Corollary 3.5 we know that \(\text{ ind }(u) \ge 1\), since \({J}\) has been chosen generically.

  22. Note b could be 0 since the result holds if \(u_1\) does not have any branch points.

References

  1. Bourgeois, F.: A Morse–Bott approach to contact homology. Ph.D. thesis, Stanford University (2002)

  2. Bourgeois, F.: Contact homology and homotopy groups of the space of contact structures. Math. Res. Lett. 13(1), 71–85 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bourgeois, F.: A survey of contact homology. In: New Perspectives and Challenges in Symplectic Field Theory, pp. 45–71. CRM Proceedings and Lecture Notes, vol. 49. American Mathematical Society, Providence (2009)

  4. Bourgeois, F., Cieliebak, K., Ekholm, T.: A note on Reeb dynamics on the tight 3-sphere. J. Mod. Dyn. 1(4), 597–613 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bourgeois, F., Colin, V.: Homologie de contact des variétés toroïdales. Geom. Topol. 9, 299–313 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bourgeois, F., Eliashberg, Y., Hofer, H., Wysocki, K., Zehnder, E.: Compactness results in symplectic field theory. Geom. Topol. 7, 799–888 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bourgeois, F., Mohnke, K.: Coherent orientations in symplectic field theory. Math. Z. 248(1), 123–146 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bourgeois, F., Oancea, A.: An exact sequence for contact and symplectic homology. Invent. Math. 175(3), 611–680 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bourgeois, F., Oancea, A.: Symplectic homology, autonomous Hamiltonians, and Morse–Bott moduli spaces. Duke Math. J. 146, 71–174 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bourgeois, F., Oancea, A.: \(S^1\)-equivariant symplectic homology and linearized contact homology. arXiv:1212.3731

  11. Chen, B., Li, A.-M., Wang, B.-L.: Virtual neighborhood technique for pseudo-holomorphic spheres. arXiv:1306.3276

  12. Colin, V., Honda, K.: Constructions contrôlées de champs de Reeb et applications. Geom. Topol. 9, 2193–2226 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Colin, V., Honda, K.: Reeb vector fields and open book decompositions. J. Eur. Math. Soc. (JEMS) 15(2), 443–507 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dragnev, D.: Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations. Commun. Pure Appl. Math. 57, 726–763 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Eliashberg, Y., Givental, A., Hofer, H.: Introduction to symplectic field theory. Geom. Funct. Anal. (2000) (special volume, Part II, pp. 560–673, GAFA, 2000)

  16. Floer, A.: The unregularized gradient flow of the symplectic action. Commun. Pure Appl. Math. 41, 775–813 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  17. Floer, A., Hofer, H.: Coherent orientations for periodic orbit problems in symplectic geometry. Math. Z. 212(1), 13–38 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. Floer, A., Hofer, H., Salamon, D.: Transversality in elliptic Morse theory for the symplectic action. Duke Math. J. 80(1), 251–292 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Technical detail on Kuranishi structure and virtual fundamental chain. arXiv:1209.4410

  20. Hofer, H.: Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. Invent. Math. 114, 515–563 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hofer, H., Kriener, M.: Holomorphic curves in contact dynamics. Proc. Symp. Pure Math. 66, 77–131 (1999)

    MathSciNet  Google Scholar 

  22. Hofer, H., Wysocki, K., Zehnder, E.: A characterization of the tight 3-sphere. II. Commun. Pure Appl. Math. 52(9), 1139–1177 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hofer, H., Wysocki, K., Zehnder, E.: Properties of pseudoholomorphic curves in symplectisations I. Asymptotics. Ann. Inst. H. Poincaré Anal. Nonlinéaire 13(3), 337–379 (1996)

  24. Hofer, H., Wysocki, K., Zehnder, E.: Properties of pseudo-holomorphic curves in symplectisations. II. Embedding controls and algebraic invariants. Geom. Funct. Anal. 5(2), 270–328 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hofer, H., Wysocki, K., Zehnder, E.: Properties of pseudoholomorphic curves in symplectizations III: Fredholm Theory. In: Topics in Nonlinear Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol. 35. Birkhäuser, Boston (1999)

  26. Hofer, H., Wysocki, K., Zehnder, E.: Properties of pseudoholomorphic curves in symplectisation. IV. Asymptotics with degeneracies. In: Contact and Symplectic Geometry (Cambridge, 1994), pp. 78–117. Publications of the Newton Institute, vol. 8, Cambridge University Press, Cambridge (1996)

  27. Hofer, H., Wysocki, K., Zehnder, E.: Finite energy cylinders of small area. Ergod. Theory Dyn. Syst. 22(5), 1451–1486 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  28. Hofer, H., Wysocki, K., Zehnder, E.: Sc-smoothness, retractions and new models for smooth spaces. Discrete Contin. Dyn. Syst. 28 (2), 665–788 (2010)

  29. Hofer, H., Wysocki, K., Zehnder, E.: Integration theory on the zero set of polyfold Fredholm sections. Math. Ann. 336(1), 139–198 (2010)

    Article  MathSciNet  Google Scholar 

  30. Hofer, H., Wysocki, K., Zehnder, E.: Applications of polyfold theory I: Gromov–Witten theory (2011). arXiv:1107.2097

  31. Hutchings, M.: Taubes’ proof of the Weinstein conjecture. Bull. AMS 47, 73–125 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  32. Hutchings, M.: Lecture notes on embedded contact homology. Contact and Symplectic Topology, pp. 389–484. In: Bolyai Society Mathematical Studies, vol. 26. Springer, New York (2014)

  33. Hutchings, M., Nelson, J.: Cylindrical contact homology for dynamically convex contact forms in three dimensions. J. Symplectic Geom.arXiv:1407.2898 (to appear)

  34. Hutchings, M., Nelson, J.: An integral lift of cylindrical contact homology for contact forms without contractible Reeb orbits (in preparation)

  35. Hutchings, M., Nelson, J.: Invariance and an integral lift of cylindrical contact homology for dynamically convex contact forms (in preparation)

  36. Hutchings, M., Taubes, C.: Gluing pseudoholomorphic curves along branched covered cylinders I. J. Symplectic Geom. 5(1), 43–137 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  37. Hutchings, M., Taubes, C.: Gluing pseudoholomorphic curves along branched covered cylinders II. J. Symplectic Geom. 7(1), 29–133 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  38. Ionel, E., Parker, T.: A natural Gromov–Witten fundamental class. arXiv:1302.3472

  39. van Koert, O.: Contact homology of Brieskorn manifolds. Forum Math. 20(2), 317–339 (2008)

    MATH  MathSciNet  Google Scholar 

  40. Lazzarini, L.: Relative frames on J-holomorphic curves. J. Fixed Point Theory Appl. 9(2), 213–256 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  41. Long, Y.: Index Theory for Symplectic Paths with Applications. Birkhäuser, Basel (2002)

  42. McDuff, D., Salamon, D.: Introduction to Symplectic Topology. Oxford University Press, Oxford (1995)

  43. McDuff, D., Salamon, D.: \(J\)-Holomorphic Curves and Symplectic Topology. AMS Colloquium Publications, New York (2004)

  44. McDuff, D., Wehrheim, K.: Smooth Kuranishi structures with trivial isotropy (2012). arXiv:1208.1340v1

  45. Mclean, M.: Reeb orbits and the minimal discrepancy of an isolated singularity (2014). arXiv:1404.1857

  46. McMullen, C.: Course notes for Complex analysis on Riemann surfaces. http://www.math.harvard.edu/ctm/math213b/home/course/course

  47. Micallef, M., White, B.: The structure of branch points in minimal surfaces and in pseudoholomorphic curves. Ann. Math. 139, 35–85 (1994)

    MathSciNet  Google Scholar 

  48. Momin, A.: Contact homology of orbit complements and implied existence. J. Mod. Dyn. 5(3), 409–472 (2011)

    Article  MATH  Google Scholar 

  49. Nelson, J.: Applications of automatic transversality in contact homology. Ph.D. thesis, University of Wisconsin, Madison (2013)

  50. Nelson, J.: Automatic transversality in contact homology II: invariance and computations (in preparation)

  51. Pati, J.: Contact homology of \(S^1\)-bundles over some symplectically reduced orbifolds (2009). arXiv:0910.5934

  52. Pardon, J.: An algebraic approach to virtual fundamental cycles on moduli spaces of \(J\)-holomorphic curves. arXiv:1309.2370

  53. Robbin, J., Salamon, D.: The Maslov index for paths. Topology 32, 827–844 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  54. Schwarz, M.: Cohomology Operations from \(S^1\)-Cobordisms in Floer Homology. Ph.D. Thesis. ETH, Zürich (1995)

  55. Seidel, P.: Graded Lagrangian submanifolds. Bull. Soc. Math. France 128(1), 103–149 (2000)

    MATH  MathSciNet  Google Scholar 

  56. Seidel, P.: A biased view of symplectic cohomology. Curr. Dev. Math. 211–253 (2006)

  57. Siefring, R.: Relative asymptotic behavior of pseudoholomorphic half-cylinders. Commun. Pure Appl. Math. 61(12), 1631–1684 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  58. Taubes, C.H.: The Seiberg–Witten equations and the Weinstein conjecture. Geom. Topol. 11, 2117–2202 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  59. Ustilovsky, I.: Contact homology and contact structures on \(S^{4m+1}\). Ph.D. thesis, Stanford University (1999)

  60. Wendl, C.: Automatic transversality and orbifolds of punctured holomorphic curves in dimension four. Comment. Math. Helv. 85(2), 347–407 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  61. Wendl, C.: Lectures on holomorphic curves in symplectic and contact geometry. arXiv:1011.1690

  62. Yau, M.-L.: Contact homology of subcritical Stein-fillable contact manifolds. Geom. Topol. 8, 1243–1280 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

I thank Mohammed Abouzaid, Kai Cieliebak, Helmut Hofer, Michael Hutchings, Dusa McDuff, Joel Fish, Alex Oancea, Katrin Wehrheim, and Chris Wendl for their interest in my work and our assorted insightful discussions. I would especially like to thank Mohammed Abouzaid for being a wonderful and generous advisor as well as for his comments on my thesis, from which this paper has been extracted. I am very grateful to Michael Hutchings for showing me example 1.26 and our discussions of index calculations which gave rise to many of the results in Sect. 4; these generalized those that appeared in my thesis and spawned our joint projects [33–35]. Special thanks are due to Michael Hutchings, Janko Latschev, Dusa McDuff, Andrew McInerney, and the referee for their helpful comments on this paper.

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  1. Columbia University, New York, USA

    Jo Nelson

  2. Institute for Advanced Study, Princeton, USA

    Jo Nelson

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Correspondence to Jo Nelson.

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J. Nelson is supported by NSF grant DMS-1303903, the Bell Companies Fellowship and the Fund for Mathematics at the Institute for Advanced Study.

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Nelson, J. Automatic transversality in contact homology I: regularity. Abh. Math. Semin. Univ. Hambg. 85, 125–179 (2015). https://doi.org/10.1007/s12188-015-0112-3

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