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Contact homology of good toric contact manifolds

Part of: Symplectic geometry, contact geometry

Published online by Cambridge University Press:  09 November 2011

Miguel Abreu  and
Leonardo Macarini
Miguel Abreu
Affiliation:
Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal (email: [email protected])
Leonardo Macarini
Affiliation:
Universidade Federal do Rio de Janeiro, Instituto de Matemática, Cidade Universitária, CEP 21941-909 Rio de Janeiro, Brazil (email: [email protected])
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Abstract

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In this paper we show that any good toric contact manifold has a well-defined cylindrical contact homology, and describe how it can be combinatorially computed from the associated moment cone. As an application, we compute the cylindrical contact homology of a particularly nice family of examples that appear in the work of Gauntlett et al. on Sasaki–Einstein metrics. We show in particular that these give rise to a new infinite family of non-equivalent contact structures on S2×S3 in the unique homotopy class of almost contact structures with vanishing first Chern class.

Keywords

toric contact manifoldstoric symplectic conescontact homology

MSC classification

Secondary: 53D42: Symplectic field theory; contact homology 53D20: Momentum maps; symplectic reduction 53D35: Global theory of symplectic and contact manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 1 , January 2012 , pp. 304 - 334
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[Abr10]Abreu, M., Kähler–Sasaki geometry of toric symplectic cones in action–angle coordinates, Port. Math. (N.S.) 67 (2010), 121–153.CrossRefGoogle Scholar
[Ban99]Banyaga, A., The geometry surrounding the Arnold–Liouville theorem, in Advances in Geometry, Progress in Mathematics, vol. 172, eds Brylinski, J.-L., Brylinski, R., Nistor, V., Tsygan, B. and Xu, P. (Birkhäuser, Basel, 1999).Google Scholar
[BM93]Banyaga, A. and Molino, P., Géométrie des formes de contact complètement intégrables de type toriques, in Séminaire Gaston Darboux de Géométrie et Topologie Différentielle, 1991–1992 (Montpellier) (Université Montpellier 2, Montpellier, 1993), 1–25.Google Scholar
[BM96]Banyaga, A. and Molino, P., Complete integrability in contact geometry, Preprint PM 197, Pennsylvania State University (1996).Google Scholar
[BW58]Boothby, W. M. and Wang, H. C., On contact manifolds, Ann. of Math. (2) 68 (1958), 721–734.CrossRefGoogle Scholar
[Bou03a]Bourgeois, F., A Morse–Bott approach to contact homology, in Symplectic and contact topology: interactions and perspective, Fields Institute Communications, vol. 35, eds Eliashberg, Y., Khesin, B. and Lalonde, F. (American Mathematical Society, Providence, RI, 2003), 55–77.Google Scholar
[Bou03b]Bourgeois, F., Introduction to contact homology, Lecture notes of mini-course at Summer School in Berder on holomorphic curves and contact topology, June 2003 (2003). Available at http://homepages.ulb.ac.be/fbourgeo/pspdf/Berder.pdf.Google Scholar
[Bou09]Bourgeois, F., A survey of contact homology, in New perspectives and challenges in symplectic field theory, CRM Proceedings and Lecture Notes, vol. 49, eds Abreu, M., Lalonde, F. and Polterovich, L. (American Mathematical Society, Providence, RI, 2009), 45–71.CrossRefGoogle Scholar
[BEHWZ03]Bourgeois, F., Eliashberg, Y., Hofer, H., Wysocki, K. and Zehnder, E., Compactness results in symplectic field theory, Geom. Topol. 7 (2003), 799–888.CrossRefGoogle Scholar
[BM04]Bourgeois, F. and Mohnke, K., Coherent orientations in symplectic field theory, Math. Z. 248 (2004), 123–146.CrossRefGoogle Scholar
[BO09]Bourgeois, F. and Oancea, A., An exact sequence for contact and symplectic homology, Invent. Math. IMRN 175 (2009), 611–680.CrossRefGoogle Scholar
[BG00]Boyer, C. P. and Galicki, K., A note on toric contact geometry, J. Geom. Phys. 35 (2000), 288–298.CrossRefGoogle Scholar
[CM07]Cieliebak, K. and Mohnke, K., Symplectic hypersurfaces and transversality in Gromov–Witten theory, J. Symplectic Geom. 5 (2007), 281–356.CrossRefGoogle Scholar
[EGH00]Eliashberg, Y., Givental, A. and Hofer, H., Introduction to symplectic field theory, Geom. Funct. Anal. (2000), Special volume, Part II, 560–673.Google Scholar
[Flo88a]Floer, A., A relative Morse index for the symplectic action, Comm. Pure Appl. Math. 41 (1988), 393–407.CrossRefGoogle Scholar
[Flo88b]Floer, A., The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988), 775–813.CrossRefGoogle Scholar
[Flo89]Floer, A., Witten’s complex and infinite-dimensional Morse theory, J. Differential Geom. 30 (1989), 207–221.CrossRefGoogle Scholar
[GMSW04a]Gauntlett, J., Martelli, D., Sparks, J. and Waldram, D., Sasaki–Einstein metrics on S2×S3, Adv. Theor. Math. Phys. 8 (2004), 711–734.CrossRefGoogle Scholar
[GMSW04b]Gauntlett, J., Martelli, D., Sparks, J. and Waldram, D., A new infinite class of Sasaki–Einstein manifolds, Adv. Theor. Math. Phys. 8 (2004), 987–1000.CrossRefGoogle Scholar
[Gir94]Giroux, E., Une structure de contact, même tendue est plus ou moins tordue, Ann. Sci. Éc. Norm. Supér. (4) 27 (1994), 697–705.CrossRefGoogle Scholar
[Ham10]Hamilton, M. J. D., Inequivalent contact structures on Boothby–Wang 5-manifolds, arXiv:1001.1953.Google Scholar
[HWZ07]Hofer, H., Wysocki, K. and Zehnder, E., A general Fredholm theory. I. A splicing-based differential geometry, J. Eur. Math. Soc. (JEMS) 9 (2007), 841–876.CrossRefGoogle Scholar
[HWZ09a]Hofer, H., Wysocki, K. and Zehnder, E., A general Fredholm theory. II. Implicit function theorems, Geom. Funct. Anal. 19 (2009), 206–293.CrossRefGoogle Scholar
[HWZ09b]Hofer, H., Wysocki, K. and Zehnder, E., A general Fredholm theory. III. Fredholm functors and polyfolds, Geom. Topol. 13 (2009), 2279–2387.CrossRefGoogle Scholar
[Kan97]Kanda, Y., The classification of tight contact structures on the 3-torus, Comm. Anal. Geom. 5 (1997), 413–438.Google Scholar
[van05]van Koert, O., Open books for contact five-manifolds and applications of contact homology, PhD thesis, Universität zu Köln (2005).Google Scholar
[van08]van Koert, O., Contact homology of Brieskorn manifolds, Forum Math. 20 (2008), 317–339.Google Scholar
[Ler03a]Lerman, E., Contact toric manifolds, J. Symplectic Geom. 1 (2003), 785–828.CrossRefGoogle Scholar
[Ler03b]Lerman, E., Geodesic flows and contact toric manifolds, in Symplectic geometry of integrable hamiltonian systems (Birkhäuser, Basel, 2003).Google Scholar
[Ler03c]Lerman, E., Maximal tori in the contactomorphism groups of circle bundles over Hirzebruch surfaces, Math. Res. Lett. 10 (2003), 133–144.CrossRefGoogle Scholar
[Ler04]Lerman, E., Homotopy groups of K-contact toric manifolds, Trans. Amer. Math. Soc. 356 (2004), 4075–4083.CrossRefGoogle Scholar
[MSY06]Martelli, D., Sparks, J. and Yau, S.-T., The geometric dual of a-maximisation for toric Sasaki–Einstein manifolds, Comm. Math. Phys. 268 (2006), 39–65.CrossRefGoogle Scholar
[OU72]Ozeki, H. and Uchida, F., Principal circle actions on a product of spheres, Osaka J. Math. 9 (1972), 379–390.Google Scholar
[Pat09]Pati, J., Contact homology of S 1-bundles over some symplectically reduced orbifolds, arXiv:0910.5934.Google Scholar
[RS93]Robbin, J. and Salamon, D., The Maslov index for paths, Topology 32 (1993), 827–844.CrossRefGoogle Scholar
[Sal99]Salamon, D., Lectures on Floer homology, in Symplectic geometry and topology (Park City, 1997), IAS/Park City Mathematics Series, 7 (American Mathematical Society, Providence, RI, 1999), 143–229.Google Scholar
[Sma62]Smale, S., On the structure of 5-manifolds, Ann. of Math. (2) 75 (1962), 38–46.CrossRefGoogle Scholar
[Ust99]Ustilovsky, I., Infinitely many contact structures on S 4m+1, Int. Math. Res. Not. 14 (1999), 781–791.CrossRefGoogle Scholar
[Wen10]Wendl, C., Automatic transversality and orbifolds of punctured holomorphic curves in dimension four, Comment. Math. Helv. 85 (2010), 347–407.CrossRefGoogle Scholar