Abstract
We give a concise overview of the classification theory of symplectic manifolds equipped with torus actions for which the orbits are symplectic (this is equivalent to the existence of a symplectic principal orbit), and apply this theory to study the structure of the leaf space induced by the action. In particular we show that if M is a symplectic manifold on which a torus T acts effectively with symplectic orbits, then the leaf space M/T is a very good orbifold with first Betti number b1(M/T)=b1(M)−dim T.
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Armstrong, M.A.: The fundamental group of the orbit space of a discontinuous group. Math. Proc. Camb. Philos. Soc. 64, 299–301 (1968)
Atiyah, M.: Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc. 14, 1–15 (1982)
Benoist, Y.: Actions symplectiques de groupes compacts. Geom. Dedic. 89, 181–245 (2002)
Bochner, S.: Compact groups of differentiable transformations. Ann. Math. 46, 372–381 (1945)
Boileau, M., Maillot, S., Porti, J.: Three-Dimensional Orbifolds and Their Geometric Structures. Panoramas et Synthèses, No. 15. Société Mathématique de France, Montrouge (2003)
Chaperon, M.: Quelques outils de la théorie des actions différentiables. In: III rencontre de géométrie de Schnepfenried, vol. 1, 10–15 mai 1982. Astérisque, vols. 107–108, pp. 259–275. Société Mathématique de France, Montrouge (1983)
Dehn, M.: Die Gruppe der Abbildungsklassen. Acta Math. 69, 135–206 (1938)
Delzant, T.: Hamiltoniens périodiques et image convex de l’application moment. Bull. Soc. Math. France 116, 315–339 (1988)
Duistermaat, J.J.: On global action-angle variables. Commun. Pure Appl. Math. 33, 687–706 (1980)
Duistermaat, J.J.: The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator. Birkhäuser, Boston (1996)
Duistermaat, J.J., Pelayo, A.: Symplectic torus actions with coisotropic principal orbits. Ann. Inst. Fourier, Grenoble 57(7), 2239–2327 (2007)
Duistermaat, J.J., Pelayo, A.: Complex structures on 4-manifolds with symplectic 2-torus actions. Preprint (2009)
Guillemin, V., Sternberg, S.: Convexity properties of the moment mapping. Invent. Math. 67, 491–513 (1982)
Guillemin, V.: Moment Maps and Combinatorial Invariants of Hamiltonian T n-Spaces. Birkhäuser, Boston (1994)
Ginzburg, V.L.: Some remarks on symplectic actions of compact groups. Math. Z. 210, 625–640 (1992)
Kahn, P.J.: Symplectic torus bundles and group extensions. N. Y. J. Math. 11, 35–55 (2005). Also at http://nyjm.albany.edu:8000/j/2005/11-3.html
Karshon, Y., Tolman, S.: Complete invariants for Hamiltonian torus actions with two dimensional quotients. J. Symplectic Geom. 2(1), 25–82 (2003)
Karshon, Y., Tolman, S.: Centered complexity one Hamiltonian torus actions. Trans. Am. Math. Soc. 353(12), 4831–4861 (2001)
Kirwan, F.C.: Cohomology of Quotients in Symplectic and Algebraic Geometry. Math Notes, 31. Princeton University Press, Princeton (1984)
Kogan, M.: On completely integrable systems with local torus actions. Ann. Glob. Anal. Geom. 15(6), 543–553 (1997)
Koszul, J.L.: Sur certains groupes de Lie. In: Géometrie Différentielle. Coll. Int. CNRS, vol. 52, pp. 137–141. CNRS, Paris (1953)
Hu, S.-T.: An exposition of the relative homotopy theory. Duke Math. J. 14, 991–1033 (1947)
Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory. Presentations of Groups in Terms of Generators and Relations. Dover, Mineola (2004). Reprint of the 1976 second edition
McDuff, D.: The moment map for circle actions on symplectic manifolds. J. Geom. Phys. 5(2), 149–160 (1988)
McDuff, D., Salamon, D.A.: Introduction to Symplectic Topology, 2nd edn. Oxford University Press, Oxford (1998)
Moser, J.: On the volume elements on a manifold. Trans. Am. Math. Soc. 120, 286–294 (1965)
Novikov, S.P.: The Hamiltonian formalism and a multivalued analogue of Morse theory. Russ. Math. Surv. 37(5), 1–56 (1982)
Orlik, P., Raymond, F.: Actions of the torus on 4-manifolds, I. Trans. Am. Math. Soc. 152, 531–559 (1970)
Orlik, P., Raymond, F.: Actions of the torus on 4-manifolds, II. Topology 13, 89–112 (1974)
Ortega, J.-P., Ratiu, T.S.: A symplectic slice theorem. Lett. Math. Phys. 59, 81–93 (2002)
Pao, P.S.: The topological structure of 4-manifolds with effective torus actions, I. Trans. Am. Math. Soc. 227, 279–317 (1977)
Pao, P.S.: The topological structure of 4-manifolds with effective torus actions, II. Ill. J. Math. 21, 883–894 (1977)
Pelayo, A.: Symplectic Actions of 2-Tori on 4-Manifolds. Mem. Am. Math. Soc., vol. 204(959). Am. Math. Soc., Providence (2010)
Pelayo, A., Vũ Ngọc, S.: Semitoric integrable systems on symplectic 4-manifolds. Invent. Math. 177, 571–597 (2009)
Pelayo, A., Vũ Ngọc, S.: Constructing integrable systems of semitoric type. Acta Math., to appear
Poincaré, H.: Analysis situs. A series of articles on topology from 1895 until 1904. Reproduced on pp. 193–498. In: Œuvres de H. Poincaré, t. VI, Gauthier-Villars, Paris (1953)
Scott, P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15, 401–487 (1983)
Seifert, H., Threlfall, W.: Lehrbuch der Topologie. Teubner, Leipzig (1934)
Spanier, E.: Algebraic Topology. McGraw-Hill, New York (1966) xiv+528 pp.
Thurston, W.P.: Three-Dimensional Geometry and Topology. MSRI, Berkeley, (1991). This book is based on the course notes The geometry and topology of 3-manifolds, Princeton Math. Dept., 1979. The discussion of orbifolds is not contained in the book Three-Dimensional Geometry and Topology, vol. 1, published by Princeton University Press in 1997, of which we could not find vol. 2
Weinstein, A.: Symplectic manifolds and their Lagrangian submanifolds. Adv. Math. 6, 329–346 (1971)
Zung, N.T.: Symplectic topology of integrable Hamiltonian systems, I. Arnold-Liouville with singularities. Compos. Math. 101, 179–215 (1996)
Zung, N.T.: Symplectic topology of integrable Hamiltonian systems, II. Topological classification. Compos. Math. 138(2), 125–156 (2003)
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A. Pelayo was partially supported by an NSF fellowship.
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Duistermaat, J.J., Pelayo, A. Topology of symplectic torus actions with symplectic orbits. Rev Mat Complut 24, 59–81 (2011). https://doi.org/10.1007/s13163-010-0028-5
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DOI: https://doi.org/10.1007/s13163-010-0028-5