$\pi _1$ of Hamiltonian $S^1$ manifolds
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- by Hui Li
- Proc. Amer. Math. Soc. 131 (2003), 3579-3582
- DOI: https://doi.org/10.1090/S0002-9939-03-06881-3
- Published electronically: February 14, 2003
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Abstract:
Let $(M, \omega )$ be a connected, compact symplectic manifold equipped with a Hamiltonian $S^1$ action. We prove that, as fundamental groups of topological spaces, $\pi _1(M)=\pi _1(\mathrm {minimum})=\pi _1(\mathrm {maximum})=\pi _1(M_{red})$, where $M_{red}$ is the symplectic quotient at any value in the image of the moment map $\phi$.References
- M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1–15. MR 642416, DOI 10.1112/blms/14.1.1
- Michèle Audin, The topology of torus actions on symplectic manifolds, Progress in Mathematics, vol. 93, Birkhäuser Verlag, Basel, 1991. Translated from the French by the author. MR 1106194, DOI 10.1007/978-3-0348-7221-8
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- Michel Brion and Claudio Procesi, Action d’un tore dans une variété projective, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 509–539 (French). MR 1103602, DOI 10.1007/s101070100288
- W. Chen, A Homotopy Theory of Orbispaces, math. AT/0102020.
- V. Guillemin and S. Sternberg, Birational equivalence in the symplectic category, Invent. Math. 97 (1989), no. 3, 485–522. MR 1005004, DOI 10.1007/BF01388888
- J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331, DOI 10.1515/9781400881802
- Charles-Michel Marle, Modèle d’action hamiltonienne d’un groupe de Lie sur une variété symplectique, Rend. Sem. Mat. Univ. Politec. Torino 43 (1985), no. 2, 227–251 (1986) (French, with English summary). MR 859857
- Dusa McDuff, Examples of simply-connected symplectic non-Kählerian manifolds, J. Differential Geom. 20 (1984), no. 1, 267–277. MR 772133
- Reyer Sjamaar and Eugene Lerman, Stratified symplectic spaces and reduction, Ann. of Math. (2) 134 (1991), no. 2, 375–422. MR 1127479, DOI 10.2307/2944350
- Yoshihiro Takeuchi and Misako Yokoyama, The geometric realizations of the decompositions of $3$-orbifold fundamental groups, Topology Appl. 95 (1999), no. 2, 129–153. MR 1696444, DOI 10.1016/S0166-8641(97)00280-0
Bibliographic Information
- Hui Li
- Affiliation: Department of Mathematics, University of Illinois, Urbana-Champaign, Illinois 61801
- Email: hli@math.uiuc.edu
- Received by editor(s): January 10, 2002
- Received by editor(s) in revised form: May 23, 2002
- Published electronically: February 14, 2003
- Communicated by: Ronald A. Fintushel
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 3579-3582
- MSC (2000): Primary 53D05, 53D20; Secondary 55Q05, 57R19
- DOI: https://doi.org/10.1090/S0002-9939-03-06881-3
- MathSciNet review: 1991771