A C-symplectic free $S^1$-manifold with contractible orbits and $\mathbf {CAT} = \frac 12\mathbf {DIM}$
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- by Christopher Allday and John Oprea PDF
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Abstract:
An interesting question in symplectic geometry concerns whether or not a closed symplectic manifold can have a free symplectic circle action with orbits contractible in the manifold. Here we present a c-symplectic example, thus showing that the problem is truly geometric as opposed to topological. Furthermore, we see that our example is the only known example of a c-symplectic manifold having non-trivial fundamental group and Lusternik-Schnirelmann category precisely half its dimension.References
- C. Allday, Lie group actions on cohomology Kähler manifolds, unpublished manuscript (1978).
- J. Amorós, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental groups of compact Kähler manifolds, Mathematical Surveys and Monographs, vol. 44, American Mathematical Society, Providence, RI, 1996. MR 1379330, DOI 10.1090/surv/044
- Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanré, Lusternik-Schnirelmann category, Mathematical Surveys and Monographs, vol. 103, American Mathematical Society, Providence, RI, 2003. MR 1990857, DOI 10.1090/surv/103
- Ronald Fintushel and Ronald J. Stern, Immersed spheres in $4$-manifolds and the immersed Thom conjecture, Turkish J. Math. 19 (1995), no. 2, 145–157. MR 1349567
- J. C. Gómez-Larrañaga and F. González-Acuña, Lusternik-Schnirel′mann category of $3$-manifolds, Topology 31 (1992), no. 4, 791–800. MR 1191380, DOI 10.1016/0040-9383(92)90009-7
- D. Kotschick, Entropies, volumes and Einstein metrics, preprint 2004.
- D. Kotschick, Orientations and geometrisations of compact complex surfaces, Bull. London Math. Soc. 29 (1997), no. 2, 145–149. MR 1425990, DOI 10.1112/S0024609396002287
- Gregory Lupton and John Oprea, Cohomologically symplectic spaces: toral actions and the Gottlieb group, Trans. Amer. Math. Soc. 347 (1995), no. 1, 261–288. MR 1282893, DOI 10.1090/S0002-9947-1995-1282893-4
- Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. MR 1698616
- John Oprea and Yuli Rudyak, Detecting elements and Lusternik-Schnirelmann category of 3-manifolds, Lusternik-Schnirelmann category and related topics (South Hadley, MA, 2001) Contemp. Math., vol. 316, Amer. Math. Soc., Providence, RI, 2002, pp. 181–191. MR 1962163, DOI 10.1090/conm/316/05505
- John Oprea and John Walsh, Quotient maps, group actions and Lusternik-Schnirelmann category, Topology Appl. 117 (2002), no. 3, 285–305. MR 1874091, DOI 10.1016/S0166-8641(01)00021-9
- Yuli B. Rudyak and John Oprea, On the Lusternik-Schnirelmann category of symplectic manifolds and the Arnold conjecture, Math. Z. 230 (1999), no. 4, 673–678. MR 1686579, DOI 10.1007/PL00004709
- Clifford Henry Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994), no. 6, 809–822. MR 1306023, DOI 10.4310/MRL.1994.v1.n6.a15
Additional Information
- Christopher Allday
- Affiliation: Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Honolulu, Hawaii 96822-2273
- Email: chris@math.hawaii.edu
- John Oprea
- Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115
- MR Author ID: 134075
- Email: oprea@math.csuohio.edu
- Received by editor(s): June 20, 2004
- Received by editor(s) in revised form: September 16, 2004
- Published electronically: June 29, 2005
- Communicated by: Ronald A. Fintushel
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 599-604
- MSC (2000): Primary 57E25; Secondary 55C30, 53D05
- DOI: https://doi.org/10.1090/S0002-9939-05-07945-1
- MathSciNet review: 2176029