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A C-symplectic free $S^1$-manifold with contractible orbits and $\mathbf{CAT} = \frac12\,\mathbf{DIM}$

Authors: Christopher Allday and John Oprea
Journal: Proc. Amer. Math. Soc. 134 (2006), 599-604
MSC (2000): Primary 57E25; Secondary 55C30, 53D05
Published electronically: June 29, 2005
MathSciNet review: 2176029
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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.

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  • [A] C. Allday, Lie group actions on cohomology Kähler manifolds, unpublished manuscript (1978).
  • [ABCKT] J. Amorós, M. Burger, K. Corlette, D. Kotschick and D. Toledo, Fundamental groups of compact Kähler manifolds, Amer. Math. Soc. Surveys and Mono. 44 (1996). MR 1379330 (97d:32037)
  • [CLOT] O. Cornea and G. Lupton and J. Oprea and D. Tanré, Lusternik-Schnirelmann Category, Amer. Math. Soc. Surveys and Mono. 103 (2003). MR 1990857 (2004e:55001)
  • [FS] R. Fintushel and R. Stern, Immersed spheres in 4-manifolds and the immersed Thom conjecture, Turkish J. Math. 19 (1995) 145-157. MR 1349567 (96j:57036)
  • [GoGo] J.C. Gomez-Larranaga and F. Gonzalez-Acuna, Lusternik-Schnirelmann category of $3$-manifolds, Topology 31 (1992) 791-800. MR 1191380 (93j:55005)
  • [Ko] D. Kotschick, Entropies, volumes and Einstein metrics, preprint 2004.
  • [Ko2] D. Kotschick, Orientations and geometrisations of compact complex surfaces, Bull. London Math. Soc. 29 2 (1997) 145-149. MR 1425990 (97k:32047)
  • [LO] G. Lupton and J. Oprea, Cohomologically symplectic spaces: toral actions and the Gottlieb group, Trans. Amer. Math. Soc. 347 (1995) 261-288. MR 1282893 (95f:57056)
  • [MS] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford University Press, $2^{\rm nd}$ edition, (1998). MR 1698616 (2000g:53098)
  • [OR] J. Oprea and Y. Rudyak, Detecting elements and Lusternik-Schnirelmann category of $3$-manifolds, Contemp. Math. 316 (2002) 181-191. MR 1962163 (2004c:55003)
  • [OW] J. Oprea and J. Walsh, Quotient maps, group actions and Lusternik-Schnirelmann category, Top. Appls. 117 (2002) 285-305. MR 1874091 (2002i:55001)
  • [RO] Y. Rudyak and J. Oprea, On the Lusternik-Schnirelmann category of symplectic manifolds and the Arnold conjecture, Math. Zeit. 230 no. 4 (1999) 673-678. MR 1686579 (2000b:53115)
  • [Ta] C. Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Letters 1 (1994) 809-822. MR 1306023 (95j:57039)

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Additional Information

Christopher Allday
Affiliation: Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Honolulu, Hawaii 96822-2273

John Oprea
Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115

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
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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