A C-symplectic free -manifold with contractible orbits and

Authors:
Christopher Allday and John Oprea

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

Published electronically:
June 29, 2005

MathSciNet review:
2176029

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[A]**C. Allday,*Lie group actions on cohomology Kähler manifolds*, unpublished manuscript (1978).**[ABCKT]**J. Amorós, M. Burger, K. Corlette, D. Kotschick and D. Toledo,*Fundamental groups of compact Kähler manifolds*, Amer.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 -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, edition, (1998). MR**1698616 (2000g:53098)****[OR]**J. Oprea and Y. Rudyak,*Detecting elements and Lusternik-Schnirelmann category of -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

Email:
chris@math.hawaii.edu

**John Oprea**

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

Email:
oprea@math.csuohio.edu

DOI:
https://doi.org/10.1090/S0002-9939-05-07945-1

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.