Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Contact structures on elliptic $3$-manifolds

Author(s): Siddhartha Gadgil
Journal: Proc. Amer. Math. Soc. 132 (2004), 3705-3714.
MSC (2000): Primary 53D10, 57M50
Posted: July 22, 2004
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We show that an oriented elliptic $3$-manifold admits a universally tight positive contact structure if and only if the corresponding group of deck transformations on $S^3$ (after possibly conjugating by an isometry) preserves the standard contact structure.

We also relate universally tight contact structures on $3$-manifolds covered by $S^3$ to the isomorphism $SO(4)=(SU(2)\times SU(2))/{\pm 1}$.

The main tool used is equivariant framings of $3$-manifolds.

References:

1.
Ya. Eliashberg, Contact $3$-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier (Grenoble) 42 (1992), 165-192. MR 93k:57029

2.
Ya. Eliashberg, Classification of overtwisted contact structures on $3$-manifolds, Invent. Math. 98 (1989), 623-637. MR 90k:53064

3.
Ya. Eliashberg and W. Thurston, Confoliations, University Lecture Series, 13, American Mathematical Society, Providence, RI, 1998. MR 98m:53042

4.
J. Etnyre and K. Honda, On the non-existence of tight contact structures, Ann. of Math. (2) 153 (2001), 749-766. MR 2002d:53119

5.
S. Gadgil, Equivariant framings, lens spaces and contact structures, Pacific J. Math. 208 (2003), no. 1, 73-84. MR 2004c:57040

6.
E. Giroux, Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Invent. Math. 141 (2000), 615-689. MR 2001i:53147

7.
R. Gompf, Handlebody construction of Stein surfaces, Ann. of Math. (2) 148 (1998), 619-693. MR 2000a:57070

8.
K. Honda, On the classification of tight contact structures. I, Geom. Topol. 4 (2000), 309-368. MR 2001i:53148

9.
K. Honda, W. H. Kazez and G. Matic, Tight contact structures and taut foliations, Geom. Topol. 4 (2000), 219-242. MR 2001i:53149

10.
K. Honda, W. H. Kazez and G. Matic, Tight contact structures on fibered hyperbolic 3-manifolds. J. Differential Geom. 64 (2003), no. 2, 305-358.

11.
H. Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Ann. 95 (1925-26), 313-319.

12.
G. P. Scott, The geometries of $3$-manifolds. Bull. London Math. Soc. 15 (1983), 401-487. MR 84m:57009

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53D10, 57M50

Retrieve articles in all Journals with MSC (2000): 53D10, 57M50

Additional Information:

Siddhartha Gadgil
Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794
Address at time of publication: Stat-Math Unit, Indian Statistical Institute, 8th Mile, Mysore Road, R. V. College post, Bangalore 560059, India
Email: gadgil@math.sunysb.edu

DOI: 10.1090/S0002-9939-04-07572-0
PII: S 0002-9939(04)07572-0
Received by editor(s): March 1, 2002
Received by editor(s) in revised form: August 20, 2002
Posted: July 22, 2004
Communicated by: Ronald A. Fintushel
Copyright of article: Copyright 2004, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google