A note on the geometry of \(M^3\) and symplectic structures on \(S^1 \times M^3\). (English) Zbl 1121.57013
Summary: We investigate the relationship between the geometry of a closed, oriented 3-manifold \(M\) and the symplectic structures on \(S^1\times M\). In most cases the existence of a symplectic structure on \(S^1\times M\) and Thurston’s geometrization conjecture imply the existence of a geometric structure on \(M\). This observation together with the existence of geometric structures on most 3-manifolds which fiber over the circle suggests a different approach to the problem of finding a fibration of a 3-manifold over the circle in case its product with the circle admits a symplectic structure.
MSC:
| 57R57 | Applications of global analysis to structures on manifolds |
| 57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |
| 57R17 | Symplectic and contact topology in high or arbitrary dimension |
Keywords:
symplectic 4-manifolds; surface fibration over the circle; Taubes’ Conjecture; geometrization conjectureReferences:
| [1] | T. Etgü, Lefschetz fibrations, complex structures and Seifert fibrations on S 1 {\(\times\)} M, Algebr. Geom. Topol., 1 (2001), 469–489. · Zbl 0977.57010 · doi:10.2140/agt.2001.1.469 |
| [2] | E. Ionel and T. Parker, The Gromov invariants of Ruan-Tian and Taubes, Math. Res. Lett., 4 (1997), 521–532. · Zbl 0889.57030 |
| [3] | J. McCarthy, On the asphericity of a symplectic M 3 {\(\times\)} S 1, Proc. Amer. Math. Soc., 129 (2001), 257–264. · Zbl 0972.53045 · doi:10.1090/S0002-9939-00-05571-4 |
| [4] | C. Taubes, SW Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves, J. Amer. Math. Soc., 9 (1996), 845–918. · Zbl 0867.53025 · doi:10.1090/S0894-0347-96-00211-1 |
| [5] | C. Taubes, Gr SW: from pseudo-holomorphic curves to Seiberg-Witten solutions, J. Differential Geom., 51 (1999), 203–334. · Zbl 1036.53066 |
| [6] | C. Taubes, Gr = SW: counting curves and connections, J. Differential Geom., 52 (1999), 453–609. |
| [7] | W. Thurston, Hyperbolic structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle, preprint, ArXiv:GT/9801045. |
| [8] | W. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc., 6 (1982), 357–381. · Zbl 0496.57005 · doi:10.1090/S0273-0979-1982-15003-0 |
| [9] | S. Vidussi, The Alexander norm is smaller than the Thurston norm: a Seiberg-Witten proof, Prépublication École Polytechnique, 99–6. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
