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:
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