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Twisted Alexander Polynomials and Symplectic Structures

2008; Johns Hopkins University Press; Volume: 130; Issue: 2 Linguagem: Inglês

10.1353/ajm.2008.0014

1080-6377

Stefan Friedl, Stefano Vidussi,

Homotopy and Cohomology in Algebraic Topology

Let N be a closed, oriented 3-manifold. A folklore conjecture states that S 1 × N admits a symplectic structure only if N admits a fibration over the circle. The purpose of this paper is to provide evidence to this conjecture studying suitable twisted Alexander polynomials of N , and showing that their behavior is the same as of those of fibered 3-manifolds. In particular, we will obtain new obstructions to the existence of symplectic structures and to the existence of symplectic forms representing certain cohomology classes of S 1 × N . As an application of these results we will show that S 1 × N ( P ) does not admit a symplectic structure, where N ( P ) is the 0-surgery along the pretzel knot P = (5, -3, 5), answering a question of Peter Kronheimer.