Twisted Alexander polynomials and symplectic structures

Details Twisted Alexander polynomials and symplectic structures Twisted Alexander polynomials and symplectic structures

2006-04-18

arxiv.org/abs/math/0604398v3

Mathematics

Geometric Topology

26 pages, final version, to appear in Amer. J. Math

Scientific paper

Let N be a closed, oriented 3-manifold. A folklore conjecture states that S^1 x 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 x N. As an application of these results we will show that S^1 x 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.

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