### A characterization of osculating maps

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In this paper we show that given any 3-manifold $N$ and any non-fibered class in ${H}^{1}(N;Z)$ there exists a representation such that the corresponding twisted Alexander polynomial is zero. We obtain this result by extending earlier work of ours and by combining this with recent results of Agol and Wise on separability of 3-manifold groups. This result allows us to completely classify symplectic 4-manifolds with a free circle action, and to determine their symplectic cones.

Let $\Sigma $ be a surface with a symplectic form, let $\phi $ be a symplectomorphism of $\Sigma $, and let $Y$ be the mapping torus of $\phi $. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in $\mathbb{R}\times \mathbb{Y}$, with cylindrical ends asymptotic to periodic orbits of $\phi $ or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spaces. This paper establishes some of the foundations for a program with Michael Thaddeus, to understand...

The classification of class VII surfaces is a very difficult classical problem in complex geometry. It is considered by experts to be the most important gap in the Enriques-Kodaira classification table for complex surfaces. The standard conjecture concerning this problem states that any minimal class VII surface with b₂ > 0 has b₂ curves. By the results of [Ka1]-[Ka3], [Na1]-[Na3], [DOT], [OT] this conjecture (if true) would solve the classification problem completely. We explain a new approach...