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We develop techniques to construct explicit symplectic Lefschetz fibrations over the 2-sphere with any prescribed signature σ and any spin type when σ is divisible by 16. This solves a long-standing c...onjecture on the existence of such fibrations with positive signature. As applications, we produce symplectic 4-manifolds that are homeomorphic but not diffeomorphic to connected sums of S^2×S^2, with the smallest topology known to date, as well as larger examples as symplectic Lefschetz fibrations.続きを見る
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