This thesis introduces a new technique for constructing symplectic4-manifolds, generalizing the 3- and 4-fold sums introduced bySymington, and by McDuff and Symington.We first define relative connect normal sums. This method allows oneto join concave (or convex) fillings along complements of properlyembedded symplectomorphic surfaces with boundary.We then define the k-fold sum as follows. Given k pairs ofsymplectic surfaces, such that pairs are disjoint from one another,and the surfaces in each pair intersect ω-orthogonally once,we may remove neighbourhoods of the intersection points. We may thenperform the relative connect normal sum k times to obtain aconcave filling of a manifold that fibers over S1 with torusfibers. We study when the resulting contact structure on theboundary is convexly fillable.As an application of k-fold sums, we construct seven closed exoticsymplectic manifolds, two of which violate the Noether inequality.
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