On symplectic 4-manifolds and contact 5-manifolds

In this thesis we prove some results on symplectic structures on 4-dimensional manifolds and contact structures on 5-dimensional manifolds. We begin by discussing the relation between holomorphic and symplectic minimality for Kähler surfaces and the irreducibility of minimal simply-connected symplectic 4-manifolds under connected sum. We also prove a result on the conformal systoles of symplectic 4-manifolds. For the generalized fibre sum construction of 4-manifolds we calculate the integral homology groups if the summation is along embedded surfaces with trivial normal bundle. In the symplectic case we derive a formula for the canonical class of the generalized fibre sum and give several applications, in particular to the geography of simply-connected symplectic 4-manifolds whose canonical class is divisible by a given integer. We also use branched coverings of complex surfaces of general type to construct simply-connected algebraic surfaces with divisible canonical class. In the second part of the thesis we show that these geography results together with the Boothby-Wang construction of contact structures on circle bundles over symplectic manifolds imply that certain simply-connected 5-manifolds admit inequivalent contact structures in the same (non-trivial) homotopy class of almost contact structures.