**Stable Geometric Topology of 4-Manifolds**

Geometric topology of smooth 4-manifolds up to stablization with S^{2}xS^{2}

Consider a separating 3-dimensional submanifold of a smooth 4-dimensional manifold whose fundamental group is mapped injectively into the fundamental group of the 4-manifold. The Seifertâ€“van Kampen theorem implies the fundamental group of the 4-manifold has an amalgamated product structure. This work attempts to characterize when the converse holds, allowing for stabilization of the 4-manifold by forming the connected sum with S^{2}xS^{2} factors. When such decompositions can be realized geometrically, uniqueness of the decomposition is characterized.