Relative trisections of smooth 4-manifolds with boundary

We show that every smooth, compact, connected, oriented 4-manifold with non-empty (connected or disconnected) boundary can be decomposed into three diffeomorphic 4-dimensional 1-handlebodies, $natural^k S^1 times B^3,$ for some k. The pairwise intersections are compression bodies diffeomorphic to $natural^k S^1 times D^2$ and the triple intersection is a surface with boundary. Such a decomposition is called a relative trisection. Additionally, we define a stabilization technique for relative trisections which provides a more general uniqueness statement for relative trisections than that provided by Gay and Kirby. We also show that relatively trisected 4-manifolds can be glued together along diffeomorphic boundary components to induce a trisection of the resulting 4-manifold.