Holomorphic triangles and invariants for smooth four-manifolds

In earlier articles, the authors introduced invariants for closed, oriented three-manifolds, defined using a variant of Lagrangian Floer homology in the symmetric products of Riemann surfaces. The aim of this article is to introduce invariants of oriented, smooth four-manifolds, built using these Floer homology groups. This four-dimensional theory also endows the corresponding three-dimensional theories with additional structure: an absolute grading of certain of its Floer homology groups.