  This article analyzes the interplay between symplectic geometry in dimension
four and the invariants for smooth four-manifolds constructed using holomorphic
triangles introduced in math.SG/0110169. Specifically, we establish a
non-vanishing result for the invariants of symplectic four-manifolds, which
leads to new proofs of the indecomposability theorem for symplectic
four-manifolds and the symplectic Thom conjecture. As a new application, we
generalize the indecomposability theorem to splittings of four-manifolds along
a certain class of three-manifolds obtained by plumbings of spheres. This leads
to restrictions on the topology of Stein fillings of such three-manifolds.
