This article analyzes the interplay between symplectic geometry in
dimension $4$ and the invariants for smooth four-manifolds
constructed using holomorphic triangles introduced in [20].
Specifically, we establish a nonvanishing 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.