Holomorphic triangle invariants and the topology of symplectic four-manifolds

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.