A note on embeddings of 3-manifolds in symplectic 4-manifolds

We show that any closed oriented 3-manifold can be topologically embedded in some simply connected closed symplectic 4-manifold and that it can be made a smooth embedding after one stabilization. As a corollary of the proof, we show that the homology cobordism group is generated by Stein fillable 3-manifolds. We also find obstructions on smooth embeddings: there exist 3-manifolds that cannot smoothly embed in a way that appropriately respects orientations in any symplectic manifold with a weakly convex boundary.