A note on embeddings of 3-manifolds insymplectic 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. 57K43; 53D05Corollary 1.12 Given any 3-manifold Y there exists a Stein fillable 3-manifold Y 0 and a degree one map f W Y 0 !Y .The distinction between smooth and topological embeddings serves as a tool for detecting exotic structures on compact manifolds.If we encounter two homeomorphic 4-manifolds such that a 3-manifold embeds smoothly in one but not the other, then they are not diffeomorphic; they form an exotic pair.Corollary 1.13, which we will present shortly, was initially demonstrated by Akbulut [3] and subsequently proven by many others.However, we will offer an alternative proof stemming from the study of embeddings of 3-manifolds into 4-manifolds.Corollary 1.13There exist compact 4-manifolds with boundary X and X 0 such that b 2 .X / D b 2 .X 0 / D 1 that are homeomorphic but not diffeomorphic.