# Embedding questions for 3-manifolds

# Embedding questions for 3-manifolds

In this talk I will discuss 3-manifold embedding questions into 4-dimensional manifolds from the perspectives of smooth topology as well as symplectic and complex geometry. The talk will have two parts: In the first part the focus will be on embedding questions into 4-dimensional Euclidean space (R^4). Given a closed, orientable 3-manifold Y, it is of great interest but often a difficult problem to determine whether Y may be smoothly embedded in R^4. This is the case even for integer homology spheres (where usual obstructions coming from homology disappear), and restricting to special classes such as Seifert manifolds, the problem is open in general. On the other hand, under additional geometric considerations coming from symplectic geometry (such as hypersurfaces of contact type in R^4) and complex geometry (such as the boundaries of pseudo-convex or rationally convex domains in complex Euclidean space C^2), the problems become tractable and in certain cases a uniform answer is possible. For example, recent work shows no correctly oriented Seifert homology spheres admits an embedding as a hypersurface of contact type in R^4. In the second part, I will consider general closed 4-manifolds as target manifolds and mention some recent work in progress. I will provide further context and motivations for these results, and give some details of the proofs. This is joint work with Tom Mark.