Embedding every 3-manifold in a 4-manifold with non-trivial gauge-theoretic invariants

Determine whether every closed oriented 3-manifold Y admits a smooth embedding as a separating hypersurface in some closed smooth 4-manifold with non-trivial Donaldson invariants; similarly, determine whether every closed oriented 3-manifold Y embeds (as a separating hypersurface) in some closed smooth 4-manifold with non-trivial Seiberg–Witten invariants, and whether such embeddings exist into 4-manifolds with non-trivial Bauer–Furuta invariants.

Background

The authors’ method for constructing exotic families of embeddings requires starting with an embedding of the given 3-manifold Y into a 4-manifold V that has a non-trivial gauge-theoretic invariant (Donaldson, Seiberg–Witten, or Bauer–Furuta). They therefore pose the general existence problem of whether such embeddings are possible for all 3-manifolds.

They emphasize that the embedding should be separating to avoid trivial constructions like Y ↪ (S^1 × Y) # X when X itself has non-trivial invariants. They also note known obstructions: certain 3-manifolds do not embed smoothly in any symplectic 4-manifold, which blocks an easy route via symplectic geometry to positive answers (Daemi–Lidman–Miller).