Embeddings of 3-manifolds in S4 from the point of view of the 11-tetrahedron census

This is a collection of notes on embedding problems for 3-manifolds. The main question explored is ‘which 3-manifolds embed smoothly in S4?’ The terrain of exploration is the Burton/Martelli/Matveev/Petronio census of triangulated prime closed 3-manifolds built from 11 or less tetrahedra. There are 13766 manifolds in the census, of which 13400 are orientable. Of the 13400 orientable manifolds, only 149 of them have hyperbolic torsion linking forms and are thus candidates for embedability in S4 . The majority of this paper is devoted to the embedding problem for these 149 manifolds. At present 28 are known to embed in S4 . Among the remaining manifolds, embeddings into homotopy 4-spheres are constructed for 3 manifolds in the list. 51 manifolds in the list are known to not embed in S4 . This leaves 55 unresolved cases, of which only 14 are geometric manifolds ie: having a trivial JSJ-decomposition.