# New 3-manifolds in the 4-sphere

I've been kicking around a project, trying to discover which 3-manifolds embed in the 4-sphere for five years now. This preprint represents the state of the art. V5 should appear on Thursday, Sept. 27th. There's two big changes in the preprint from V4.

1) Andrew Donald read it, and found a mistake. The version of the paper he was working off was the arXiv version 4. In that paper, the mistake appears in section 5, item 37. The manifold is SFS[S2 : 1/2, 1/2, 1/2, -5/3]. I miscomputed a characteristic link, resulting in a miscomputed mu invariant. So now (V5) the manifold is listed in Section 4, item 23, as Donald shows it embeds in S4. Donald also found several other embeddings in his preprint.

2) Ben Burton and I have been finding embeddings of 3-manifolds in 4-spheres via a new technique. What we're doing is the analogue of normal surface theory in triangulated 3-manifolds. But we're computing normal 3-manifolds in triangulated 4-manifolds. The idea is we take a triangulation of a (homotopy) S4, and search for 3-manifolds that appear to be linear in each top-dimensional simplex and transverse to the triangulation. This is an integer linear programming problem. The setup in 4-manifold theory is quite similar to the 3-manifold theory formalism. In 3-manifold theory you have normal triangles and quadrilaterals. Normal 3-manifolds in triangulated 4-manifolds consist of tetrahedra and "prisms" -- things that look like triangles cross an interval. Interestingly, Ben and I recover most of Andrew Donald's embeddings this way, as well as most of the embeddings appearing in our paper that were constructed via other techniques. At present we've found two completely novel embeddings. These are 3-manifolds that have one incompressible torus, separating the 3-manifold into two torus knot complements. In one situation it's two trefoil complements, and the 3-manifold is a homology sphere. In the 2nd case it's a (3,5)-torus knot complement glued to a Seifert fibered manifold SFS[D2 : 1/2, 1/2], and this 3-manifold has homology (Z2)2. See the preprint for details.

At present I've only verified that these embeddings are into homotopy 4-spheres. In the next few months I hope to confirm these are triangulations of the standard S4. I'm also hopeful that we'll find plenty of other new and interesting 3-manifolds in the 4-sphere via these techniques. But the techniques are fairly heavy handed. As I write this, a cluster of 96 CPUs is busily hunting for 3-manifold embeddings around-the-clock.