Embeddings of 3-manifolds in S^4 from the point of view of the 11-tetrahedron census

TL;DR

This paper investigates which prime closed 3-manifolds from the 11-tetrahedron census embed smoothly in the 4-sphere, providing new results and classifications for a subset of 149 hyperbolic candidates.

Contribution

It offers the first systematic analysis of embedding problems for these 3-manifolds within the specific census, identifying known, constructed, and unresolved cases.

Findings

41 manifolds are known to embed in S^4.

Embeddings into homotopy 4-spheres are constructed for 4 manifolds.

67 manifolds are proven not to embed in S^4.

Abstract

This is a collection of notes on embedding problems for 3-manifolds. The main question explored is `which 3-manifolds embed smoothly in the 4-sphere?' 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 the 4-sphere. The majority of this paper is devoted to the embedding problem for these 149 manifolds. At present 41 are known to embed. Among the remaining manifolds, embeddings into homotopy 4-spheres are constructed for 4. 67 manifolds are known to not embed in the 4-sphere. This leaves 37 unresolved cases, of which only 3 are geometric manifolds i.e. having a trivial JSJ-decomposition.