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

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

2008-10-14

arxiv.org/abs/0810.2346v4

Mathematics

Geometric Topology

48 pages, 80+ figures. V4 add a Gilmer attribution, update questions at end of paper

Scientific paper

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 31 are known to embed. Among the remaining manifolds, embeddings into homotopy 4-spheres are constructed for two. 63 manifolds are known to not embed in the 4-sphere. This leaves 53 unresolved cases, of which only 12 are geometric manifolds i.e. having a trivial JSJ-decomposition.

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