Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Ryan Budney, Benjamin A. Burton

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.

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

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.

Paper Structure

This paper contains 8 sections, 12 theorems, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Obstructions and embedding constructions
  3. Manifolds from the census which embed smoothly in $S^4$
  4. Manifolds which embed in homotopy $4$-spheres
  5. Manifolds in the census known to not embed in $S^4$
  6. Manifolds for which embeddability is not known
  7. Notes on computations & notation
  8. Observations and questions from the data

Key Result

Theorem 2.2

HanKK (Hantzsche Test) If $M$ is a compact, boundaryless, connected, oriented $3$-manifold which embeds in a homology $S^4$ then there is a splitting $\tau H_1(M,{\mathbb Z}) = A \oplus B$, inducing a splitting which is reversed by Poincaré duality, in the sense that the P.D. isomorphism restricts to isomorphisms $A \to {\mathrm{Hom}}_{\mathbb Z}(B,{\mathbb Q}/{\mathbb Z})$ and $B \to {\mathrm{H

Figures (3)

  • Figure 1: A $0$-surgical embedding.
  • Figure 2: A $1$-surgical embedding.
  • Figure 3: Dual polyhedral bits inside a tetrahedron $\Delta_3$

Theorems & Definitions (25)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • ...and 15 more