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One-ended 3-manifolds without locally finite toric decompositions

Sylvain Maillot

Abstract

We introduce a class of one-ended open 3-manifolds which can be `recursively' defined from two compact 3-manifolds, and construct examples of manifolds in this class which fail to have a toric decomposition in the sense of Jaco-Shalen and Johannson.

One-ended 3-manifolds without locally finite toric decompositions

Abstract

We introduce a class of one-ended open 3-manifolds which can be `recursively' defined from two compact 3-manifolds, and construct examples of manifolds in this class which fail to have a toric decomposition in the sense of Jaco-Shalen and Johannson.

Paper Structure

This paper contains 11 sections, 10 theorems, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. The orbifold case
  4. Definition of the orbifold $\mathcal{O}_0$
  5. The pillows $P_n$ and their first properties
  6. Case 1
  7. Case 2
  8. The $P_n$'s are canonical
  9. Case 1
  10. Case 2
  11. The manifold $M_0$

Key Result

Theorem 1.1

There is an open 3-manifold $M_0$ in the class $\mathcal{C}$ with the following properties.

Figures (3)

  • Figure 1: General configuration of $\Sigma_{\mathcal{O}_0}$
  • Figure 2: Linking numbers of components of $\Sigma_{\mathcal{O}_0}$
  • Figure 3: The canonical pillows $P_n$, the compact set $K'$, and the plane $\Pi$

Theorems & Definitions (23)

  • Definition
  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4: Schubert schubert:vollringe
  • Lemma 2.5
  • proof
  • ...and 13 more