One-ended 3-manifolds without locally finite toric decompositions

TL;DR

This paper introduces a new class of one-ended open 3-manifolds constructed recursively from compact 3-manifolds, demonstrating that some lack the traditional toric decompositions defined by Jaco-Shalen and Johannson.

Contribution

It defines a novel class of 3-manifolds and provides explicit examples that do not admit the classical toric decomposition, challenging existing decomposition theories.

Findings

Existence of 3-manifolds without toric decompositions

Construction method for these manifolds

Counterexamples to classical decomposition conjectures

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.