A Structure Theorem for Bad 3-Orbifolds

TL;DR

This paper provides a detailed construction and characterization of bad 3-orbifolds, showing how they can be systematically built from good orbifolds through specific operations, and establishing a structure theorem for these spaces.

Contribution

It explicitly constructs a collection of bad 3-orbifolds and demonstrates how all bad 3-orbifolds can be obtained from good ones via a finite sequence of cut-and-cap operations.

Findings

Explicit construction of bad 3-orbifolds with specific boundary properties

Any bad 3-orbifold contains a sub-orbifold from the constructed collection

Finitely many cut-and-cap operations transform bad orbifolds into good orbifolds

Abstract

We explicitly construct a collection of bad 3-orbifolds, \(\mathcal{X}\), satisfying the following properties: \begin{enumerate} \item The underlying topological space of any \(X \in \mathcal{X}\) is homeomorphic to $S^2\times I$ or $(S^2\times S^1)\backslash B^3$. \item The boundary of any \(X \in \mathcal{X}\) consists of one or two spherical 2-orbifolds. \item Any bad 3-orbifold is obtained from a good 3-orbifold by repeating, finitely many times, the following operation: remove one or two orbifold-balls, and glue in some \(X \in \mathcal{X}\). \end{enumerate} Conversely, any bad 3-orbifold \(\OO\) contains some \(X \in \mathcal{X}\) as a sub-orbifold; we call removing \(X\) and capping the resulting boundary \em cut-and-cap.\em\ Then by cutting-and-capping finitely many times we obtain a good orbifold.