Fibered spherical 3-orbifolds

TL;DR

This paper explores the classification of fibered spherical 3-orbifolds by linking algebraic subgroup classifications of SO(4) with their topological properties, providing explicit formulas for understanding their structure.

Contribution

It establishes explicit formulas connecting finite subgroups of SO(4) with invariants of fibered spherical 3-orbifolds, enhancing the algebraic-topological understanding.

Findings

Explicit formulas relating subgroups of SO(4) to orbifold invariants

Topological properties deduced from algebraic classification

Enhanced understanding of fibered spherical 3-orbifolds

Abstract

In early 1930s Seifert and Threlfall classified up to conjugacy the finite subgroups of $\mathrm{SO}(4)$, this gives an algebraic classification of orientable spherical 3-orbifolds. For the most part, spherical 3-orbifolds are Seifert fibered. The underlying topological space and singular set of non-fibered spherical 3-orbifolds were described by Dunbar. In this paper we deal with the fibered case and in particular we give explicit formulae relating the finite subgroups of $\mathrm{SO}(4)$ with the invariants of the corresponding fibered 3-orbifolds. This allows to deduce directly from the algebraic classification topological properties of spherical 3-orbifolds.