Finite, fiber-preserving group actions on elliptic 3-manifolds

TL;DR

This paper classifies finite fiber-preserving group actions on elliptic 3-manifolds, showing restrictions based on Euler class and providing explicit descriptions of quotient spaces for these actions.

Contribution

It extends previous work by focusing on elliptic 3-manifolds, proving non-existence of certain diffeomorphisms, and classifying possible group actions with explicit quotient space presentations.

Findings

Orientation-reversing fiber-preserving diffeomorphisms do not exist for nonzero Euler class.

Most elliptic 3-manifolds admit specific finite fiber-preserving group actions.

Explicit presentations of quotient spaces under these actions are constructed.

Abstract

In two previous papers the author presented a general construction of finite, fiber- and orientation-preserving group actions on orientable Seifert manifolds. In this paper we restrict our attention to elliptic 3-manifolds. A proof is given that orientation-reversing and fiber-preserving diffeomorphisms of Seifert manifolds do not exist for nonzero Euler class, in particular elliptic 3-manifolds. Each type of elliptic 3-manifold is then considered and the possible group actions that fit the given construction. This is shown to be all but a few cases that have been considered elsewhere. Finally, a presentation for the quotient space under such an action is constructed and a specific example is generated.