Fiber-preserving and orientation-reversing involutions of Seifert fibered 3-manifolds

TL;DR

This paper classifies fiber-preserving, orientation-reversing involutions on orientable Seifert fibered 3-manifolds, showing they can be decomposed into known involutions and extending results to non-orientable base orbifolds.

Contribution

It introduces a class of involutions $ ext{ extbackslash Psi}$ and proves that all such involutions factor through these, simplifying the classification problem.

Findings

$ ext{ extbackslash Psi}$ forms a single conjugacy class under fiber-preserving diffeomorphisms

Any involution factors as $ ext{ extbackslash psi} ext{ extbackslash circ}g$ with known involutions

Classification extends to non-orientable base orbifolds via double covering

Abstract

We consider fiber-preserving, orientation-reversing involutions on orientable Seifert fibered 3-manifolds and the conditions on a manifold for admissibility of such involutions. We construct a class $\Psi$ of fiber-preserving, orientation-reversing involutions that act trivially on the base. Each element of $\Psi$ is obtained by extending a product involution across Seifert pieces of type $V(2,2;-1)$ - a solid torus with three fibers filled according to Seifert invariants $(2,1)$, $(2,1)$, and $(1,-1)$. We show that $\Psi$ forms a single conjugacy class under fiber-preserving diffeomorphisms. Our main result establishes that any fiber-preserving, orientation-reversing involution factors as $\psi\circ g$, where $g$ is fiber-preserving and orientation-preserving and $\psi\in\Psi$, thus reducing the problem to the previously known orientation-preserving case. Through the orientable base-space double covering, we further extend the classification to manifolds with non-orientable base orbifold.