Finite actions on the Klein four-orbifold and prism manifolds

Abstract

We describe the finite group actions, up to equivalence, which can act on the orbifold $\Sigma(2,2,2)$, and their quotient types. This is then used to consider actions on prism manifolds $M(b,d)$ which preserve a longitudinal fibering, but do not leave any Heegaard Klein bottle invariant. If $\varphi\colon G\rightarrow \text{Homeo} (M(b,d))$ is such an action, we show that $M(b,d) = M(b,2)$ and $M(b,2)/\varphi$ fibers over a certain collection of 2-orbifolds with positive Euler characteristic which are covered by $\Sigma(2,2,2)$. For the standard actions, we compute the fundamental group of $M(b,2)/\varphi$ and indicate when it is a Seifert fibered manifold.