Orbifolds having Euler number zero Heegaard decomposition (Q2039735)

This paper classifies closed 3-dimensional orbifolds admitting Heegaard decompositions into orbifold handlebodies with Euler number zero. It also computes the fundamental groups. More precisely, the authors classify \(H(O, O')\) for handlebody orbifolds \(O\) and \(O'\) with Euler characteristic zero where \begin{itemize} \item \(H(O, O')\) denotes the equivalence classes of 3-orbifolds with \(\partial O\) and \(\partial O'\) identified by orbifold homeomorphisms \item where two elements with corresponding identified orbifolds \(M\) and \(M'\) are equivalent if there is an orbifold homeomorphism \(f:M \rightarrow M'\) preserving \(O\) and \(O'\). \end{itemize} They also use the slightly unusual notation of \(H\circ G\) for the semidirect product \(H \rtimes G\) of two groups \(H\) and \(G\). \(H\circ_{-1} G\) means \(H\rtimes G\) where the \(G\)-action \(G \rightarrow \mathrm{Aut}(H)\) factors to a nontrivial \(\mathbb{Z}_2\)-action \(\mathbb{Z}_2 \rightarrow \mathrm{Aut}(H)\) on \(H\). Finally, \(\mathrm{Dih}(G)\) means \(G \circ_{-1} \mathbb{Z}_2\) where \(\mathbb{Z}_2\) acts nontrivially on \(G\).