Complexity of 3-orbifolds (Q2493883)

The author extends Matveev's theory of complexity for \(3\)-manifolds to closed, orientable, locally orientable \(3\)-orbifolds. A special polyhedron (fake surface) \(P\) is a \textit{spine} of a \(3\)-orbifold \(X\) if \(P\) is embedded in \(X\) such that (1) \(P\) intersects the singular set \(S(X)\) transversely only at surface points of \(P\) and non-vertex points of \(S(X)\) and (2) each component of \(| X| \) cut along \(P\) is isomorphic to a \(3\)-discal orbifold \(D_{\star}^3\). The complexity \(c(P,S(X))\) of \(P\) relative to \(S(X)\) is defined to be (the number of vertices of \(P) + \Sigma (p -1)\), where the sum is over all points \(x\) of \(P\cap S(X)\) and the order of \(S(X)\) at \(x\) is \(p\). The \textit{complexity} \(c(X)\) is then defined to be the minimum of \(c(P,S(X))\) over all spines \(P\) of \(X\). The author shows that, with some exceptions, an irreducible orbifold \(X\) has special minimal spines \(P\) (i.e. \(c(P,S(X)) = c(X)\)). A corollary is that for any given \(n\) there are only finitely many irreducible \(3\)-orbifolds \(X\) of complexity \(n\). He then considers \textit{normal} \(2\)-suborbifolds with respect to \textit{handle decompositions} of \(3\)-orbifolds and shows that essential spherical \(2\)-orbifolds in \(X\) can be normalized (thereby extending the corresponding \(3\)-manifold result of Haken). This is applied to obtain results about additivity of the complexity under orbifold connected sum.