Polyhedral Deformations of Cone Manifolds

Abstract

Two single parameter families of polyhedra $P(ψ)$ are constructed in three dimensional spaces of constant curvature $C(ψ)$. Identification of the faces of the polyhedra via isometries results in cone manifolds $M(ψ)$ which are topologically $S^1\timesS^2$, $S^3$ or singular $S^2$. The singular set of $M(ψ)$ can have self intersections for some values of $ψ$ and can also be the Whitehead link or form other configurations. Curvature varies continuously with $ψ$. At $ψ=0$ spontaneous surgery occurs and the topological type of $M(ψ)$ changes. This phenomenon is described.