Polyhedral Deformations of Cone Manifolds A Aalam

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 × S, S or singular S . 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. 0 Introduction We study continuous families of cone manifolds Mψ parametrised by cone angle which begin at cone angle zero with the complement of the Whitehead link in S. We consider the case of equal cone angles on all singular link components. Increasing cone angles the families trace different paths in Dehn surgery space joined by what we call a Dehn surgery transition point. The cone structures for certain non-zero values of cone angles exist in projective models or in S. In one Dehn surgery direction the cone manifolds are for certain cone angles, obtained by surgery on the Whitehead link in S resulting in a topologically distinct singular set in S × S. As cone angle is increased the topological type of the singular set changes and the hyperbolic cone manifold develops two cusps and becomes S at cone angle 2 3 π. The topological type of the singular set and the structure of Mψ remain unaltered as cone angle increases beyond 2 3 π until we reach a cone angle ω where Mψ becomes R 3 with topologically the same type of singularity. Increasing cone angle past ω the singular set reverts back to its pre2 3 π cone angle topological type andMψ becomes spherical in S 2 ×S1. At cone angle π the underlying polyhedron becomes a lens in S from which the cone manifold is obtained by suitable identifications. For cone angles in the interval [π, ζ], Mψ is spherical and the topological type of its singular set is unchanged but it is now in S. At cone angle ζ, Mψ becomes the suspension of a sphere with four cone points. It remains the well understood sphere with four singularities for cone angles larger than ζ. Investigating the deformation on the other side of Dehn surgery we obtain the Whitehead link in S for certain non-zero cone angles. A complete investigation will be carried out later.