The construction from initial data of spacetimes with nontrivial spatial and bundle topology

We show how to use the 3 + 1 construction program to build globally heperbolic spacetimes with topologically nontrivial Cauchy surfaces. Spacetimes in which the classical fields are sections of a nontrivial bundle are handled as well. In evolving the initial data in these spacetimes, one must work with an atlas of overlapping patches. Data must be transfered from patch to patch during the evolution, so the transition functions on patch intersections must be evolved as well. We describe how to do this. Often the evolution of the Cauchy data is considerably simplified by choosing the coordinate-shift field $\text{M}$ and the gaugeshift field A_↓ to be patch dependent. We give examples of this phenomenon and show how to incorporate the patch dependence of $\text{M}$ and A_↓ into a consistent evolution program for the spacetime.