  We report on simulations of DT simplicial gravity for manifolds with the
topology of the 4-disk. We find evidence for four phases in a two-dimensional
parameter space. In two of these the boundary plays no dynamical role and the
geometries are equivalent to those observed earlier for the sphere $S^4$. In
another phase the boundary is maximal and the quantum geometry degenerates to a
one dimensional branched polymer. In contrast we provide evidence that the
fourth phase is effectively three-dimensional. We find discontinuous phase
transitions at all the phase boundaries.
