  An asymptotically AdS geometry connecting two or more boundaries is given by
a entangled state, that can be expanded in the product basis of the Hilbert
spaces of each CFT living on the boundaries. We derive a prescription to
compute this expansion for states describing spacetimes with general spatial
topology in arbitrary dimension. To large N, the expansion coincides with the
Schmidt decomposition and the coefficients are given by $n$-point correlation
functions on a particular Euclidean geometry.
  We show that this applies to all spacetime that admits a Hartle-Hawking type
of wave functional, which via a standard hypothesis on the spatial topology,
can be (one to one) mapped to CFT states defined on the asymptotic boundary. It
is also observed that these states are endowed with quantum coherence
properties.
  Applying this as holographic engineering, one can to construct an emergent
space geometry with certain predetermined topology by preparing an entangled
state of the dual quantum system. As an example, we apply the method to
calculate the expansion and characterize a spacetime whose initial spatial
topology is a (genus one) handlebody.
