  The configuration-space topology in canonical General Relativity depends on
the choice of the initial data 3-manifold. If the latter is represented as a
connected sum of prime 3-manifolds, the topology receives contributions from
all configuration spaces associated to each individual prime factor. There are
by now strong results available concerning the diffeomorphism group of prime
3-manifolds which are exploited to examine the topology of the configuration
spaces in terms of their homotopy groups. We explicitly show how to obtain
these for the class of homogeneous spherical primes, and communicate the
results for all other known primes except the non-sufficiently large ones of
infinite fundamental group.
