John asks:

<<
     Is there a shape for space-time that makes space compact but
that doesn't have an invariant factorization as  space x time?
>>

The lens spaces L(7,1) and L(7,2) (compact 3-manifolds) are known to be
topologically distinct but homotopy equivalent.  I wonder if

          L(7,1) x R is homeomorphic to L(7,2) x R.

IF this is the case, then L(7,1) x R = L(7,2) x R is a topological 4-manifold with a non-unique factorization as (compact 3-manifold) x R. 

(Is this what you meant, John?)

--Dan