  Studying spacetimes with continuous symmetries by dimensional reduction to a
lower dimensional spacetime is a well known technique in field theory and
gravity. Recently, its use has been advocated in numerical relativity as an
efficient computational technique for the numerical study of axisymmetric
asymptotically flat 4-dimensional spacetimes. We prove here that if the
dimensionally reduced spacetime is a physically reasonable 3-dimensional
asymptotically flat or asymptotically anti-de Sitter spacetime, then,
surprisingly, the topology of the higher dimensional spacetime must be one of
two product topologies. Reductions of other topologies result in physically
pathological spacetimes. In particular, reduction of asymptotically flat
4-dimensional spacetimes must lead to pathologies. These results use only the
topological censorship theorem and topological methods and consequently are
independent of the field equations and reduction method.
