The Isotropy of Compact Universes

We discuss the problem of the stability of the isotropy of the universe in the space of ever-expanding spatially homogeneous universes with a compact spatial topology. The anisotropic modes which prevent isotropy being asymptotically stable in Bianchi-type V IIh universes with non-compact topologies are excluded by topological compactness. Bianchi type V and type V IIh universes with compact topologies must be exactly isotropic. In the flat case we calculate the dynamical degrees of freedom of Bianchi-type I and V II0 universes with compact 3-spaces and show that type V II0 solutions are more general than type I solutions for systems with perfect fluid, although the type I models are more general than type V II0 in the vacuum case. For particular topologies the 4-velocity of any perfect fluid is required to be non-tilted. Various consequences for the problems of the isotropy, homogeneity, and flatness of the universe are discussed.