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 VIIh universes with non-compact topologies are excluded by topological compactness. Blanch type V and Wh universes with compact topologies must be exactly isotropic. In the flat case we calculate the dynamical degrees of freedom of Blanch type I and Vile universes with compact 3-spaces and show that type VII0 solutions are more general than type I solutions for systems with a perfect fluid, although the type I models are more general than type VII0 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.