Ashtekar and Samuel have shown that Bianchi cosmological models with compact spatial sections must be of Bianchi class A. Motivated by general results on the symmetry reduction of variational principles, we show how to extend the Ashtekar-Samuel results to the setting of weakly locally homogeneous spaces as defined, e.g., by Singer and Thurston. In particular, it is shown that any m-dimensional homogeneous space G/K admitting a G-invariant volume form will allow a compact discrete quotient only if the Lie algebra cohomology of G relative to K is non-vanishing at degree m.Spatially homogeneous spacetimes have been studied extensively as both classical and quantum cosmological models (see, e.g., [1,2]). In 3+1 dimensions all spatially homogeneous models but one (the Kantowski-Sachs model) admit a freely acting three-dimensional Lie group of isometries. These models are known as 'Bianchi models' since they can be classified-up to topology-according to Bianchi's classification of three-dimensional Lie algebras. It was noted by Ashtekar and Samuel [3] that the orbit manifolds (the preferred spatial slices) for the Bianchi models can be compact only if the Lie algebra of the homogeneity group is Bianchi class A. They also consider a class of locally homogeneous geometries and show that the restriction to Bianchi class A Lie algebras is still necessary for compact spatial sections in this more general setting. It is straightforward to generalize the Ashtekar-Samuel results to orbit manifolds of any dimension. The result is the same provided one generalizes 'Bianchi class A' to 'unimodular' 1 . However, to our knowledge it is unknown how to extend these results to the large class of 'weakly locally homogeneous spaces' 2 of Singer, Thurston,

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