Cosmology, cohomology, and compactification

TL;DR

This paper extends previous results on cosmological models with compact spatial sections to weakly locally homogeneous spaces, linking the existence of compact quotients to non-vanishing Lie algebra cohomology.

Contribution

It generalizes the Ashtekar-Samuel results to a broader class of spaces using symmetry reduction and cohomology theory.

Findings

Compact quotients exist only if Lie algebra cohomology is non-zero.

Extension of Bianchi class A results to weakly locally homogeneous spaces.

Provides conditions for the existence of compact spatial sections in cosmological models.

Abstract

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.