Euclidean-valued group cohomology is always reduced
If G is a countable discrete group acting linearly on a finite-dimensional vector space over any topological field, then the groups of coboundaries are closed for the product topology in all degrees, and hence the cohomology is reduced in all degrees. This is deduced from a more general automatic-closure theorem for continuous linear transformations between inverse limits of finite-dimensional vector spaces.
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Group Theory
Functional Analysis
