Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology

Consider a compact group G acting on a real or complex Banach Lie group U, by automorphisms in the relevant category, and leaving a central subgroup K ≤ U invariant. We define the spaces _KZⁿ(G, U) of K-relative continuous cocycles as those maps Gⁿ → U whose coboundary is a K-valued (n + 1)-cocycle; this applies to possibly non-abelian U, in which case n = 1. We show that the _KZⁿ(G, U) are analytic submanifolds of the spaces C(Gⁿ, U) of continuous maps Gⁿ → U and that they decompose as disjoint unions of fiber bundles over manifolds of K-valued cocycles. Applications include: (a) the fact that Zⁿ(G, U) ⊂ C(Gⁿ, U) is an analytic submanifold and its orbits under the adjoint of the group of U-valued (n−1)-cochains are open; (b) hence the cohomology spaces Hⁿ(G, U) are discrete; (c) for unital C^*-algebras A and B with A finite-dimensional the space of morphisms A → B is an analytic manifold and nearby morphisms are conjugate under the unitary group U(B); (d) the same goes for A and B Banach, with A finite-dimensional and semisimple; (e) and for spaces of projective representations of compact groups in arbitrary C^* algebras (the last recovering a result of Martin’s).