  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
$K\le U$ invariant. We define the spaces ${}_KZ^n(G,U)$ of $K$-relative
continuous cocycles as those maps ${G^n\to U}$ 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 ${}_KZ^n(G,U)$ are analytic submanifolds of the spaces
$C(G^n,U)$ of continuous maps $G^n\to U$ and that they decompose as disjoint
unions of fiber bundles over manifolds of $K$-valued cocycles. Applications
include: (a) the fact that ${Z^n(G,U)\subset C(G^n,U)}$ 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 $H^n(G,U)$ are
discrete; (c) for unital $C^*$-algebras $A$ and $B$ with $A$ finite-dimensional
the space of morphisms $A\to B$ 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).
