Group 1-cohomology is complemented
  We show a structural property of cohomology with coefficients in an isometric representation on a uniformly convex Banach space: if the cohomology group $H^1(G,\pi)$ is reduced, then, up to an isomorphism, it is a closed complemented, subspace of the space of cocycles and its complement is the subspace of coboundaries. 
Piotr W. Nowak
math.GR math.AT math.FA