Multiple period integrals and cohomology

We give a version of the Eichler-Shimura isomorphism with a nonabelian H-1 in group cohomology. Manin has given a map from vectors of cusp forms to a noncommutative cohomology set by means of iterated integrals. We show that Manin's map is injective but far from surjective. By extending Manin's map we are able to construct a bijective map and remarkably this establishes the existence of a nonabelian version of the Eichler-Shimura map.