#### Vol. 10, No. 3, 2016

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Multiple period integrals and cohomology

### Roelof W. Bruggeman and Youngju Choie

Vol. 10 (2016), No. 3, 645–664
##### Abstract

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.

##### Keywords
cusp form, iterated integral, noncommutative cohomology
Primary: 11F67
Secondary: 11F75
##### Milestones
Received: 29 June 2015
Revised: 26 February 2016
Accepted: 28 February 2016
Published: 12 June 2016
##### Authors
 Roelof W. Bruggeman Mathematisch Instituut Universiteit te Utrecht Postbus 80010 3508 TA Utrecht Netherlands Youngju Choie Department of Mathematics and PMI Postech Pohang 790-784 South Korea