Hodge Theory and Iterated Integrals

Abstract

In this chapter we give Hain's construction of a mixed Hodge structure on homotopy groups. His results are explained in § 8.2 after a first section in which we collect some basic material from homotopy theory that we need later on. A central result in this section is the Borel-Serre theorem which (under suitable assumptions) relates the homotopy groups of a topological space to the homology of its loop space. Loop spaces are no longer finite dimensional manifolds and so we can not hope to put directly a mixed Hodge structure on their cohomologygroups. However, dually, the cohomology of a loop space can be calculated by means of an integration procedure which associates to an ordered set of forms on a differential manifold a single form on its loop space, which is called an "iterated integral". It is explained in § 8.3. In §8.4 and §8.5 we explain how to deal with the fundamental group of a smooth complex projective variety. The mixed Hodge structure given there depends on the choice of base points.