Hodge Theory and Minimal Models

Abstract

This chapter is devoted to Sullivan's theory of the minimal model and Morgan's construction of a mixed Hodge structure on the homotopy groups using minimal models. A priori this mixed Hodge structure might differ from Hain's. But in fact, for the higher homotopy groups the two are equal (as communicated to us by Hain). However, since the base point is absent in Morgan's construction for the fundamental group, it cannot be the same as Hain's. On the other hand, Morgan's construction is more powerful since it gives a mixed Hodge structure on the cohomology level of the constituents of the (rational) Postnikov tower (which contains all information from rational homotopy).