DGAs and rational homotopy theory

Abstract

This chapter connects the two themes of the book—rational homotopy theory and differential forms. There is an equivalence between Hirsch extensions of the algebra of p.l. forms on a simplicial complex and principal fibrations over the space with fiber an Eilenberg–MacLane space. This equivalence is seen by identifying the cohomology groups that classify each. The basic theorem is that under this correspondence, the Hirsch extension maps to the p.l. forms on the total space of the principal fibration, extending the map on the base and inducing an isomorphism on cohomology. Once this result is established, inductively one shows that the rational Postnikov tower of a space is read off from the minimal model of the p.l. forms on the space. The proof of the main inductive result, the Hirsch lemma, is postponed until Chap. 16.