The fundamental group

Abstract

Until now we have been working with simply connected spaces, but this chapter takes up the question of the fundamental group and its relation to differential forms. The notion of a 1-minimal model is introduced, and it is shown that every connected DGA has a 1-minimal model, unique up to isomorphism. The dual of this is a tower of nilpotent Lie algebras. We then turn to the nilpotent quotients of the fundamental group. The main result is that the tower of nilpotent rational Lie algebras associated with tower of nilpotent Lie group quotients of the fundamental group of a space is dual to the 1-minimal model of the p.l. forms on the space.