Of Sullivan models, Massey products, and twisted Pontrjagin products

Abstract

Associated to every connected, topological space $X$ there is a Hopf algebra - the Pontrjagin ring of the based loop space of the configuration space of two points in X. We prove that this Hopf algebra is not a homotopy invariant of the space. We also exhibit interesting examples of H-spaces, which are homotopy equivalent as spaces, which either lead to isomorphic rational Hopf algebras or not, depending crucially on the existence of Whitehead products. Moreover, we investigate a (naturally motivated) twisted version of these Pontrjagin rings in the various aforementioned contexts. In all of these examples, Massey products abound and play a key role.