Of Sullivan models, Massey products, and twisted Pontrjagin products

TL;DR

This paper investigates the properties of Pontrjagin rings associated with loop spaces of configuration spaces, revealing their non-invariance under homotopy and highlighting the role of Massey products and Whitehead products.

Contribution

It demonstrates that Pontrjagin rings are not homotopy invariants and explores twisted versions influenced by Massey and Whitehead products, providing new insights into their algebraic structures.

Findings

Pontrjagin rings are not homotopy invariants.

Whitehead products influence the isomorphism types of rational Hopf algebras.

Massey products are central to the structure of these algebraic invariants.

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.