Lie coalgebras and rational homotopy theory II: Hopf invariants

Abstract

We develop a new framework which resolves the homotopy periods problem. We start with integer-valued homotopy periods defined explicitly from the classic bar construction. We then work rationally, where we use the Lie coalgebraic bar construction to get a sharp model for H o m ( π ∗ X , Q ) \mathrm {Hom}(\pi _* X, {\mathbb Q}) for simply connected X X . We establish geometric interpretations of these homotopy periods, to go along with the good formal properties coming from the Koszul-Moore duality framework. We give calculations, applications, and relationships with the numerous previous approaches.