Lie coalgebras and rational homotopy theory II: Hopf invariants

TL;DR

This paper introduces a new approach to rational homotopy theory using Lie coalgebras, providing explicit formulas for homotopy groups as generalized linking invariants, and explores applications to spheres, homogeneous spaces, and configuration spaces.

Contribution

It offers a novel solution to the homotopy periods problem, connecting geometric invariants with algebraic formalism, and generalizes classical linking number concepts.

Findings

Homotopy groups are rationally described by generalized linking invariants.

Provides criteria for homotopy equivalence of maps from spheres.

Extends classical linking number concepts to complex spaces.

Abstract

We give a new solution of the "homotopy periods" problem, as highlighted by Sullivan, which places explicit geometrically meaningful formulae first dating back to Whitehead in the context of Quillen's formalism for rational homotopy theory and Koszul-Moore duality. Geometrically, we show that homotopy groups are rationally given by "generalized linking/intersection invariants" of cochain data. Moreover, we give a method for determining when two maps from $S^n$ to $X$ are homotopic after allowing for multiplication by some integer. For applications, we investigate wedges of spheres and homogeneous spaces (where homotopy is given by classical linking numbers), and configuration spaces (where homotopy is given by generalized linking numbers); also we propose a generalization of the Hopf invariant one question.