  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.
