  Let M be a closed Riemannian manifold. We extend the product of
Goresky-Hingston, on the cohomology of the free loop space of M relative to the
constant loops, to a nonrelative product. It is graded associative and
commutative, and compatible with the length filtration on the loop space, like
the original product. We prove the following new geometric property of the dual
homology coproduct: the nonvanishing of the k-th iterate of the coproduct on a
homology class ensures the existence of a loop with a (k+1)-fold
self-intersection in every representative of the class. For spheres and
projective spaces, we show that this is sharp, in the sense that the k-th
iterated coproduct vanishes precisely on those classes that have support in the
loops with at most k-fold self-intersections. We study the interactions between
this cohomology product and the more well-known Chas-Sullivan product. We give
explicit integral chain level constructions of these loop products and
coproduct, including a new construction of the Chas-Sullivan product, which
avoid the technicalities of infinite dimensional tubular neighborhoods and
delicate intersections of chains in loop spaces.
