Products and coproducts in string topology

Nancy Hingston, Nathalie Wahl

Published 2017-09-20Version 1

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 non-relative product. It is associative, graded 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 non-vanishing of the $k$--th iterate of the dual coproduct detects the presence of loops with $(k+1)$-fold self-intersections in the image of chain representatives of homology classes in the loop space. For spheres and projective spaces, we show that this is sharp, in the sense that the $k$--iterated coproduct vanishes precisely when a class has 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, and show that both structures are preserved by degree 1 maps, proving in particular that the Goresky-Hingston product is homotopy invariant. 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 or delicate intersections of chains in loop spaces.