TY - JOUR

T1 - Product and coproduct in string topology

AU - Hingston, Nancy

AU - Wahl, Nathalie

N1 - Publisher Copyright: © 2023 Société Mathématique de France. Tous droits réservés.

PY - 2023

Y1 - 2023

N2 - 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 C 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 better-known Chas-Sullivan product. We give explicit integral chain level constructions of the loop product and coproduct, including a new construction of the Chas-Sullivan product, which avoids the technicalities of infinite dimensional tubular neighborhoods and delicate intersections of chains in loop spaces.

AB - 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 C 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 better-known Chas-Sullivan product. We give explicit integral chain level constructions of the loop product and coproduct, including a new construction of the Chas-Sullivan product, which avoids the technicalities of infinite dimensional tubular neighborhoods and delicate intersections of chains in loop spaces.

UR - http://www.scopus.com/inward/record.url?scp=85188821464&partnerID=8YFLogxK

U2 - 10.24033/asens.2558

DO - 10.24033/asens.2558

M3 - Journal article

AN - SCOPUS:85188821464

VL - 56

SP - 1381

EP - 1447

JO - Annales Scientifiques de l'Ecole Normale Superieure

JF - Annales Scientifiques de l'Ecole Normale Superieure

SN - 0012-9593

IS - 5

ER -