Product and coproduct in string topology
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Product and coproduct in string topology. / Hingston, Nancy; Wahl, Nathalie.
In: Annales Scientifiques de l'Ecole Normale Superieure, Vol. 56, No. 5, 2023, p. 1381-1447.Research output: Contribution to journal › Journal article › Research › peer-review
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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 -
ID: 387701794