String topology and configuration spaces of two points

Thursday, 14 November, 2019

Published in: 

arXiv:1911.06202

Given a closed manifold M. We give an algebraic model for the Chas-Sullivan product and the Goresky-Hingston coproduct. In the simply-connected case, this admits a particularly nice description in terms of a Poincaré duality model of the manifold, and involves the configuration space of two points on M. We moreover, construct an IBL_\infty-structure on (a model of) cyclic chains on the cochain algebra of M, such that the natural comparison map to the S^1-equivariant loop space homology intertwines the Lie bialgebra structure on homology. The construction of the coproduct/cobracket depends on the perturbative partition function of a Chern-Simons type topological field theory. Furthermore, we give a construction for these string topology operations on the absolute loop space (not relative to constant loops) in case that M carries a non-vanishing vector field and obtain a similar description. Finally, we show that the cobracket is sensitive to the manifold structure of M beyond its homotopy type. More precisely, the action of {\rm Diff}(M) does not (in general) factor through {\rm aut}(M).

Author(s): 

Florian Naef
Thomas Willwacher