We relate the Batalin-Vilkovisky (BV) algebra structure of the string topology of a manifold to the homological algebra of the singular chains of the based loop space of that manifold, showing that its Hochschild cohomology carries a BV algebra structure isomorphic to that of string topology. Furthermore, this structure is compatible with the usual cup product and Lie bracket on Hochschild cohomology. This isomorphism arises from a derived form of Poincare duality using modules over the based loop space as local coefficient systems. This derived Poincare duality also comes from a form of fibrewise Atiyah duality on the level of fibrewise spectra, and we use this perspective to connect the algebraic constructions to the Chas-Sullivan loop product.
String topology and the based loop space
2010
en
665049539
Malm, Eric James.
Malm
Eric James
Stanford University. Department of Mathematics.
Carlsson, G. (Gunnar), 1952-
Carlsson
G.
1952
Kerckhoff, Steve,
Kerckhoff
Steve
Cohen, Ralph L., 1952-
Cohen
Ralph L.
1952
Galatius, Søren, 1976-
Galatius
Søren
1976