String topology for loop stacks

Behrend, Kai; Ginot, Grégory; Noohi, Behrang; Xu, Ping

doi:10.1016/j.crma.2006.10.006

Topology/Geometry

String topology for loop stacks

Presented by: Charles-Michel Marle

Behrend, Kai ¹; Ginot, Grégory ²; Noohi, Behrang ³; Xu, Ping ⁴

¹ Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2
² École normale supérieure de Cachan et université Paris 13, CMLA, 61, avenue du Président Wilson, 94230 Cachan cedex, France
³ Max Planck Institut für Mathematik, Vivastsgasse 7, 53111 Bonn, Germany
⁴ Pennsylvania State University, 210 McAllister Building, University Park, PA 16802, USA

Comptes Rendus. Mathématique, Volume 344 (2007) no. 4, pp. 247-252.

Abstract
Résumé

We prove that the homology groups of the free loop stack of an oriented stack are equipped with a canonical loop product and coproduct, which makes it into a Frobenius algebra. Moreover, the shifted homology $H_{•} (L X) = H_{• + d} (L X)$ admits a BV algebra structure.

On munit les groupes d'homologie du champ des lacets libres d'un champ orienté d'un produit et d'un coproduit induisant une structure d'algèbre de Frobenius. De plus, l'homologie en degrés décalés $H_{•} (L X) = H_{• + d} (L X)$ est une algèbre BV.

Received: 2006-08-09
Accepted: 2006-10-11
Published online: 2007-02-06

DOI: 10.1016/j.crma.2006.10.006

Author's affiliations:

Behrend, Kai ¹; Ginot, Grégory ²; Noohi, Behrang ³; Xu, Ping ⁴

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Behrend, Kai; Ginot, Grégory; Noohi, Behrang; Xu, Ping. String topology for loop stacks. Comptes Rendus. Mathématique, Volume 344 (2007) no. 4, pp. 247-252. doi : 10.1016/j.crma.2006.10.006. https://www.numdam.org/articles/10.1016/j.crma.2006.10.006/

References
Cited by

[1] K. Behrend, G. Ginot, B. Noohi, P. Xu, String product for inertia stacks, preprint

[2] K. Behrend, G. Ginot, B. Noohi, P. Xu, Frobenius structure for inertia stacks, preprint

[3] Chas, M.; Sullivan, D. String topology | arXiv

[4] Cohen, R.; Godin, V. A polarized view of string topology, Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser., vol. 308, 2004, pp. 127-154

[5] Cohen, R.; Jones, J. A homotopy theoretic realization of string topology, Math. Ann., Volume 324 (2002) no. 4, pp. 773-798

[6] Cohen, R.; Voronov, A. Notes on string topology, String Topology and Cyclic Homology, Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel, 2006, pp. 1-95

[7] Lupercio, E.; Uribe, B.; Xicoténcatl, M. Orbifold string topology | arXiv

[8] Moerdijk, I.; Mrčun, J. Lie groupoids, sheaves and cohomology, Poisson Geometry, Deformation Quantisation and Group Representations, London Math. Soc. Lecture Note Ser., vol. 323, 2005, pp. 145-272

[9] Noohi, B. Foundations of topological stacks, I | arXiv

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