String topology of classifying spaces
type: Item
instance of: scholarly article
title: String topology of classifying spaces
zbMATH Open document ID: 1253.55008
DOI: 10.1515/CRELLE.2011.140
author: David Chataur
author: Luc Menichi
publication date: 2012-09-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/0801.0174
Mathematics Subject Classification ID: 55P50
Mathematics Subject Classification ID: 55P48
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 6081357
zbMATH Keywords: string topology
zbMATH Keywords: classifying space
zbMATH Keywords: Hochschild cohomology
zbMATH Keywords: conformal field theory
MaRDI profile type: Publication
arXiv ID: 0801.0174
cites work: String topology for loop stacks
cites work: Cyclic homology and algebraic K-theory of spaces. II
cites work: On a theorem of Kontsevich
cites work: Topological conformal field theories and Calabi-Yau categories
cites work: Transfer maps for fibrations
cites work: Gerstenhaber duality in Hochschild cohomology
cites work: Batalin-Vilkovisky algebras and two-dimensional topological field theories
cites work: Invariant functions on Lie groups and Hamiltonian flows of surface group representations
cites work: Cyclic homology, derivations, and the free loopspace
cites work: String topology prospectra and Hochschild cohomology
cites work: Orbifold string topology
cites work: Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras
cites work: The Hochschild Cohomology Ring of a Group Algebra
cites work: \(K(n)\)-local duality for finite groups and groupoids
cites work: Equivariant operads, string topology, and Tate cohomology
OpenAlex ID: W2963235379
published in: Journal für die Reine und Angewandte Mathematik
review text: A new and very rich algebraic structure has been defined by Chas and Sullivan on the homology of the free loop space of a compact closed oriented manifold. This structure is known under the generic term of string topology and the nicest definition uses 2-dimensional topological field theories.NEWLINENEWLINEIn this paper Chataur and Menichi extend this theory to the classifying spaces of finite groups and compact Lie groups. More precisely they consider groups \(G\) that are either discrete finite groups or connected topological groups whose singular homologies with coefficients in a field are finite dimensional. For such a group \(G\) they prove that the free loop space on the classifying space of \(G\), \(LBG\), is a homological conformal field theory. They deduce that, with coefficients in a principal ideal domain \(\Bbbk\), the cohomology \(H^*(LBG,\Bbbk)\) is a Batalin-Vilkovisky algebra. When \(G\) is a compact Lie group of dimension \(d\), they also construct an isomorphism between \(H^{*+d}(LBG)\) and the Hochschild cohomology \(H\!H^*(S_*(G), S_*(G))\) where \(S_*(G)\) denotes the singular chains of \(G\).
reviewed by: Yves Félix
description: scientific article; zbMATH DE number 6081357