String topology of classifying spaces
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 \(G\).
scientific article; zbMATH DE number 6081357
scientific article