Computational examples of rational string operations on Gorenstein spaces
String topology is the study of algebraic structures on the homology \(H_*(LX)\), where \(LX\) denotes the free loop space of \(X\). The first result was given by \textit{M. Chas} and \textit{D. Sullivan} [String Topology (1999; \url{arXiv:math/9911159})]. They took \(X=M\) a closed oriented manifold and showed that \(H_*(LX)\) is a Batalin-Vilkovisky algebra. This result was extended by \textit{D. Chataur} and \textit{L. Menichi} [J. Reine Angew. Math. 669, 1--45 (2012; Zbl 1253.55008)] when \(X=M\) (resp. \(BG\) the classifying space of a compact Lie group \(G\)). They proved that \(H_*(LX)\) is a homological conformal field theory of degree \(\dim M\) (resp. \(-\text{rank}\,G\)). Closed oriented simply-connected manifolds and classifying spaces of a compact connected Lie group are both Gorenstein spaces. As proved by \textit{Y. Félix} and \textit{J.-C. Thomas} [Bull. Soc. Math. Fr. 136, No. 2, 311--327 (2008; Zbl 1160.55006)], string operations can be defined in this more general situation. The purpose of this paper is to provide an explicit way to compute these operations for Gorenstein spaces with the aid of Sullivan minimal models when the coefficients field is of characteristic zero. These methods allow to rediscover the few known results as well as to make explicit computations with the Borel space associated with a group action of a compact connected Lie group.
scientific article; zbMATH DE number 6538888