Computational examples of rational string operations on Gorenstein spaces
type: Item
instance of: scholarly article
title: Computational examples of rational string operations on Gorenstein spaces
zbMATH Open document ID: 1333.55010
author: Takahito Naito
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2016-02-03T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bbms/1447856058
review text: 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.
Mathematics Subject Classification ID: 55P50
Mathematics Subject Classification ID: 55P62
zbMATH DE Number: 6538888
zbMATH Keywords: string topology
zbMATH Keywords: Gorenstein space
zbMATH Keywords: rational homotopy theory
reviewed by: Jean-Claude Thomas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6538888