String topology for stacks (Q2897216)

String topology is a term introduced by Chas and Sullivan to describe the algebraic structure of the homology of the free loop space \(LM\) on an oriented manifold \(M\). Later string topology has been extended to spaces that are not manifolds, for instance to classifying spaces of compact connected Lie groups or finite discrete groups by Chataur and Menichi, and to orbifolds by Lupercio, Uribe and Xicotencatl. Those two generalizations are examples of geometric stacks. Stacks appear in many situations. Among the most interesting examples of stacks we can also mention the global quotients of a manifold by a Lie group.NEWLINENEWLINEThe book of Behrend, Ginot, Noohi and Xu is a complete and nice description of the Chas Sullivan theory for stacks. This book establishes the general machinery of string topology to stacks, in 17 sections and an appendix. The appendix concerns on the one hand generalized Fulton-MacPherson bivariant theories and on the other hand categories fibered in groupoids. The book contains a lot of examples, and in particular the last section is devoted only to detailed examples.NEWLINENEWLINEThe theory of topological stacks is recalled briefly in the first three sections. The next 7 sections lead to the definition of the loop product and the loop coproduct for stacks. The authors explain how they solve the different steps leading to the definition of the string operations. The steps concern the definition of the free loop stack \(L\mathcal X\) of a stalk \(\mathcal X\), the construction of a bivariant theory in the sense of Fulton-MacPherson for stacks, the construction of an orientation class \(\mathcal O\) in the bivariant cohomology of the diagonal \({\mathcal X}\to {\mathcal X}\times {\mathcal X}\) and the construction of Gysin maps.NEWLINENEWLINEThe next six sections are devoted to the properties of the string topology. The authors prove that the homology of the free loop space of a stack is a Frobenius algebra and that its shifted version is a BV algebra. Using then Sullivan's chord diagrams and the formalism introduced of Gysin maps, they put all those constructions in the general framework of a homological conformal field theory.NEWLINENEWLINEThis theory is then applied to different particular situations. First of all this enables the definition of a loop product for \(\text{Map}(S^n,{\mathcal X})\). Concerning almost complex orbifolds, recall that Chen and Ruan have defined a cup product on the orbifold cohomology \(H^*_{orb} ({\mathcal X})\). Here the authors define an orbifold intersection product in homology that is identified, in the compact case, with the Chen-Ruan product via orbifold intersection duality.