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String Topology for Stacks
Kai Behrend, Grégory Ginot, Behrang Noohi and Ping Xu
Publisher: Société Mathématique de France (2012)
Details: 169 pages, Paperback
Series: Astérisque 343
Topics: Quantum Field Theory, Category Theory, Algebraic Topology
MAA Review[Reviewed by Felipe Zaldivar, on 12/11/2012]
In the late 1990s, Chas and Sullivan constructed a graded product on the homology of the free loop space of a closed oriented smooth manifold that induces a graded Lie bracket on the homology of its non-parametrized loops. The construction is beautiful and simple: At each point of intersection of two closed oriented curves on a smooth oriented manifold, Chas and Sullivan considered the closed curve given by going around the first curve and then going around the second. This gluing process induces a loop product on the homology of the loop manifold. The construction of the graded Lie bracket involves the evaluation of loops at zero and then pulling back along a tubular neighborhood of the diagonal to obtain a normal bundle and then applies a collapse map.
The homology of the free loop space has several natural structures, e.g., it is a Frobenius algebra. All of these, and more, were captured in the suggestive term used by Chas and Sullivan to describe this construction, string topology. The fertility of these ideas has also influenced adjacent areas, such as algebraic and differential geometry. Here, many of the geometric objects do not have a structure of a smooth manifold, but it would be desirable to have an analogue of string topology in this realm.
This brings us to the monograph under review, which is actually a long research paper. Its main goal is to establish a general framework and machinery for string topology on differentiable stacks. Differentiable stacks are not that far from manifolds; for example, quotients of actions of Lie groups on manifolds, orbifolds, and classifying spaces of compact Lie groups are all included.
This generalization is not straightforward, and the authors have to borrow some ideas from Fulton-McPherson to construct the Gysin map and be able to compute it in some crucial examples. The formalism introduced by the authors allows them to obtain, for example, the Frobenius algebra structure of the homology of the free loop stack of an oriented stack and extend it into a homological conformal field theory. They are also able to treat the case of the homology of hidden loops and relate this to the homology of the free loop stack. As pointed out by the authors, some of these topics were independently studied by other authors.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is email@example.com.