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The loop orbifold of the symmetric product. (English) Zbl 1148.55005

When taking a quotient of a manifold by a group action, unless the action is free, the object may not be as nice as a manifold. The theory of orbifolds was developed to extend manifold properties to such quotient spaces. Orbifolds can be modeled in other ways—as a groupoid, or as a category. In the case of a symmetric product orbifold \([X^n/\Sigma_n]\), this structure is explicit: the objects in the category are \((x_1, \ldots, x_n)\) where \(x_i \in X\), a topological space, and the morphisms are given by \((x_1, \ldots, x_n; \sigma)\colon (x_1, \ldots, x_n) \to (x_{\sigma(1), \ldots, x_\sigma(n)})\) for \(\sigma\in \Sigma_n\), the symmetric group on \(n\) letters. The loop orbifold \(L[X^n/\sigma_n]\) is the space of functors from \([{\mathbb R}/{\mathbb Z}] \to [X^n/\Sigma_n]\) with morphisms given by natural transformations. In this setting the role of centralizers of group elements in the decomposition of the loop space is especially transparent.
The authors explicate a ring structure on \(H_*(L[M^n/\Sigma_n]; {\mathbb R})\), for \(M\) a manifold, which is analogous to the Chas-Sullivan ring. Given an orbifold \([Y/G]\) there is its inertia suborbifold given by \[ I[Y/G] = \left[\left(\bigsqcup_{g\in G} Y^g\times \{g\}\right)/G\right], \] which defines the orbifold cohomology of \([Y/G]\), a shifted version of the cohomology of \(I[Y/G]\). In the case of the loop orbispace, \(I[M^n/\Sigma_n]\), there is a ring structure on the homology making it a subring of \(H_*(L[M^n/\Sigma_n]; {\mathbb R})\) analogous to the subring of the homology of the free loop space given by inclusion of the constant loops. This product on \(H_*(I[M^n/\Sigma_n];{\mathbb R})\) shares formal properties with the dual of the Chen-Ruan orbifold cohomology product. When \(M\) is an almost complex orbifold, the authors construct an explicit product called the virtual intersection product and compare it to the other structures—it is essentially the pairwise transversal intersection of cycles and under Poincaré duality is isomorphic to the generalized Chas-Sullivan product on the free loop orbispace.

MSC:

55P35 Loop spaces
55N91 Equivariant homology and cohomology in algebraic topology
55N45 Products and intersections in homology and cohomology
55S15 Symmetric products and cyclic products in algebraic topology
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