The loop orbifold of the symmetric product (Q995590)

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.