Loop homology of some global quotient orbifolds

Abstract

We determine the ring structure of the loop homology of some global quotient orbifolds. We can compute by our theorem the loop homology ring with suitable coefficients of the global quotient orbifolds of the form $\left[M∕G\right]$ for $M$ being some kinds of homogeneous manifolds, and $G$ being a finite subgroup of a path-connected topological group $\mathcal{G}$ acting on $M$. It is shown that these homology rings split into the tensor product of the loop homology ring ${ℍ}_{\ast }\left(LM\right)$ of the manifold $M$ and that of the classifying space of the finite group, which coincides with the center of the group ring $Z\left(k\left[G\right]\right)$.