# Loop homology of some global quotient orbifolds

## TL;DR

This paper computes the loop homology ring of certain global quotient orbifolds, revealing a tensor product structure involving the manifold's loop homology and the group's classifying space.

## Contribution

It provides a method to explicitly compute the loop homology ring of quotient orbifolds of the form [M/G], combining manifold and group data.

## Key findings

- Loop homology rings split into tensor products.
- Explicit computation for orbifolds [M/G] with homogeneous M.
- Homology rings relate to the center of the group ring Z(k[G]).

## 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 $[M/G]$ for $M$ being some kinds of homogeneous manifolds, and $G$ being a finte 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 $\mathbb{H}_{*}(LM)$ of the manifold $M$ and that of the classifying space of the finite group, which coincides with the center of the group ring $Z(k[G])$.