ABSTRACT: The group $P\Sigma_n$ is something like a higher dimensional version of the pure braid group. Just as the pure braid group can be thought of as the group of motions of $n$ points in the plane, $P\Sigma_n$ is the group of motions of $n$ unknotted, unlinked circles in $3$-space. I will present a computation of the homology groups of $P\Sigma_n$. The bad news is this is a spectral sequence computation. The good news is all the hard work boils down to the combinatorics of planted forests. (Joint with Craig Jensen and Jon McCammond)