The first homology of square-section braid groups

Abstract

We compute the first homology of the unordered configuration space of $n$ open unit-squares in the $w$ by $h$ rectangle for all positive integers $n$, $w$, and $h$. Letting $w$ and $h$ vary, these square configuration spaces form a bi-filtration of the unordered configuration space of $n$ points in the plane, so their fundamental groups approximate the $n$-strand braid group. Using a pair of spectral sequences, we prove that these square and point configuration spaces have naturally isomorphic first homology groups if $w,h\ge 3$ and $n\le wh-3$. In the $h=2$ case, we show that the first homology groups of these square configuration spaces are free of rank $n-1$ for $n\le 2w-2$, whereas they are free of rank $w-1$ for $n=2w-1$.