On Belk's classifying space for Thompson's group F

The space of configurations of n ordered points in the plane serves as a classifying space for the pure braid group PB_n. Elements of Thompson's group F admit a model similar to braids, except instead of braiding the strands split and merge. In Belk's thesis, a space CF was considered, of configurations of points on the real line allowing for splitting and merging, and a proof was sketched that CF is a classifying space for F. The idea there was to build the universal cover and construct an explicit contraction to a point. Here we start with an established CAT(0) cube complex X on which F acts freely, and construct an explicit homotopy equivalence between X/F and CF, proving that CF is indeed a K(F,1).