K-homology and K-theory of pure Braid groups

We produce an explicit description of the K-theory and K-homology of the pure braid group on $n$ strands. We describe the Baum--Connes correspondence between the generators of the left- and right-hand sides for $n=4$. Using functoriality of the assembly map and direct computations, we recover Oyono-Oyono's result on the Baum--Connes conjecture for pure braid groups. We also discuss the case of the full braid group $B_3$.
Comment: 39 pages, revised version to appear on New York J. Math