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 [24]. We also discuss the case of the full braid group on 3-strands.