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.

We thank Alain Valette for the suggestion to examine K-theory and K-homology of pure braid groups. We thank the organizers of the Women in Operator Algebras Conference that took place at BIRS where this project started. HW acknowledges the support from Science and Technology Commission of Shanghai Municipality (STCSM), grant No.18dz2271000. MGA was partially supported by ANR project Singstar.