Homomorphisms of commutator subgroups of braid groups with small number of strings

For any n, we describe all endomorphisms of the braid group B n and of its commutator subgroup B n ′ , as well as all homomorphisms B n ′ →B n . These results are new only for small n because endomorphisms of B n are already described by Castel for n≥6, and homomorphisms B n ′ →B n and endomorphisms of B n ′ are already described by Kordek and Margalit for n≥7. We use very different approaches for n=4 and for n≥5.