Automorphisms of pure braid groups

In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for n> 3 , Aut (P _n ) is generated by the subgroup Aut _c (P _n ) of central automorphisms of P _n , the subgroup Aut (B _n ) of restrictions of automorphisms of B _n on P _n and one extra automorphism w _n . We also investigate the lifting and extension problem for automorphisms of some well-known exact sequences arising from braid groups, and prove that that answers are negative in most cases. Specifically, we prove that no non-trivial central automorphism of P _n can be extended to an automorphism of B _n .