Braid groups and regular elements

I,et W be a complex reflection group, and w a Springer regular element of W . Let B be the braid group corresponding to W, and B(w) the braid group cor responding to the centralizer of w in W. Answering a question raised by Bro ue and Michel, we prove, except in a small number of cases, that the natura l morphism B(w) --> B is injective. The proof given is mainly topological. In order to derive consequences about presentations, we discuss the connect ion between the topological and combinatorial definitions of Artin braid gr oups.