Springer theory in braid groups and the Birman–Ko–Lee monoid

Abstract

Abstract We state a conjecture about centralizers of certain roots of central elements in braid groups, and check it for Artin braid groups and some other cases. Our proof makes use of results by Birman-Ko-Lee. We give a new intrinsic account of these results. Notations If G is a group acting on a set X, we denote by X G the subset of X of elements fixed by all elements of G. If (X, x) is a pointed topological space, we denote by Ω(X, x) the corresponding loop space, by ∼ the homotopy relation on Ω(X, x) and by π1(X, x) the fundamental group. For all n ∈ N, we denote by µn the set of n-th roots of unity in C.