On a canonical lift of Artin's representation to loop braid groups

Abstract

Each pointed topological space has an associated π-module , obtained from action of its first homotopy group on its second homotopy group. For the 3-ball with a trivial link with n -components removed from its interior, its π -module M n is of free type . In this paper we give an injection of the (extended) loop braid group into the group of automorphisms of M n . We give a topological interpretation of this injection, showing that it is both an extension of Artin's representation for braid groups and of Dahm's homomorphism for (extended) loop braid groups.