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

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 $\mathcal{M}_n$ is of free type. In this paper we give an injection of the (extended) loop braid group into the group of automorphisms of $\mathcal{M}_n$. 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.

No reviews yet.

No comments yet.