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

TL;DR

This paper constructs a canonical lift of Artin's braid group representation to loop braid groups, using topological and algebraic methods involving $c ext{pi}$-modules and automorphisms, extending classical representations.

Contribution

It introduces an injection of the extended loop braid group into automorphisms of a $c ext{pi}$-module, extending Artin's and Dahm's representations with a topological interpretation.

Findings

Established an injection of loop braid groups into automorphism groups.

Provided a topological interpretation linking classical and extended braid representations.

Extended classical braid group representations to loop braid groups.

Abstract

Each pointed topological space has an associated $\pi$-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 $\pi$-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.