The Stabilizer of 1 in Habegger-Lin's Action for String Links

Abstract

Let $H(k)$ be the group of all homotopy classes of $k$-string links. It has been proved that $f,g\in H(k)$ have the same closure if and only if there is $\beta\in S_k(1)$ such that $\beta\cdot f=g$, where $S_k(1)$ is the stabilizer of 1 for a certain action of the group $H(2k)$ on the set $H(k)$. If $\beta\in S_k(1)$, Artin's automorphism $\overline{\beta}$, induced by $\beta$ on $RF(2k)$, the reduced free group in $2k$ generators, induces an automorphism $\overline{\overline{\beta}}\in A(RF(k))$, the group of all automorphisms of $RF(k)$ that send each generator to one of its conjugates. This can be used to compare the homotopy classes of links obtained by closing $f$ and $g$. The association $\beta\mapsto \overline{\overline{\beta}}$ is a homomorphism from $S_k(1)$ to $A(RF(k))$. In this paper we determine its kernel.