Lower order quotients in the $n$-solvable filtration

We establish several results about two short exact sequences involving lower terms of the $n$-solvable filtration, $\{\mathcal{F}_n^m\}$ of the string link concordance group $\mathcal{C}^m$, which was introduced by Cochran, Orr, and Teichner in the late 90's. We show that the short exact sequence
\[0\to \mathcal{F}_0^m/\mathcal{F}_{0.5}^m\to \mathcal{F}_{-0.5}^m/\mathcal{F}_{0.5}^m\to\mathcal{F}_{-0.5}^m/\mathcal{F}_{0}^m\to  0\]
 does not split for links of two or more components, in contrast to the fact that it splits for knots. Considering lower terms of the filtration $\{\mathcal{F}_n^m\}$ in the short exact sequence
 \[0\to \mathcal{F}_{-0.5}^m/\mathcal{F}_0^m\to \mathcal{C}^m/\mathcal{F}_0^m\to \mathcal{C}^m/\mathcal{F}_{-0.5}^m\to 0,\] we show that while the sequence does not split for $m\geq 3$, it does indeed split for $m=2$. This allows us to determine that the quotient $\mathcal{C}^2/\mathcal{F}_0^2$ is an abelian group. This is joint work with Carolyn Otto.