Splittings of link concordance groups

TL;DR

This paper investigates the structure of link concordance groups using algebraic and geometric tools, revealing non-splitting phenomena and classifying certain quotient groups within the $n$-solvable filtration.

Contribution

It establishes new non-splitting results for exact sequences in link concordance groups and characterizes the quotient structure for 2-component links.

Findings

The sequence involving $rac{^m_0}{^m_{0.5}}$ does not split for links with two or more components.

The sequence involving $rac{^m_{-0.5}}{^m_0}$ splits for $m=2$ but not for $m extgreater 2$.

The quotient $rac{^2}{^2_0}$ is isomorphic to $bZ_2 imes bZ_2 imes bZ_2 imes bZ$.

Abstract

We establish several results about two short exact sequences involving lower terms of the $n$-solvable filtration, $\{\mathcal{F}^m_n\}$ of the string link concordance group $\mathcal{C}^m$. We utilize the Thom-Pontryagin construction to show that the Sato-Levine invariants $\bar{\mu}_{(iijj)}$ must vanish for 0.5-solvable links. Using this result, we show that the short exact sequence $0\rightarrow \mathcal{F}^m_0/\mathcal{F}^m_{0.5} \rightarrow \mathcal{F}^m_{-0.5}/\mathcal{F}^m_{0.5} \rightarrow \mathcal{F}^m_{-0.5}/\mathcal{F}^m_0 \rightarrow 0$ 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 $\{\mathcal{F}^m_n\}$ in the short exact sequence $0\rightarrow \mathcal{F}^m_{-0.5}/\mathcal{F}^m_{0} \rightarrow \mathcal{C}^m/\mathcal{F}^m_{0} \rightarrow \mathcal{C}^m/\mathcal{F}^m_{-0.5} \rightarrow 0$, we show that while the sequence does not split for $m\ge 3$, it does indeed split for $m=2$. We conclude that the quotient $\mathcal{C}^2/\mathcal{F}^2_0 \cong \mathbb{Z}_2\oplus \mathbb{Z}_2\oplus\mathbb{Z}_2 \oplus \mathbb{Z}$.