Knot groups and slice conditions

Jonathan A.Hillman

Abstract

We introduce the notions of "$k$-connected-slice" and "${\pi }_1$-slice", interpolating between "homotopy ribbon" and "slice". We show that every high-dimensional knot group $\pi$ is the group of an $(n-1)$-connected-slice $n$-knot for all $n\ge 3$. However if $\pi$ is the group of an $n$-connected-slice $n$-knot the augmentation ideal $I(\pi )$ must have deficiency 1 as a module. If moreover $n=2$ and ${\pi }^{\prime }$ is finitely generated then ${\pi }^{\prime }$ is free. In this case $\mathrm{def}(\pi )=1$ also.

Keywords: deficiency, knot, ribbon, slice.

AMS Subject Classification: Primary 57Q45.