Knot groups and slice conditions

Jonathan A.Hillman

Abstract

We introduce the notions of "k-connected-slice" and "π₁-slice", interpolating between "homotopy ribbon" and "slice". We show that every high-dimensional knot group π is the group of an (n-1)-connected- slice n-knot for all n≥3. However if π is the group of an n-connected-slice n-knot the augmentation ideal I(π) must have deficiency 1 as a module. If moreover n=2 and π' is finitely generated then π' is free. In this case \mathrm{def}(π)=1 also.

Keywords: deficiency, knot, ribbon, slice.

AMS Subject Classification: Primary 57Q45.