## 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 $$\pi'$$ is finitely generated then $$\pi'$$ is free. In this case $$\mathrm{def}(\pi)=1$$ also.

Keywords: deficiency, knot, ribbon, slice.

: Primary 57Q45.

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 Friday, July 14, 2006