On the kernel of the zero-surgery homomorphism from knot concordance

Dongsoo Lee

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arXiv · 2004.05967

On the kernel of the zero-surgery homomorphism from knot concordance

Dongsoo Lee

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Abstract

Kawauchi defined a group structure on the set of homology $S^1$$\times$$S^2$'s under an equivalence relation called $\widetilde{H}$-cobordism. This group receives a homomorphism from the knot concordance group, given by the operation of zero-surgery. It is natural to ask whether the zero-surgery homomorphism is injective. We show that this question has a negative answer in the smooth category. Indeed, using knot concordance invariants derived from knot Floer homology we show that the kernel of the zero-surgery homomorphism contains a $\mathbb{Z}^\infty$-subgroup.

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Dongsoo Lee. 2020-04-13. On the kernel of the zero-surgery homomorphism from knot concordance. https://arxiv.org/abs/2004.05967

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