Knot concordance in homology cobordisms
type: Item
instance of: scholarly article
title: Knot concordance in homology cobordisms
DOI: 10.1215/00127094-2021-0110
author: Jennifer Hom
author: Tye Lidman
author: Adam Simon Levine
published in: Duke Mathematical Journal
publication date: 2022-10-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1801.07770
review text: Two integer homology spheres, \(Y_0, Y_1\) are said to be homology cobordant if there exists an oriented 4-manifold \(W\) such that \(\partial W = -Y_0 \sqcup Y_1\), and the inclusions \(Y_i \rightarrow W\) induce isomorphisms in homology. Two knots \(K_0 \subset Y_0\), \(K_1 \subset Y_1\) are \textit{homology concordant} if there exists a homology cobordism \(W\) between \(Y_0\) and \(Y_1\), and a smoothly embedded annulus \(A \subset W\) such that \(\partial A = -K_0 \sqcup K_1\), generalizing the concept of concordance between two knots in \(S^3\). This paper addresses a structural question for the group of homology concordances for knots in homology spheres (the group operation here is the connected sum of the manifolds and the knots). In particular, if \(\mathscr{C}_{\mathbb{Z}}\) denotes the group of knots in \(S^3\), modulo homology concordance, and if \(\widehat{\mathscr{C}}_{\mathbb{Z}}\) denotes the group of knots \(K \subset Y\) where \(Y\) is a homology 3-sphere that bounds a homology 4-ball, modulo homology concordance. Then the main theorem (Theorem 1.1) of the paper states that the subgroup \(\mathscr{C}_{\mathbb{Z}} \subset \widehat{\mathscr{C}}_{\mathbb{Z}}\) is of infinite index, precisely, \(\widehat{\mathscr{C}}_{\mathbb{Z}} / \mathscr{C}_{\mathbb{Z}}\) is infinitely generated, and contains a subgroup isomorphic to \(\mathbb{Z}\). The authors provide an infinite family of knots contained in homology spheres \((K_j, Y_j) \in \widehat{\mathscr{C}}_{\mathbb{Z}}\) for which a Heegaard Floer theoretic invariant \(\theta(Y_j,K_j)\) is unbounded, which proves the infinite generation result. They provide a knot \((Y,K)\) such that \(Y\) is a homology sphere bounding a homology 4-ball, such that \(K\) is not homology concordant to any knot in \(S^3\), and such that the same holds for \((\#_n(Y,K))\), which provides them with the \(\mathbb{Z}\) summand in the quotient as mentioned above. The crux of the arguments lies on finding obstructions leveraging sev…
reviewed by: Subhankar Dey
Mathematics Subject Classification ID: 57K10
Mathematics Subject Classification ID: 57K18
Mathematics Subject Classification ID: 57R58
zbMATH DE Number: 7608259
zbMATH Keywords: Heegaard Floer homology
zbMATH Keywords: knot concordance
zbMATH Keywords: low-dimensional topology
zbMATH Keywords: homology concordance
MaRDI profile type: Publication
OpenAlex ID: W2784410842
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description: scientific article; zbMATH DE number 7608259