Correction to the article ``An infinite-rank summand of topologically slice knots''
type: Item
instance of: scholarly article
title: Correction to the article ``An infinite-rank summand of topologically slice knots''
title: An infinite-rank summand of topologically slice knots
zbMATH Open document ID: 1447.57029
zbMATH Open document ID: 1315.57029
author: Jennifer Hom
published in: Geometry \& Topology
publication date: 2019-10-24T00:00:00Z
publication date: 2015-05-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1310.4476
Mathematics Subject Classification ID: 57N70
Mathematics Subject Classification ID: 57R58
Mathematics Subject Classification ID: 57K18
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 7121760
zbMATH DE Number: 6433032
zbMATH Keywords: knot concordance
zbMATH Keywords: knot Floer homology
zbMATH Keywords: Heegaard Floer homology
zbMATH Keywords: concordance
MaRDI profile type: Publication
OpenAlex ID: W2980393439
OpenAlex ID: W3102315352
arXiv ID: 1310.4476
review text: Let \(\mathcal{C}\) denote the smooth concordance group and let \(\mathcal{C}_{\mathrm{TS}}\) be the subgroup of \(\mathcal{C}\) consisting of topologically slice knots. The main result of the paper under review is the existence of a \(\mathbb{Z}^{\infty}\) summand in \(\mathcal{C}_{\mathrm{TS}}\). The existence of a \(\mathbb{Z}^{\infty}\) \textit{subgroup} in \(\mathcal{C}_{\mathrm{TS}}\) was known before: see [\textit{H. Endo}, Topology Appl. 63, No. 3, 257--262 (1995; Zbl 0845.57006)] and [\textit{J. Hom}, Comment. Math. Helv. 89, No. 3, 537--570 (2014; Zbl 1312.57008)]. It is instructive to compare this with \(\mathbb{Q}\) -- it has a \(\mathbb{Z}^{\infty}\) subgroup, but not a summand. \textit{M. Hedden}, \textit{S.-G. Kim} and \textit{C. Livingston} [``Topologically slice knots of smooth concordance order two'' (2012), \url{arXiv:1212.6628}] have established the existence of a \(\mathbb{Z}_2^{\infty}\) subgroup in \(\mathcal{C}_{\mathrm{TS}}\). Given a null-homologous knot \(K\) in a three-manifold \(Y\), \textit{P. Ozsváth} and \textit{S. Szabó} [Adv. Math. 186, No. 1, 58--116 (2004; Zbl 1062.57019)] and independently \textit{J. Rasmussen} [Floer homology and knot complements. PhD thesis. Harvard University (2003)] define a doubly-filtered chain complex \(CFK^{\infty}(Y, K)\) whose doubly-filtered chain homotopy type is an invariant of \(K\). When \(Y = S^3\), this complex is denoted by \(CFK^{\infty}(K)\). For knots in \(S^3\), using \(CFK^{\infty}\) one may define a smooth concordance invariant \(\tau\) [\textit{P. Ozsváth} and \textit{S. Szabó}, Geom. Topol. 7, 615--639 (2003; Zbl 1037.57027)] which gives a surjective homomorphism from \(\mathcal{C}\) to \(\mathbb{Z}\). \textit{C. Livingston} [Geom. Topol. 8, 735--742 (2004; Zbl 1067.57008)] has found topologically slice knots with \(\tau\)-invariant \(1\) thus establishing the existence of a \(\mathbb{Z}\) summand in \(\mathcal{C}_{\mathrm{TS}}\). This result was improved in [\textit{C. Manolescu} and \t…
reviewed by: Fyodor Gainullin
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Wikidata QID: Q126998417
DOI: 10.2140/GT.2019.23.2699
DOI: 10.2140/GT.2015.19.1063
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description: scientific article; zbMATH DE number 6433032
alternative label: An infinite-rank summand of topologically slice knots