Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants
type: Item
instance of: scholarly article
title: Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants
zbMATH Open document ID: 1067.57003
author: Stefan Friedl
published in: Algebraic \& Geometric Topology
publication date: 2005-02-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0305402
full work available at URL: https://eudml.org/doc/124909
full work available at URL: agt-4-39.abs.html
review text: A knot \(K\) in the three-sphere \(S^{3}\) is called slice if it bounds a smooth two-disk in the four-ball. It is well known [\textit{J. Levine}, Invent. Math. 8, 98--110 (1969); addendum ibid. 8, 355 (1969; Zbl 0179.52401)] that the Seifert pairing of a slice knot is metabolic, that is, there exists a subspace (metabolizer) \(H\) of \(H_{1}(F)\) whose rank is half of that of \(H_{1}(F)\) such that the Seifert pairing vanishes on \(H\), where \(F\) is a Seifert surface (an oriented surface in \(S^{3}\) bounding the knot) and the Seifert pairing of \(g_{i},g_{j}\in H_1(F)\) is defined to be the linking number of \(g_{i}\) and \(g_{j}^{+}\), the push-off of \(g_{j}\) to the positive normal direction. A knot with metabolic Seifert pairing is called algebraically slice. [\textit{A.~Casson} and \textit{C.~Gordon}, Cobordism of classical knots. With an appendix by P. M. Gilmer. Progr. Math., 62, A la recherche de la topologie perdue, 181--199, Birkhäuser (Boston, Boston, MA), (1986; Zbl 0597.57001)] showed that the converse is not true. \textit{T. Cochran, K.~Orr}, and \textit{P.~Teichner} [Ann. Math. (2) 157, No.~2, 433--519 (2003; Zbl 1044.57001)] introduced a notion of \((n)\)-solvability for any non-negative half integer \(n\), giving a sliceness filtration (an \((n)\)-solvable knot is \((n-0.5)\)-solvable). They showed (i) a knot is \((0)\)-solvable if and only if it has trivial Arf invariant, (ii) a knot is \((0.5)\)-solvable if and only if it is algebraically slice, (iii) a \((1.5)\)-solvable knot has trivial Casson--Gordon invariants, and (iv) there exist non-slice \((2.0)\)-solvable knots. Moreover T.~Kim showed that there exist \((1.0)\)-solvable knots with trivial Casson--Gordon invariant that are not \((1.5)\)-solvable [\textit{T. Kim}, Math. Proc. Camb. Philos. Soc. 137, No.~2, 293--306 (2004; Zbl 1059.57015)]. In the paper under review, the author studies metabelian representations \(\pi_{1}(M_{K})\to U(n)\) and their eta invariants, where \(M_{K}\) is the c…
reviewed by: Hitoshi Murakami
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 57Q60
Mathematics Subject Classification ID: 58J28
zbMATH DE Number: 2139843
zbMATH Keywords: slice knot
zbMATH Keywords: knot concordance
zbMATH Keywords: Casson--Gordon invariant
zbMATH Keywords: Cochran--Orr--Teichner filtration
zbMATH Keywords: eta-invariant
MaRDI profile type: Publication
OpenAlex ID: W3102846673
arXiv ID: math/0305402
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DOI: 10.2140/AGT.2004.4.893
description: scientific article