Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants
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 closed three-manifold obtained from \(S^{3}\) by \(0\)-surgery along the knot \(K\) (remove the tubular neighborhood of \(K\) and replace it back so that \(H_{1}(M_{K};\mathbb{Z})=\mathbb{Z}\)) and a representation is called metabelian if it vanishes on the second derived subgroup. Note that a metabelian representation of \(\pi_{1}(M_{K})\) corresponds to a representation of \(\mathbb Z\ltimes H_{1}(M_{K},\mathbb{Z}[t,t^{-1}])\). Let \(M_{n}\) be the \(n\)-fold cyclic cover of \(M_{K}\) and \(TH_{1}(M_{n})\) be the torsion subgroup of \(H_{1}(M_{n})\). The main result of this paper is that for a slice knot \(K\) and a prime power \(n\), there exists a metabolizer \(P_{n}\subset TH_{1}(M_{n})\) for the linking pairing such that for any irreducible representation \(\alpha\colon\pi_{1}(M_{K})\to U(n)\) that factors through \(\mathbb Z\ltimes H_{1}(M_{K},\mathbb{Z}[t,t^{-1}])/(t^{k}-1)\) and vanishes on \(\{0\}\times P_{n}\), the eta invariant \(\eta(M_{K},\alpha)\) is zero. (Note that \(TH_{1}(M_{n})=H_{1}(M_{K},\mathbb{Z}[t,t^{-1}])/(t^{k}-1))\).) It is proved that the Casson-Gordon invariants can be interpreted as this kind of eta invariants. As applications the author studies relations between the eta invariants and \(L^{2}\)-eta invariants introduced by Cochran, Orr, and Teichner. He also introduces invariants that may detect a slice knot that is not ribbon, where a ribbon knot bounds a singular disk in \(S^{3}\) only with ribbon singularities.
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