Eta invariants as sliceness obstructions and their relation to Casson–Gordon invariants

Friedl, Stefan

doi:10.2140/agt.2004.4.893

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Eta invariants as sliceness obstructions and their relation to Casson–Gordon invariants

Stefan Friedl

Algebraic & Geometric Topology 4 (2004) 893–934

DOI: 10.2140/agt.2004.4.893

arXiv: math.GT/0305402

Abstract

We give a useful classification of the metabelian unitary representations of $π_{1} (M_{K})$ , where $M_{K}$ is the result of zero-surgery along a knot $K \subset S^{3}$ . We show that certain eta invariants associated to metabelian representations $π_{1} (M_{K}) \to U (k)$ vanish for slice knots and that even more eta invariants vanish for ribbon knots and doubly slice knots. We show that our vanishing results contain the Casson–Gordon sliceness obstruction. In many cases eta invariants can be easily computed for satellite knots. We use this to study the relation between the eta invariant sliceness obstruction, the eta-invariant ribbonness obstruction, and the $L^{2}$ –eta invariant sliceness obstruction recently introduced by Cochran, Orr and Teichner. In particular we give an example of a knot which has zero eta invariant and zero metabelian $L^{2}$ –eta invariant sliceness obstruction but which is not ribbon.

Keywords

knot concordance, Casson–Gordon invariants, Eta invariant

Mathematical Subject Classification 2000

Primary: 57M25, 57M27, 57Q45, 57Q60

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Publication

Received: 17 January 2004

Revised: 13 September 2004

Accepted: 19 September 2004

Published: 13 October 2004

Authors

Stefan Friedl
Department of Mathematics Rice University Houston TX 77005 USA

Volume 4, issue 2 (2004)