Algebraic and Geometric Topology 4 (2004), paper no. 39, pages 893-934.

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

Stefan Friedl

Abstract. We give a useful classification of the metabelian unitary representations of pi_1(M_K), where M_K is the result of zero-surgery along a knot K in S^3. We show that certain eta invariants associated to metabelian representations pi_1(M_K) --> 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

AMS subject classification. Primary: 57M25, 57M27; 57Q45, 57Q60.

DOI: 10.2140/agt.2004.4.893

E-print: arXiv:math.GT/0305402

Submitted: 17 January 2004. (Revised: 13 September 2004.) Accepted: 19 September 2004. Published: 13 October 2004.

Notes on file formats

Stefan Friedl
Department of Mathematics, Rice University, Houston, TX 77005, USA

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