Volume 4, issue 2 (2004)

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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

arXiv: math.GT/0305402


We give a useful classification of the metabelian unitary representations of π1(MK), where MK is the result of zero-surgery along a knot K S3. We show that certain eta invariants associated to metabelian representations π1(MK) 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 L2–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 L2–eta invariant sliceness obstruction but which is not ribbon.

knot concordance, Casson–Gordon invariants, Eta invariant
Mathematical Subject Classification 2000
Primary: 57M25, 57M27, 57Q45, 57Q60
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Received: 17 January 2004
Revised: 13 September 2004
Accepted: 19 September 2004
Published: 13 October 2004
Stefan Friedl
Department of Mathematics
Rice University
Houston TX 77005