#### 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

Email: friedl@rice.edu

URL: http://math.rice.edu/~friedl/

AGT home page

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