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

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 ⊂ S 3. We show that certain eta invariants associated to metabelian representations π 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.