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

By Stefan Friedl. 2003

85 citations recorded by the source corpus.

We give a useful classification of the metabelian unitary repre- sentations of π1(MK), where MK is the result of zero-surgery along a knot K ⊂ S 3 . 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 L 2 -eta invariant sliceness obstruction recently introduced by Cochran, Orr and Teichner. In partic- ular 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. AMS Classification 57M25, 57M27; 57Q45, 57Q60

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/81214bfe922bfae947d095200b3bbecc0c4a46a8

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Citations in the corpus

Primary decomposition and the fractal nature of knot concordance (2009, background): “…(B) · S. Harvey Department of Mathematics, MS-136, Rice University, P. O. Box 1892, Houston, TX 77251-1892, USA e-mail: cochran@rice.edu S. Harvey e-mail: shelly@rice.edu C. Leidy Department of Mathematics, Wesleyan University, Wesleyan Station, Middletown, CT 06459, USA e-mail: cleidy@wesleyan.edu”
Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation (2008, unknown): “A few more recent articles include [4,5,9,32,34,35].”
A Survey of Classical Knot Concordance (2003, result): “Other work in this realm includes that of Letsche [59] and recent work of Friedl [28, 29].”
The structure of the rational concordance group of knots (2006, unknown): “18 were considered by Kim [28] and Friedl [18].”
Knot concordance and higher-order Blanchfield duality (2007, background): “The question of which knots are slice knots was first considered by Kervaire and Milnor in the early 60′s in their study of isolated singularities of 2-spheres in 4-manifolds in the context of a surgery-theoretic scheme for classifying 4- dimensional manifolds.”