A second order algebraic knot concordance group

Abstract

Let C be the topological knot concordance group of knots S1⊂S3 under connected sum modulo slice knots. Cochran, Orr and Teichner defined a filtration: [C \supset F_{(0)} \supset F_{(0.5)} \supset F_{(1)} \supset F_{(1.5)} \supset F_{(2)} \supset\cdots] The quotient C/F(0.5) is isomorphic to Levine’s algebraic concordance group; F(0.5) is the algebraically slice knots. The quotient C/F(1.5) contains all metabelian concordance obstructions.

Using chain complexes with a Poincaré duality structure, we define an abelian group AC2, our second order algebraic knot concordance group. We define a group homomorphism C→AC2 which factors through C/F(1.5), and we can extract the two stage Cochran–Orr–Teichner obstruction theory from our single stage obstruction group AC2. Moreover there is a surjective homomorphism AC2→C/F(0.5), and we show that the kernel of this homomorphism is nontrivial.