A second order algebraic knot concordance group

Abstract

Let C be the topological knot concordance group of knots $S^{1} \subset S^{3}$ 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 $AC_{2}$, our second order algebraic knot concordance group. We define a group homomorphism $C \rightarrow AC_{2}$ 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 $AC_{2}$. Moreover there is a surjective homomorphism $AC_{2} \rightarrow C/F_{(0.5)}$, and we show that the kernel of this homomorphism is nontrivial.