A second order algebraic knot concordance group

Abstract

Let $\mathcal{C}$ 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:

$$\mathcal{C}\supset {\mathcal{ℱ}}_{\left(0\right)}\supset {\mathcal{ℱ}}_{\left(0.5\right)}\supset {\mathcal{ℱ}}_{\left(1\right)}\supset {\mathcal{ℱ}}_{\left(1.5\right)}\supset {\mathcal{ℱ}}_{\left(2\right)}\supset \cdots$$

The quotient $\mathcal{C}∕{\mathcal{ℱ}}_{\left(0.5\right)}$ is isomorphic to Levine’s algebraic concordance group; ${\mathcal{ℱ}}_{\left(0.5\right)}$ is the algebraically slice knots. The quotient $\mathcal{C}∕{\mathcal{ℱ}}_{\left(1.5\right)}$ contains all metabelian concordance obstructions.

Using chain complexes with a Poincaré duality structure, we define an abelian group ${\mathcal{A}\mathcal{C}}_2$, our second order algebraic knot concordance group. We define a group homomorphism $\mathcal{C}\to {\mathcal{A}\mathcal{C}}_2$ which factors through $\mathcal{C}∕{\mathcal{ℱ}}_{\left(1.5\right)}$, and we can extract the two stage Cochran–Orr–Teichner obstruction theory from our single stage obstruction group ${\mathcal{A}\mathcal{C}}_2$. Moreover there is a surjective homomorphism ${\mathcal{A}\mathcal{C}}_2\to \mathcal{C}∕{\mathcal{ℱ}}_{\left(0.5\right)}$, and we show that the kernel of this homomorphism is nontrivial.