#### Volume 12, issue 2 (2012)

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A second order algebraic knot concordance group

### Mark Powell

Algebraic & Geometric Topology 12 (2012) 685–751
##### Abstract

Let $\mathsc{C}$ 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:

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

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

Using chain complexes with a Poincaré duality structure, we define an abelian group  ${\mathsc{A}\mathsc{C}}_{2}$, our second order algebraic knot concordance group. We define a group homomorphism $\mathsc{C}\to {\mathsc{A}\mathsc{C}}_{2}$ which factors through $\mathsc{C}∕{\mathsc{ℱ}}_{\left(1.5\right)}$, and we can extract the two stage Cochran–Orr–Teichner obstruction theory from our single stage obstruction group ${\mathsc{A}\mathsc{C}}_{2}$. Moreover there is a surjective homomorphism ${\mathsc{A}\mathsc{C}}_{2}\to \mathsc{C}∕{\mathsc{ℱ}}_{\left(0.5\right)}$, and we show that the kernel of this homomorphism is nontrivial.

##### Keywords
knot concordance group, solvable filtration, symmetric chain complex
##### Mathematical Subject Classification 2010
Primary: 57M25, 57M27, 57N70, 57R67
Secondary: 57M10, 57R65