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A second order algebraic knot concordance group

Mark Powell

Algebraic & Geometric Topology 12 (2012) 685–751
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(0) (0.5) (1) (1.5) (2)

The quotient C(0.5) is isomorphic to Levine’s algebraic concordance group; (0.5) is the algebraically slice knots. The quotient C(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 CAC2 which factors through C(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(0.5), 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
References
Publication
Received: 29 November 2011
Revised: 11 January 2012
Accepted: 13 January 2012
Published: 8 April 2012
Authors
Mark Powell
Department of Mathematics
Indiana University
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