A second order algebraic knot concordance group

Powell, Mark

doi:10.2140/agt.2012.12.685

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A second order algebraic knot concordance group

Mark Powell

Algebraic & Geometric Topology 12 (2012) 685–751

DOI: 10.2140/agt.2012.12.685

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 \dots

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 ${A C}_{2}$ , our second order algebraic knot concordance group. We define a group homomorphism $C \to {A C}_{2}$ 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 ${A C}_{2}$ . Moreover there is a surjective homomorphism ${A C}_{2} \to C ∕ ℱ_{(0.5)}$ , 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 Rawles Hall 831 East 3rd Street Bloomington IN 47401 USA
http://mypage.iu.edu/~macp/home.html

Volume 12, issue 2 (2012)