Electronic ISBN:  9781470404895 
Product Code:  MEMO/189/885.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 189; 2007; 95 ppMSC: Primary 57;
The author studies the group of rational concordance classes of codimension two knots in rational homology spheres. He gives a full calculation of its algebraic theory by developing a complete set of new invariants. For computation, he relates these invariants with limiting behaviour of the Artin reciprocity over an infinite tower of number fields and analyzes it using tools from algebraic number theory. In higher dimensions it classifies the rational concordance group of knots whose ambient space satisfies a certain cobordism theoretic condition. In particular, he constructs infinitely many torsion elements. He shows that the structure of the rational concordance group is much more complicated than the integral concordance group from a topological viewpoint. He also investigates the structure peculiar to knots in rational homology 3spheres. To obtain further nontrivial obstructions in this dimension, he develops a technique of controlling a certain limit of the von Neumann \(L^2\)signature invariants.

Table of Contents

Chapters

1. Introduction

2. Rational knots and Seifert matrices

3. Algebraic structure of $\mathcal {G}_n$

4. Geometric structure of $\mathcal {C}_n$

5. Rational knots in dimension three


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The author studies the group of rational concordance classes of codimension two knots in rational homology spheres. He gives a full calculation of its algebraic theory by developing a complete set of new invariants. For computation, he relates these invariants with limiting behaviour of the Artin reciprocity over an infinite tower of number fields and analyzes it using tools from algebraic number theory. In higher dimensions it classifies the rational concordance group of knots whose ambient space satisfies a certain cobordism theoretic condition. In particular, he constructs infinitely many torsion elements. He shows that the structure of the rational concordance group is much more complicated than the integral concordance group from a topological viewpoint. He also investigates the structure peculiar to knots in rational homology 3spheres. To obtain further nontrivial obstructions in this dimension, he develops a technique of controlling a certain limit of the von Neumann \(L^2\)signature invariants.

Chapters

1. Introduction

2. Rational knots and Seifert matrices

3. Algebraic structure of $\mathcal {G}_n$

4. Geometric structure of $\mathcal {C}_n$

5. Rational knots in dimension three