**Memoirs of the American Mathematical Society**

2007;
95 pp;
Softcover

MSC: Primary 57;

Print ISBN: 978-0-8218-3993-5

Product Code: MEMO/189/885

List Price: $66.00

AMS Member Price: $39.60

MAA Member Price: $59.40

**Electronic ISBN: 978-1-4704-0489-5
Product Code: MEMO/189/885.E**

List Price: $66.00

AMS Member Price: $39.60

MAA Member Price: $59.40

# The Structure of the Rational Concordance Group of Knots

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*Jae Choon Cha*

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 3-spheres. 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.