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Concordance and rational knots
by
Jae Choon Cha
Indiana University
We continue the study of the structure of the concordance group of codimension two knots in rational homology spheres which was originated from Cochran and Orr, to give a full calculation. Using Seifert surfaces, we show that the rational knot concordance group is trivial in even dimensions, and is isomorphic to an algebraic rational concordance group that is defined to be a limit of ordinary algebraic concordance groups, in higher odd dimensions. We discover a complete set of invariants of the algebraic rational concordance group, and by calculating these invariants, we show that it is isomorphic to the sum of infinitely many copies of Z, Z/2, and Z/4. An investigation of norm subgroups on infinite towers of number fields using machinery of algebraic number theory plays a crucial role in calculating 4-torsion. We also show that the kernel and cokernel of the natural map of the ordinary concordance group into the rational concordance group, which measure the difference of them, are large enough to contain the sum of infinitely many copies of Z/2 and Z, Z/2, and Z/4, respectively. Our results can also be interpreted as a calculation of certain homology surgery obstruction gamma-groups.
Date received: October 1, 2002
Copyright © 2002 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cajr-07.