CHAPTER 1

Introduction

In this paper we study a classification problem of knots in rational ho-

mology spheres. More precisely, a closed manifold with the rational homol-

ogy of the sphere of the same dimension is called a rational sphere, and a

codimension two locally flat sphere embedded in a rational sphere is called

a rational knot. Two rational knots K and

Kr

with the same dimension are

said to be (rationally) concordant if there is a rational homology cobordism

between their ambient spaces which contains a locally flat annulus bounded

by K U —K'. Under connected sum, concordance classes of n-dimensional

rational knots form an abelian group which we call the rational knot con-

cordance group and denote by Cn. Our main aim is to study the structure

0fC

n

.

There are some interesting motivations of our research. We list a few of

them below. First, rational knot concordance has a close relationship with

concordance of links in the ordinary sphere. While there is a known frame-

work of the study of link concordance (e.g., see Cappell-Shaneson [3] and Le

Dimet [30]), it still remains far more elusive than knot concordance because

of a lack of our understanding of related homotopy theoretical and surgery

theoretical problems. In the remarkable work of Cochran and Orr [10, 12],

it was first proposed that problems on link concordance can be transformed

into ones on rational knot concordance. Based on this idea they proved the

long-standing conjecture that not all links are concordant to boundary links.

For some further developments and applications, see subsequent work of Ko

and the author [7, 8].

Since these successful applications, more systematic study of rational

concordance has been called on. Note that rational knot concordance is a

natural generalization of ordinary concordance of knots in the sphere which

has a deep and rich theory. For results on ordinary knot concordance par-

ticularly related to this paper, see Levine [33, 32], Kervaire [26], Cappell-

Shaneson [3], Casson-Gordon [5, 6], and Cochran-Orr-Teichner [13, 14].

Regarding this, it is natural to ask whether one can establish an analo-

gous theory for rational knots. The only result which has been known is

that the knot signature function [45, 38, 44, 37, 33] extends to rational

knots [12, 7]. Indeed all the known applications to link concordance depend

on this signature invariant.

Received by editor November 5, 2004

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