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
1
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