2 1. INTRODUCTION

We also remark that Cochran and Orr pointed out in an unpublished

note that rational concordance is closely related with the rational homology

surgery theory developed by Quinn [39] and Taylor and Williams [43]. From

this viewpoint, rational concordance can be regarded as a particular instance

of rational homology surgery which provides computational techniques and

examples.

In this paper we perform through analysis of the structure of the rational

knot concordance group Cn. As one of the results, we give a full calculation

of its algebraic theory by constructing new algebraic invariants and com-

puting them. In particular, we discover infinitely many independent finite

order elements in Cn for odd n 1. Compared with the ordinary knot con-

cordance group from a topological viewpoint, it turns out that the rational

knot concordance group has a very different structure. We also investigate

the structure peculiar to knots in rational 3-spheres. In this dimension we

develop a computational technique of further obstructions to rational con-

cordance using the von Neumann p-invariants.

In this paper we work in the category of oriented piecewise linear man-

ifolds. Submanifolds are always assumed to be locally flat. Our results

also hold in the categories of smooth and topological manifolds with minor

modifications if necessary.

1.1. Integral and rational knot concordance

We begin by recalling known results on the group of concordance classes

of codimension two knots in

£n4"2,

which we call the integral knot concor-

dance group and denote by C^. In higher dimensions, C% is classified using

abelian invariants of knots which can be extracted using several different

techniques. For n 1, Kervaire [26] and Levine [33, 32] first computed

the structure of C^ using Seifert surfaces and Seifert matrices. Cappell

and Shaneson applied their homology surgery theory to identify C^ with

a surgery obstruction T-group [3]. In [23, 24], Kearton showed that the

same classification can be obtained using the Blanchfield form [1] for odd

n 1. We remark that C% is isomorphic to the set of integral homology

concordance classes of knots in integral homology spheres for n 1. This

justifies our terminology "integral (homology) concordance".

By work of Cochran, Orr, Ko, and the author [12, 7, 8], a framework

of the rational concordance theory has been initiated. Its basic strategy

is similar to the above integral theory, but, it involves more sophisticated

topological and algebraic constructions that produce objects whose struc-

tures have not been fully understood.

The essential difference between the integral and rational theories is il-

lustrated in the following general observation which introduces the notion

of complexity. Suppose that M is a properly embedded connected subman-

ifold of codimension two in W. If Hi(W;Z) = H2(W]Z) = 0, then from

the Alexander duality, it follows that H\{W — M;Z) is the infinite cyclic