We also remark that Cochran and Orr pointed out in an unpublished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