UNRAVELING THE INTEGRAL KNOT CONCORDANCE GROUP 9

connected sums if the Alexander polynomial factors. It is interesting

to note that since the coupling invariants are trivial in the rational

concordance group, that we may split any knot as a connected sum of

knotted rational homology spheres provided the polynomial factors

suitably. We also apply our computation to the algebraic concordance

groups arising from the classification of equivariant concordances of

knots invariant under a cyclic action. As a final geometric application

we outline how to apply the algebraic techniques of this paper to the

problem of the bordism classification of diffeomorphisms, a problem

originated by W. Browder and studied by H. Winkelnkemper [Wi] and S.

Lopez de Medrano [LdM]. M. Kreck [Kr] has proved that the bordism class

of an (orientation preserving) diffeomorphism is determined by the Witt

invariant and the oriented bordism classes of the underlying manifold

and the associated mapping torus.

The formalism adopted in this paper is dictated by the geometric

problem from which it arose. We are, however, acutely aware that it is

intimately related to the setting of Hermitian algebraic K-theory,

particularly for the ring S [X] and the involution induced by X* =

1 - X . This ring is crucial in our study of the complement of a knot

or any space which is a homology circle. We also wish to record our

particular indebtedness to Pierre Conner who listened patiently to our

many musings, made pertinent suggestions and, in particular, contributed

the refinement of Hecke 's Theorem which gave the relationship of Theorem

4.15. Also significant in the development of the ideas of this paper

has been the joint work of J. Alexander, G. Hamrick and p. Conner and

J. Vick on the related problem of the Witt classification of the cyclic

isometries of integral inner products.