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.
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