The group of concordance classes of high dimensional homotopy spheres
knotted in codimension two in the standard sphere has an intricate
algebraic structure which this paper unravels. The first level of
invariants is given by the classical Alexander polynomial- By means of
a transfer construction, the integral Seifert matrices of knots whose
Alexander polynomial is a power of a fixed irreducible polynomial are
related to forms with the appropriate Hermitian symmetry on torsion free
modules over an order in the algebraic number field determined by the
Alexander polynomial. This group is then explicitly computed in terms
of standard arithmetic invariants. In the symmetric case, this computa-
tion shows there are no elements of order four with an irreducible
Alexander polynomial. Furthermore, the order is not necessarily Dedekind
and non-projective modules can occur. The second level of invariants
is given by constructing an exact sequence relating the global concordance
group to the individual pieces described above. The integral knot
concordance group is then computed by a localization exact sequence
relating it to the rational group computed by J. Levine and a group of
torsion linking forms.
This paper also develops the algebraic machinery needed to explicitly
compute the Witt group of isometries of inner product spaces arising in
the work of Lopez de Medrano and Kreck on the bordism of diffeomorphisms
and also the group of equivariant concordance of knots invariant under
a cyclic group action. Finally a geometric sequence involving codimension
two rational homology spheres and rational concordance is defined and
related to the algebraic localization sequence.
Key words and phrases: concordance of knots, Hermitian forms, Order,
transfer, localization, isometry, bordism of diffeomorphisms.
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