ABSTRACT

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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