UNRAVELING THE INTEGRAL KNOT CONCORDANCE GROUP 3

invariant under an endomorphism of a rational vector space is that the

coefficients of the minimal polynomial be integral. This condition on

the minimal polynomial of the endomorphism t defines a direct summand,

Cn(Q) , of the rational knot concordance group. We then construct an

exact sequence

0 -

C£(Z)

- CQ(Q) -

C€(Q/Z)

- 0

where C {Q/'Z) is a similarly defined concordance group of isometric

structures on finitely generated torsion abelian groups with e-symmetric

forms taking values in Q/Z . The exactness of the sequence is demon-

strated in Theorem 2.4 while the surjectivity of the boundary requires a

deeper analysis and is accomplished only in Theorem 5.9. The group of

torsion structures is then computed by first decomposing it into its

p-primary components, C (Q/Z) — © C {% [l/p]/Z) where the direct sum

P

is taken over all integral primes. The group C (Z [l/p]/Z) is then

computed by verifying that it is isomorphic to C (F ) in Theorem 2.5.

P

This isomorphism uses the idea of an anisotropic isometric structure

defined by the condition that there are no self-annihilating invariant

non-trivial subspaces. The final step in the computation for the group

of torsion isometric structures is the computation given in Theorem 2.7

which is valid for any perfect field F and reduces the problem to the

computation of the Witt group of Hermitian forms over all finite exten-

sion fields of F . Proposition 1.11 verifies the well-known fact that

the characteristic polynomial of the endomorphism t of an isometric

structure is an invariant of the concordance class under the equivalence

relation that f (X) is equivalent to f. . (X) if there is a monic poly-

nomial g(X) with f1(X) = fQ(X)g(X)g(l-X)(-l)deg g or vice versa.

The resulting monoid of equivalence classes of monic polynomials for a

ring R forms a group under polynomial multiplication which is denoted

P (R) . The map A: C~ (F) ~ * P(R) assigning to a representative isome-

tric structure in an equivalence class the equivalence class of the

characteristic polynomial of the endomorphism is then well defined and

is called the Alexander invariant. (It is related to the classical

Alexander polynomial of a knot.) For the group C (W ) , the Alexander