UNRAVELING THE INTEGRAL KNOT CONCORDANCE GROUP

7

e-symmetric isometric structures whose ^ [X]-module structure can be

reduced to S . Therefore trace induces a natural isomorphism

H€(A~ (S/^)) with C^(^) . We note that we only require that the under-

lying S-module structure be torsion-free. This differs from the usual

requirement that the underlying module be projective.

We then develop an appropriate analog of the localization exact

sequence

0 - H€(A~1(S/S)) - HE(E) - H€(E/A"~1(S/2G))

for any order S . The restriction mapping from the maximal order D

induces a mapping of the localization short exact sequences which is an

injection H (A~ (D/Z)) - » H (A~ (s/Z)) on S and is an isomorphism on

the quotient field. In a series of technical lemmas we relate the

boundary for D with that of S (At each maximal ideal M the relation

on the residue fields is ^^nc

= tr*

° ^#? where tr is the appropriate

map induced by trace for finite fields.) and finally compute the boun-

dary for D . (Lemma 4.9). Utilizing Landherr 's classical computation

for H (E) , we can then compute H (A" (D/Z)) for the maximal order

D . This is detailed in Section Four, Theorem 4.11 and 4.14. In the

case of the maximal order D , J. Lannes [La] has proven similar results

in the symmetric bilinear and quadratic cases. An interesting algebraic

feature (which does not occur for the orders S naturally arising in

the knot concordance group (by Lemma 5.1)) is a distinction between two

types of dyadic primes which are ramified in the quadratic extension of

E over the fixed field under the involution, F .

For the maximal order D , torsion-free D-modules are projective

and the question arose concerning the relation of H (A (D/z)) with

H (D) . We first define a certain arithmetic invariant SC(-l) of

order two of the field E . If this invariant is trivial, then P. E.

€ -1 —a

Conner has shown us how to prove that H (A (D/Z))— H " (D) . The

interchange of symmetry is intimately connected with the structure of

the inverse different. This Theorem is given in (4.15) and can be

viewed as a generalization of the classical theorem of Hecke on the

class of the different in the ideal class group to the Hermitian case.