UNRAVELING THE INTEGRAL KNOT CONCORDANCE GROUP 11

We call N a metabolic submodule of M or metabolizer, for short.

Proposition 1.3. (M/ , /t) is metabolic over R if and only if

(M, , ,t) & K is metabolic.

The proof is standard. Utilizing this fact, we may weaken the con-

ditions of Definition 1.2 to the following:

Proposition 1.4. (M, , /t) is metabolic if and only if there is

an R-submodule N satisfying:

i) rank M = 2 rank N

ii) N c

N"

iii) N is invariant under t

Proof: We may assume that our metabolic submodule is pure (that is,

if nx is in N for some n in R and x ^ 0 in M then x is in

x ad|N

N) . Then we have the following exact sequence: 0 ~ * N - M — Horn (N,R)

R

"- * 0 since any form on the pure submodule N can be lifted to M and

by the non-singularity of the bilinear form represented as the adjoint of

some element in M . Now, by the additivity of rank over short exact

sequences and condition i), rank N = rank N""" . Since N

c

N by ii)

and both are pure submodules of the same rank , N = N

The orthogonal direct sum of two isometric structures, (Mo#^ ' ' ntn^

and (M]_, , 1»t1) is (MQ S Mx, , © , 1»tQ © t^ . Finally we

define the appropriate equivalence relation:

Definition 1.5 . Two isometric structures, MQ and M-, , are Witt-

equivalent (or concordant) if there are metabolic isometric structures,

H^ and H, , so t h a t IVL © H„ i s i s o m o r p h i c t o M, © H, .

U 1 u u l l

The resulting set of equivalence classes, C*~ (R) , form a group

under (orthogonal) direct sum with the inverse of {M, , ,t} given by

{M,- , ,t} (where { } will denote the appropriate equivalence class.)

Proposition 1.6. {M, , ,t} = 0 in C"(R) if and only if

(M, , #t) is metabolic.

Proof: The assumption gives a metabolic structure H such that

M © H is metabolic with metabolizers Hn for H and N for M © H .