k

NEAL W. STOLTZFUS

invariant is almost an isomorphism. In Theorem 2.8, we show that the

Alexander invariant has a cokernel of order two (generated by X - 1/2)

if e = -1 and has a kernel of order two if e = +1 and p is odd.

In all other cases the kernel and cokernel are zero. Furthermore/ the

group P (W ) is infinitely generated as a F9-vector space. Similar

p z

localization sequences related to Witt groups of bilinears forms and

surgery obstruction groups have been studied by A. Bak and W. Scharlau

[BS], A. Dress [D], M. Karoubi [Ka], Lannes, Latour, Barge and Vogel

[La,BLLV], Milnor and Husemoeller [MH], W. Neumann [N]/ W. Pardon [P]

and CTC Wall [WaJ.

There is a geometric long exact sequence involving the spherical

knot concordance group which mimics the algebraic localization exact

sequence defined above. We first define a rational geometric knot con-

cordance group, £ (Q) , as follows: The objects are oriented codimen-

sion two manifold pairs, (s ,N ) , where N is only required to be

a rational homology sphere, i.e. H (N;Q) — H (Sn;Q) . The equivalence

relation is then given by the requirement that two rational knots are

equivalent if they are the oriented boundary of a rational homology

cobordism H in S x I . More prosaically, a rational knot is null

concordant if it bounds a rational homology disc in Dn , the unit

disc which bounds S . The torsion geometric knot concordance group

is the appropriate relative gadget: the objects are oriented codimension

two manifold pairs (D ,M ) , where M is a rational homology disc

whose boundary is an integral homotopy sphere contained in dD = S

The equivalence relation is given by the requirement that there is a

rational homology cobordism H in D x I which restricts to an

h-cobordism on the boundary. This group is denoted C (Q,^) . There

is a long exact sequence (see Section Six)

... - Sn(B) - en(Q)-

Vl(Q'B)

*

Cn-lW

- •••

where i is the inclusion, j removes a standard unknotted disc pair

(Dn ,Dn) and d takes the geometric boundary. In Section Six, we

will relate the localization exact sequence of Theorem 2.4 to the above

geometric exact sequence and prove that the subgroup, &\ _-, (Q) , of