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
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