L-theory of fields and rational localization

Abstract

The computation of the knot cobordism groups reduces to the computation of the L-theory of various fields and integral domains. The main properties of the L-groups of a field with involution F are summarized in 37A. The computations of the L-groups of ℤ, ℤ _m , ${\widehat {\Bbb Z}_m}$, ℚ and the localization exact sequence are used in 37B to prove a general result: the natural maps L* (A) → L* (ℚ ⊗_ℤ A) are isomorphisms modulo 8-torsion for any additively torsion-free ring involution A (e.g. a group ring A = ℤ[π]). In Chap. 42 the version of this result for torsion L-groups will be used to prove that the natural maps

$${C_*} = LIs{o^{* + 1}}({\Bbb Z}) \to LIso_{{U_{\Bbb Z}}}^{* + 1}({\Bbb Q})$$

are isomorphisms modulo 8-torsion, for an appropriately defined U _ℤ.