ON THE RELATIVE WALL-WITT GROUPS

Let R be a simplicial involutive ring. According to certain involutio ns on K(R) and epsilon L(R*), there are 1/2-local splittings K(R*) si milar or equal to K-s(R) X K-alpha(R*) and epsilon L(R*) similar or e qual to epsilon L(s)(R) X epsilon L(a)(R*). It is known [2] that epsi lon L(n)(a)(R) congruent to epsilon L(n)(alpha)(pi(0)R*) congruent to epsilon W(pi(0)R), the Wall-Witt group. Suppose I --> R --> S is a s pilt extension of discrete involutive rings with I-2 = 0, and I is a f ree S-biomodule. Then we have K-n+1(f) circle times Q congruent to Pri m(n) Lambda M(I circle times Q) and epsilon L(n+1)(f) circle times Q congruent to Prim(n) epsilon O(I circle times Q). The trace map Tr: Pr im(n) Lambda M(I circle times Q) --> ($) over bar W-0(rho(n); I circl e times Q) is an isomorphism. We prove in Lemma 1 that the trace map T r is Z/2-equivariant. In Theorem 2 we show that under a certain assump tion the rational relative Wall-Witt group vanishes. Theorem 2 can be extended to a more general case (Theorem 3) by employing Goodwillie's reduction technique [3].