On the relative Wall-Witt groups

Abstract

LetR* be a simplicial involutive ring. According to certain involutions onK(R*) andeLR∗, there are 1/2-local splittings\(K(R_ * ) \simeq K^S (R_ * ) \times K^a (R_ * )\) and\(K(R_ * ) \simeq K^S (R_ * ) \times K^a (R_ * )\). It is known [2] thateL\gaαR∗, the Wall-Witt group. SupposeI→R\(I \to R \mathop \to \limits^f S\)S is a split extension of discrete involutive rings withI2=0, andI is a freeS-bimodule. Then we have\(K_n + 1(f) \otimes \mathbb{Q} \cong Prim_n \wedge ^ * M(I \otimes \mathbb{Q})\) and\(K_n + 1(f) \otimes \mathbb{Q} \cong Prim_n \wedge ^ * M(I \otimes \mathbb{Q})\). The trace map Tr: Primn∧*M(I ⊗ ℚ)→\(\bar W\)0ρn;I ⊗ ℚ) is an isomorphism. We prove in Lemma 1 that the trace map Tr is ℤ/2-equivariant. In Theorem 2 we show that under a certain assumption 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].