The transfer maps induced in the algebraic K0- and K1-groups by a fibration II

Abstract

In this paper we continue the study of the algebraic transfer p ∗: K n( Zπ 1 (B)) → K n ( Zπ 1 (E)) for n = 0, 1 defined in [12] for a fibration p: E → B. The algebraic transfer p ∗ agrees with the geometric transfers p !: K 0( Zπ 1(B)) → K 0( Zπ 1 (E)) and p !: Wh( π 1( B)) → Wh( π 1( E)) constructed in [7,8] and [4] respectively. The geometric K 0-transfer sends Wall's finiteness obstruction of B to the one of E. The Whitehead torsion of a homotopy equivalence f: B 0 → B is mapped by the Whitehead transfer to the one of f E 0 → E given by the pullback. An algebraic vanishing theorem for p ∗ is a vanishing theorem for p ! and is thus gemoetrically meaningful. Such algebraic vanishing theorems are obtained in the last three sections.