Surgery obstructions of fibre bundles

Abstract

In a previous paper we obtained an algebraic description of the transfer maps p∗:L n( Z[π 1(B)]→L n+d( Z[π 1(e)]) induced in the Wall surgery obstruction groups by a fibration ▪ with the fibre F a d-dimensional Poincaré complex. In this paper we define a Π 1( B)-equivariant symmetric signature σ∗(F, ω)ϵL d(π 1(b), Z) depending only on the fibre transport ω: Π 1( B)→[ F, F], and prove that the composite p ∗p∗:L n( Z[π 1(B)])→L n+d( Z[π 1(b)] is the evaluation σ∗(F, ω)⊗? of the product ⊗:L d(π 1(B), Z⊗L n( Z[π 1(B)])→L n+d( Z[π 1(B)]) . This is applied to prove vanishing results for the surgery transfer, such as p∗=0 if F= G is a compact connected d-dimensional Lie group which is not a torus, and ▪ is a G-principal bundle. An appendix relates this expression for p∗p ∗ to the twisted signature formula of Atiyah, Lusztig and Meyer.