Surgery obstructions of fibre bundles
type: Item
instance of: scholarly article
title: Surgery obstructions of fibre bundles
zbMATH Open document ID: 0755.57013
DOI: 10.1016/0022-4049(92)90003-X
author: Andrew Ranicki
author: Wolfgang Lück
published in: Journal of Pure and Applied Algebra
publication date: 1992-09-27T00:00:00Z
review text: For a fibre bundle \(F\to E@>p>> B\), the signature formula \[ \text{sign}(E)=\text{sign}(F)\cdot\text{sign}(B)\in\mathbb{Z} \] holds provided the fundamental group \(\pi_ 1(B)\) acts trivially on \(H^*(F;\mathbb{R})\) [see \textit{S. S. Chern, F. Hirzebruch} and \textit{J.-P. Serre} [Proc. Am. Math. Soc. 8, 587-596 (1957; Zbl 0083.178)]. In general, even if the action is nontrivial, \textit{M. F. Atiyah} [Global Analysis, Pap. Hon. K. Kodaira, 73-84 (1969; Zbl 0193.523)] presents a characteristic class formula for the signature \(\text{sign}(E)\) involving the action of \(\pi_ 1(B)\) on \(H^*(F;\mathbb{R})\) in the case \(B\) and \(F\) are even-dimensional manifolds. For a fibration \(F\to E @>p>> B\), where the fibre \(F\) is a Poincaré complex of dimension \(d\), the authors apply their algebraic surgery transfer [Lect. Notes Math. 1361, 167-246 (1988; Zbl 0677.57012)] \[ p^*:\;L_ n(\mathbb{Z}\pi_ 1(B))\to L_{n+d}(\mathbb{Z}\pi_ 1(E)) \] induced in the Wall surgery obstruction groups to study the Wall surgery obstruction and the Mishchenko symmetric signature in fibrations. These invariants are generalizations of the equivariant signature, and the characteristic classes are replaced by more general \(L\)-theory invariants. Instead of dealing with the action of \(\pi_ 1(B)\) on \(H^* (F;\mathbb{R})\), the authors consider the chain homotopy action of \(\pi_ 1(B)\) on the chain complex \(C(F)\) induced by the homotopy action of \(\pi_ 1(B)\) on \(F\) given by the fibre transport \(\omega: \pi_ 1(B) \to [F,F]\). The main result expresses the composition \[ L_ n(\mathbb{Z}\pi_ 1(B))@>p^*>> L_{n+d}(\mathbb{Z}\pi_ 1(E))@>p_ *>>L_{n+d}(\mathbb{Z}\pi_ 1(B)) \] as the evaluation \(\sigma^*(F,\omega)\otimes -\) for the \(\pi_ 1(B)\)-equivariant symmetric signature invariant \(\sigma^*(F,\omega)\allowbreak \in L^ d(\pi_ 1(B),\mathbb{Z})\). This expression is applied to prove vanishing results for the surgery transfer, such as \(p^*= 0\) if \(F\) is a compact connected nonab…
Mathematics Subject Classification ID: 57R67
Mathematics Subject Classification ID: 55R12
Mathematics Subject Classification ID: 19J25
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 55R10
zbMATH DE Number: 64643
zbMATH Keywords: fibre bundle
zbMATH Keywords: signature
zbMATH Keywords: fundamental group
zbMATH Keywords: characteristic class formula for the signature
zbMATH Keywords: algebraic surgery transfer
zbMATH Keywords: Wall surgery obstruction groups
zbMATH Keywords: Mishchenko symmetric signature
zbMATH Keywords: chain homotopy action
zbMATH Keywords: surgery transfer
zbMATH Keywords: twisted signature formula
reviewed by: Krzysztof Pawałowski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 64643