Surgery obstructions of fibre bundles
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 nonabelian Lie group \(G\) of dimension \(d\), and \(F \to E@>p>> B\) is a \(G\)-principal bundle. In the appendix, the above expression is related to the twisted signature formula of Atiyah, Lusztig, and Meyer.
characteristic class formula for the signature
scientific article; zbMATH DE number 64643
scientific article