The family signature theorem

TL;DR

The paper develops a unified framework for the Family Signature Theorem across three settings: rational cohomology, KO[1/2]-theory, and symmetric L-theory L^s(ℤ). It integrates Meyer's twisted signatures, Sullivan’s real K-theory orientation, and Ranicki’s algebraic surgery to treat oriented topological block bundles and their fibrations, including Poincaré complexes. Key contributions include explicit twisted-signature formulas, modul 4 multiplicativity results for fibrations, a Ranicki orientation yielding a universal L-theory formulation, and detailed 2-local/integrality analyses that connect de Rham invariants, Stiefel–Whitney classes, and Grothendieck–Witt theory. The work provides 2-local refinements and precise multiplicativity/integrality statements with broad implications for fiberwise topology, linking geometric bundles to algebraic L-theory via a coherent, spectrum-level framework.

Abstract

We discuss several versions of the Family Signature Theorem: in rational cohomology using ideas of Meyer, in $KO[\tfrac{1}{2}]$-theory using ideas of Sullivan, and finally in symmetric $L$-theory using ideas of Ranicki. Employing recent developments in Grothendieck--Witt theory, we give a quite complete analysis of the resulting invariants. As an application we prove that the signature is multiplicative modulo 4 for fibrations of oriented Poincaré complexes, generalising a result of Hambleton, Korzeniewski and Ranicki, and discuss the multiplicativity of the de Rham invariant.

Theorems & Definitions (56)

• Remark 2.1
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Remark 2.4
• Remark 2.5
• Theorem 2.6: Family signature theorem over $\mathbb{Q}$
• proof
• Theorem 3.1: Family signature theorem over ${\mathrm{KO}[\tfrac{1}{2}]}$