The family signature theorem

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, generalizing a result of Hambleton, Korzeniewski and Ranicki, and discuss the multiplicativity of the de Rham invariant.

Some rational homology computations for diffeomorphisms of odd‐dimensional manifolds

“…The classical family signature theorem for smooth fiber bundles (which uses families of elliptic operators in its proof) holds more generally for block bundles, as shown by Randal–Williams in [69, Theorem 2.6]. It has two cases, the odd‐dimensional and the even‐dimensional case.…”

“…This fact produces a map $$h: B\mathrm{hAut}_\partial ^+ (M) \rightarrow B \mathrm{Aut}(H_n (M;\mathbb {Q});I_M)$$; the latter is the classifying space of a symplectic or an orthogonal group, depending on the parity of $$n$$. Randal–Williams defines classes $$\sigma _i \in H^i(B \mathrm{Aut}(H_n (M;\mathbb {Q});I_M);\mathbb {Q})$$, which live in degrees $$i\equiv 2 \pmod 4$$ if $$n$$ is odd and $$i \equiv 0 \pmod 4$$ if $$n$$ is even, and shows in [69, Theorem 2.6] that $$$$\begin{equation} \kappa _{L_m} = h^* \sigma _{4m-2n} \in H^{4m-2n}(B \widetilde{\mathrm{Diff}}_\partial ^+ (M);\mathbb {Q}). \end{equation}$$$$Proposition Let $$M$$ be a compact oriented …”

“…References By Lemma 2.8, it is enough to show the theorem for closed $$M$$, and this is done in [69, Theorem 2.6] for block diffeomorphisms, and in [13] for diffeomorphisms.…”

“…In §2, we will show/review the vanishing theorems for the tautological classes that are entailed by Theorem 1.4 and Theorem B, namely, that the listed elements lie in the kernel of $$\Phi _{U_{g,1}^n}$$ and $$\Gamma$$. We will actually only use the vanishing of $$\kappa _{L_m}$$ on $$B \widetilde{\mathrm{Diff}}_\partial (U_{g,1}^n)$$ which we can just quote from [69]. The other two relations follow (for the manifolds $$U_{g,1}^n$$) from the subsequent calculations; we give the proper context for them in Propositions 2.10 and 2.13 for sake of completeness.…”