Higher \(G\)-signatures for Lipschitz manifolds
The paper is a continuation of an article of the same authors [Ann. Sci. Éc. Norm. Supér., IV. Sér. 21, No. 4, 479-495 (1988; Zbl 0693.58024)]. First, a proof of the Novikov conjecture on higher signatures for topological manifolds is given. Using the Teleman signature operator and Kasparov's KK-theory a strong de Rham theorem and a higher \(G\)-signature theorem for Lipschitz manifolds are proved in order to study nonsmooth transformation groups. In the last section, the authors give a number of applications. A topologically invariant characteristic class for certain \(G\)-manifolds is constructed; this generalizes the \(L\)-class of smooth manifolds whose topological invariance is a famous theorem of Novikov. As a consequence, a direct proof of the known result that nonlinear similarities preserve renormalized Atiyah-Bott numbers is obtained. Similarly the authors prove the following theorem: A topological orientation-preserving action of a finite group \(G\) on a compact simply connected topological manifold \(M\), such that for all subgroups \(H \subseteq K \subseteq G\), the fixed sets \(M^ K\) and \(M^ H\) are simply connected submanifolds of dimension \(\neq 3\) and the inclusion of \(M^ K\) in \(M^ H\) is locally flat and of codimension \(\geq 3\), is determined by its isovariant homotopy type and the classes of the equivariant signature operators on all the sets \(M^ H\).
scientific article; zbMATH DE number 458843
scientific article