Asymptotic Lipschitz cohomology and higher signatures (Q1919139)

The paper deals with the study of the Novikov conjecture on homotopy invariance of higher signatures. The higher signature of a closed oriented manifold \(M\) with fundamental class \([M]\) is a characteristic number of the form \((L(M)f^*(x))[M]\), where \(L(M)\) denotes the Hirzebruch L-class of \(M\), \(f:M\to K(\Gamma,1)\) is a continuous map and \(x\in H^*(K(\Gamma,1);\mathbb{Q})\). The Novikov conjecture was proved for special classes of groups \(\Gamma\) by several authors. The main result of the present paper is, that the higher signatures are oriented homotopy invariants, if \(\Gamma\) is a torsionfree quasi-geodesic bicombing group. The assumption on the group is a technical one and we will not give the definition here. The author extends in his proof methods of \textit{A. Connes}, \textit{M. Gromov} and \textit{H. Moscovici} [Geom. Funct. Anal. 3, 1-78 (1993; Zbl 0789.58069)]. The basic idea of the author's proof is to approximate \(K(\Gamma,1)\) by finite-dimensional simplicial complexes. In a final step, the proof uses results of \textit{A. S. Mishchenko's} paper [Izv. Akad. Nauk SSSR, Ser. Mat. 38, 81-106 (1974; Zbl 0292.57017)].