The Novikov conjecture for low degree cohomology classes
Let \(c\) be an \(\mathbb R\)-valued group 2-cocycle on a discrete group \(\Gamma\), \(\sigma^s={\text e}^{2\pi isc}\), \(s\in[0,1]\), \(\sigma^1=\sigma\), and let \(C^*_r(\Gamma,\sigma)\) denote the twisted group \(C^*\)-algebra. The author constructs a \(C^*_r(\Gamma,\sigma)\)-vector bundle \(V_\sigma\to B\Gamma\) over a classifying space \(B\Gamma\) that defines a canonical class \([V_\sigma]\in K_0(C_0(B\Gamma)\otimes C^*_r(\Gamma,\sigma))\). This construction generalizes, for the twisted case, the canonical class of \textit{A. S. Mishchenko} [Izv. Akad. Nauk SSSR Ser. Mat. 34, 501-514 (1970; Zbl 0216.19701)]. Using this, a twisted Kasparov map \(\mu_\sigma:K_*(B\Gamma)\to K_*(C^*_r(\Gamma,\sigma))\) is defined as the intersection product by \([V_\sigma]\). The main result is homotopy invariance of \(\mu_{\sigma^s}(D)\in K_0(C^*_r(\Gamma,\sigma^s))\) for small enough \(s\), where \(D\) denotes the signature operator. This allows the author to give a short proof of the Novikov conjecture for all homotopy classes generated by those of degree \(\leq\) 2, a result that was first established by \textit{A. Connes, M. Gromov} and \textit{H. Moscovici} [Geom. Funct. Anal. 3, 1-78 (1993; Zbl 0789.58069)].
scientific article; zbMATH DE number 1977885