Local System and Smooth Correspondence in de Rham Theory with Twisted Coefficients

Abstract

Let \(\mathcal L\) be a local system, i.e., a flat vector bundle, on a manifold M. We denote by \((\Omega ^{\bullet }(M;{\mathcal L})=\Gamma (M;\bigwedge ^{\bullet }T^*M \otimes {\mathcal L}), d=d_{\mathcal L})\) the de Rham complex with coefficients in \(\mathcal L\). We recall some basic operations on the de Rham complex with twisted coefficients.