Characteristic classes for flat bundles and their formulas
Let \(G\) denote a complex reductive Lie group, \({\mathcal G}\) its Lie algebra and \(G^\delta\) the underlying discrete group. Using homology with rational coefficients, one has for any rational vector space \(V\) the well-known isomorphism \(H^*(X;V)= \Hom_\mathbb{Q} (H_*(X),V)\). This means that constructing characteristic classes with coefficients in \(V\) for flat \(G\)-bundles is equivalent to defining characteristic homomorphisms \(H_*(BG^\delta)\to V\), where \(BG^\delta\) denotes the classifying space of the group \(G^\delta\). The paper deals with the construction of such characteristic homomorphisms, which lead to some more characteristic classes. The author uses a fibre sequence of homotopy types \(G\to FG\to BG^\delta @>Bj>> BG\), where \(BG\) is the classifying space of the Lie group \(G\), and \(Bj\) is the natural map between the classifying spaces. The main ingredient of the construction is the definition of a homomorphism of \(H_*(FG)\) to the Lie algebra homology of \({\mathcal G}\) with complex coefficients. The definition uses integration over chains in the singular complex of \(G\), and simplicial models of the space \(FG\). The author gives some explicit formulae, and in the last section he discusses the relation of his characteristic homomorphism to the secondary characteristic classes of Cheeger-Chern-Simons.
scientific article; zbMATH DE number 633636