Connections up to homotopy and characteristic classes

Abstract

The aim of this note is to clarify the relevance of connections up to homotopy to the theory of characteristic classes and to present an application to the characteristic classes of algebroids and of Poisson manifolds in particular

We have already remarked that such connections up to homotopy can be used to compute the classical Chern characters Here we present a slightly dierent argument for this and then proceed with the discussion of the at characteristic classes In contrast with we do not only recover the classical characteristic classes of at vector bundles but we also obtain new ones The reason for this is that Zgraded nonat vector bundles may have at connections up to homotopy As we shell explain here in this category fall e g the characteristic classes of Poisson manifolds

As already mentioned in one of our motivations is to understand the intrinsic characteristic classes for Poisson manifolds and algebroids of and the connection with the characteristic classes of representations Conjecturally Fernandes intrinsic characteristic classes are the characteristic classes of the adjoint representation The problem is that the adjoint representation is a representation up to homotopy only Applied to algebroids our construction immediately solves this problem it extends the characteristic classes of from representations to representations up to homotopy and shows that the intrinsic characteristic classes are indeed the ones associated to the adjoint representation