0 Connections up to homotopy and characteristic classes ∗

The aim of this note is to clarify the relevance of “connections up to homotopy” [4, 5] to the theory of characteristic classes, and to present an application to the characteristic classes of Lie algebroids [3, 5, 7] (and of Poisson manifolds in particular [8, 13]). We have already remarked [4] that such connections up to homotopy can be used to compute the classical Chern characters. Here we present a slightly different argument for this, and then proceed with the discussion of the flat characteristic classes. In contrast with [4], we do not only recover the classical characteristic classes (of flat vector bundles), but we also obtain new ones. The reason for this is that (Z2-graded) non-flat vector bundles may have flat connections up to homotopy. As we shall explain here, in this category fall e.g. the characteristic classes of Poisson manifolds [8, 13]. As already mentioned in [4], one of our motivations is to understand the intrinsic characteristic classes for Poisson manifolds (and Lie algebroids) of [7, 8], and the connection with the characteristic classes of representations [3]. Conjecturally, Fernandes’ intrinsic characteristic classes [7] are the characteristic classes [3] of the “adjoint representation”. The problem is that the adjoint representation is a “representation up to homotopy” only. Applied to Lie algebroids, our construction immediately solves this problem: it extends the characteristic classes of [3] from representations to representations up to homotopy, and shows that the intrinsic characteristic classes [7, 8] are indeed the ones associated to the adjoint representation [5]. I would like to thank J. Stasheff and A. Weinstein for their comments on a preliminary version of this paper.