OBSTRUCTIONS TO REPRESENTATIONS UP TO HOMOTOPY AND IDEALS

This paper considers the Pontryagin characters of graded vector bundles of finite rank, in the cohomology vector spaces of a Lie algebroid over the same base. These Pontryagin characters vanish if the graded vector bundle carries a representation up to homotopy of the Lie algebroid. In other words, strong obstructions to the existence of a representation up to homotopy on a graded vector bundle of finite rank are found. In particular, if a graded vector bundle E-0[0] ? E-1[1] ? M carries a 2-term representation up to homotopy of a Lie algebroid A ? M, then all the (classical) A-Pontryagin classes of E0 and E1 must coincide.This paper generalises as well Bott's vanishing theorem to the setting of Lie algebroid representations (up to homotopy) on arbitrary vector bundles. As an application, the main theorems induce new obstructions to the existence of infinitesimal ideal systems in a given Lie algebroid.