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 $E_0[0] \oplus E_1[1] \to M$ carries a $2$-term representation up to homotopy of a Lie algebroid $A \to M$, then all the (classical) $A$-Pontryagin classes of $E_0$ and $E_1$ 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.

Keywords

Lie algebroids, representations up to homotopy, connections up to homotopy, Pontryagin classes, graded vector bundles, Bott vanishing theorem, infinitesimal ideal systems, fibrations of Lie algebroids

2010 Mathematics Subject Classification

Primary 53B05, 57R20, 57R22. Secondary 16D25, 53D17.

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Received 23 May 2021

Accepted 23 September 2021

Published 6 March 2023