SIGMA 22 (2026), 107, 13 pages      arXiv:2607.23859      https://doi.org/10.3842/SIGMA.2026.107
Contribution to the Special Issue on Interactions of Poisson Geometry, Lie Theory and Symmetry in honor of Rui Loja Fernandes

A Note on the Cohomology of Pullback Lie Algebroids

Eckhard Meinrenken
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S 2E4, Canada

Received July 28, 2026, in final form October 28, 2026; Published online November 04, 2026

Abstract
We give a direct differential-geometric proof of a result of Crainic on the cohomology of pullbacks of Lie algebroids under surjective submersions: If the fibers are connected and have vanishing cohomology through degree $n$, pullback with coefficients in any representation is an isomorphism through degree $n$ and is injective in degree $n+1$. The proof reduces to a vanishing result for leafwise cohomology and uses an exhaustion-and-patching argument due to Buchdahl. No local triviality or finite-type assumption on the fibers is required. We also prove the analogous result for holomorphic Lie algebroids.

Key words: Lie algebroids; Lie algebroid cohomology; foliated cohomology; van Est theory.

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