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arXiv · 2607.23859

A note on the cohomology of pullback Lie algebroids

Eckhard Meinrenken
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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.

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Eckhard Meinrenken. 2026-07-26. A note on the cohomology of pullback Lie algebroids. https://arxiv.org/abs/2607.23859

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