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Peer-reviewedCC BY-SA 4.0Symmetry, Integrability and Geometry: Methods and Applications

A Note on the Cohomology of Pullback Lie Algebroids

  • Eckhard Meinrenken1
  1. 1University of Toronto, Canada
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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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Cite

BibTeXRIS
BibTeX
@article{meinrenken2026note,
  title = {{A Note on the Cohomology of Pullback Lie Algebroids}},
  author = {Eckhard Meinrenken},
  journal = {Symmetry, Integrability and Geometry: Methods and Applications},
  year = {2026},
  month = nov,
  doi = {10.3842/sigma.2026.107},
  url = {https://sigma-journal.com/2026/107/},
  note = {CC BY-SA 4.0},
}
RIS
TY  - JOUR
TI  - A Note on the Cohomology of Pullback Lie Algebroids
AU  - Eckhard Meinrenken
PY  - 2026
DA  - 2026/11/04
JO  - Symmetry, Integrability and Geometry: Methods and Applications
DO  - 10.3842/sigma.2026.107
UR  - https://sigma-journal.com/2026/107/
AB  - 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.
SN  - 1815-0659
ER  -