Gelfand–Fuks cohomology

In mathematics, Gelfand–Fuks cohomology, introduced in (Gel'fand & Fuks 1969–70), is a cohomology theory for Lie algebras of smooth vector fields. It differs from the Lie algebra cohomology of Chevalley-Eilenberg in that its cochains are taken to be continuous multilinear alternating forms on the Lie algebra of smooth vector fields where the latter is given the $C^{\infty }$ topology.

References

• Gel'fand, I. M.; Fuks, D. B. (1969). "Cohomologies of Lie algebra of tangential vector fields of a smooth manifold". Funct Anal Its Appl. 3: 194–210. doi:10.1007/BF01676621.
• Gel'fand, I. M.; Fuks, D. B. (1970). "Cohomologies of Lie algebra of tangential vector fields. II". Funct Anal Its Appl. 4: 110–6. doi:10.1007/BF01094486.
• Gel'fand, I. M.; Fuks, D. B. (1970). "The cohomology of the Lie algebra of formal vector fields". Mathematics of the USSR-Izvestiya. 2 (2): 327–342. doi:10.1070/im1970v004n02abeh000908.
• Shigeyuki Morita (2001). "§2.4 Gel'fand–Fuks cohomology". Geometry of Characteristic Classes. Translations of Mathematical Monographs. Vol. 199. American Mathematical Society. pp. 75ff. ISBN 978-0-8218-2139-8.

Further reading

• Gelfand–Fuks cohomology at the nLab