Gel’fand-Fuks cohomology is the cohomology of contionuous alternating chains on the topological algebra of smooth vector fields on smooth manifold, equipped; where the topology on the algebra is given by uniform convergence of all (higher) partial derivative on compacts (sometimes called -topology on that algebra).
I. M. Gel'fand, D. B. Fuks, The cohomology of the Lie algebra of vectro fields on a smooth manifold, J. Funct. Analysis 33, 1969, 194–210, II, J. Funct. Anal. 4 (1970) 110-116; The cohomology of the Lie algebra of formal vector fields, Izv. AN SSR 34 (1970), 110-116