nLab Gelfand-Fuks cohomology

Idea

Gel’fand-Fuks cohomology is the cochain cohomology of continuous alternating chains on the topological algebra of smooth vector fields on smooth manifolds, where the topology on the algebra is given by uniform convergence of all (higher) partial derivative on compact subsets (sometimes called $C^\infty$-topology on that algebra).

References

• I. M. Gel'fand, D. B. Fuks, The cohomology of the Lie algebra of vector 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

• Shigeyuki Morita, Geometry of characteristic classes, Transl. Math. Monographs 199, AMS 2001

• Jean Celeyrette, Catégories internes et fibrations & Cohomologie de Gel’fand-Fuks, PhD thesis, Paris (1975) [pdf]