For a finite-dimensional Lie algebra , let be the space of alternating -forms on . Taking cues from the Lie algebra of left-invariant vector fields on a Lie group, one defines the differential
When is the infinite-dimensional Lie algebra of vector fields on a manifold , the formula (4) still makes sense, but the space of all alternating forms is too large for its cohomology to be computable. Gelfand and Fuks proposed putting a topology, the topology, on , and computing instead the cohomology of the continuous alternating forms on . The Gelfand-Fuks cohomology of is the cohomology of the complex of continuous forms. They hoped to find in this way new invariants of a manifold. As an example, they computed the Gelfand-Fuks cohomology of a circle.
It is not clear from the definition that the Gelfand-Fuks cohomology is a homotopy invariant. In  Bott and Segal proved that the Gelfand-Fuks cohomology of a manifold is the singular cohomology of a space functorially constructed from . Haefliger [H] and Trauber gave a very different proof of this same result. The homotopy invariance of the Gelfand-Fuks cohomology follows. At the same time it also showed that the Gelfand-Fuks cohomology produces no new invariants.