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

by

As usual, the hat over means that is to be omitted. This makes into a differential complex, whose cohomology is by definition the cohomology of the Lie algebra .

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 [71] 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.