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# The cohomology of the vector fields on a manifold

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

 (4)

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.

Next: Localization in equivariant cohomology Up: The life and works Previous: Obstruction to integrability
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