The cohomology of the vector fields on a manifold

For a finite-dimensional Lie algebra $L$ , let $A^q(L)$ be the space of alternating $q$ -forms on $L$ . Taking cues from the Lie algebra of left-invariant vector fields on a Lie group, one defines the differential

$$d: A^q(L) \to A^{q+1}(L)$$

by

$$(d\omega)(X_0, \dots, X_q) = \sum_{i < j} (-1)^{i+j} \omega([X_i, X_j], X_0, \dots, \hat{X_i}, \dots, \hat{X_j}, \dots, X_q).$$ (4)

As usual, the hat $\hat{}$ over $X_i$ means that $X_i$ is to be omitted. This makes $A^*(L)$ into a differential complex, whose cohomology is by definition the cohomology of the Lie algebra $L$ .

When $L$ is the infinite-dimensional Lie algebra $L(M)$ of vector fields on a manifold $M$ , the formula (4) still makes sense, but the space of all alternating forms $A^*(L(M))$ is too large for its cohomology to be computable. Gelfand and Fuks proposed putting a topology, the $C^{\infty}$ topology, on $L(M)$ , and computing instead the cohomology of the continuous alternating forms on $L(M)$ . The Gelfand-Fuks cohomology of $M$ is the cohomology of the complex $\{ A_c^*(L(M)), d\}$ 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 $M$ is the singular cohomology of a space functorially constructed from $M$ . 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.