The smooth vector fields on a smooth manifold \( M \) form a Lie algebra \( \operatorname{Vect}(M) \) under the bracket. Gelfand and Fuchs [1968, 1969, 1970a, 1970b] have studied the Lie algebra cohomology of \( \operatorname{Vect}(M) \), which they define by means of a cochain algebra \( A(M) \), where \( A^k(M) \) is the vector space of continuous \( \mathbb{R} \)-multilinear maps
\[ \operatorname{Vect}(M) \stackrel{\leftarrow k \rightarrow}{\times \cdots \times} \operatorname{Vect}(M) \to \mathbb{C} \]
and the differential \( \operatorname{d}:A^k(M)\to A^{k+1}(M) \) is defined by the formula
\[ \operatorname{d}\alpha(\xi_1,\dots,\xi_{k+1}) = \sum_{i < j} (-1)^{i+j-1} \alpha([\xi_i,\xi_j],\xi_1,\dots,\hat{\xi}_i,\dots,\hat{\xi}_j,\dots,\xi_{k+1}). \]
‘Continuous’ refers to the usual \( C^{\infty} \) topology on \( \operatorname{Vect}(M) \). (Actually Gelfand and Fuchs considered the cohomology with real coefficient, but we have found it convenient to change from \( \mathbb{R} \) to \( \mathbb{C} \).).
In this paper we shall prove that when \( M \) is either a compact manifold or the interior of a compact manifold with boundary the cohomology of \( \operatorname{Vect}(M) \) is the same as that of the space of continuous cross-sections of a certain natural fibre bundle \( E_M \) on \( M \) associated to its tangent bundle. The fibre of \( E_M \) is an open manifold \( F \) whose cohomology is that of \( \operatorname{Vect}(\mathbb{R}^n) \). The result was conjectured independently by Fuchs and the first author, and has also been proved by Haefliger [1976] and Trauber by different methods.
