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.