The cohomology of the vector fields on a manifold

Abstract

‘Continuous’ refers to the usual c topology on Vect(M). (Actually Gelfand and Fuchs considered the cohomology with real coefficients, but we have found it convenient to change from R to 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 Vect(M) is the same as that of the space of continuous cross-sections of a certain natural fibre bundle EM on M associated to its tangent bundle. The fibre of EM is an open manifold F whose cohomology is that of Vect(R”). The result was conjectured independently by Fuchs and the first author, and has also been proved by Haefliger [ 1 l] and Trauber by different methods. The history of the present proof is roughly as follows. Gelfand and Fuchs in their first highly original papers considered only vector fields with compact supports. They observed that the cochains of the Lie algebra can be regarded as distributions, and they obtained their results by studying the natural filtration of the cochains by the dimension of their supports. Our plan has been to work with general vector fields on non-compact manifolds, so that they can be restricted to open submanifolds. This enables us to prove the theorem by first considering the case of R” and then using a patching-together process. We treat R” by a contraction argument which we think illuminates the essential homotopy-invariance of the Lie algebra cohomology (cf. Prop. (2.1)). It turns out that the Lie algebra of all vector fields on R” has the same cohomology as the jets of vector fields at a single point, i.e., the “formal vector fields” at the origin in R”. Given the validity of the local result it seemed, especially in view of Anderson’s calculation of the cohomology of mapping spaces [l], that a fairly simple patching argument should prove the conjecture. But there were two difficulties: first the technical difhculty that the patching involved surprisingly delicate questions of convergence, and secondly the fundamental dithculty that there seemed to be no map relating the Lie algebra cohomology to that of a space of sections. A solution of these difhculties along the lines presented here was devised, more or less, by the second author in 1974, and has been gradually improved by the authors jointly since then. It is still not very satisfying. It may be worth commenting on the relationship between our approach and others. Haefliger’s framework is Sullivan’s explicit algebraic model of the rational homotopy category in terms of minimal ditrerential graded algebras. Roughly one can say that he constructs on the one hand a minimal model for the Lie algebra cochains and on the other hand a minimal model for the homotopy type of the space of sections, and proves that the two models are actually isomorphic. An advantage of his approach is that it incorporates an explicit calculation of the