INTRODUCTION The main purpose of this paper is to compute H(LM, FM), the continuous cohomology of thetopological Liealgebra LM of allthesmooth vector fields on a manifold M associated totheusual representation on FM, where FM is thealgebra of smooth functions on M. H(LM, FM) is by definition thecohomology of thedifferential graded algebra C(LM, FM)= (©CP(LM, FM), d},where CP(LM, FM) denotes thespace of all the continuous alternating multilinear maps from LMx...xLM (ptimes) to FM and d is defined by (dca)(X1,...,Xp+1) = P £ (-l)i+1XiW(X1,...,Xi,...,Xp+1) + Z (-i)1+j([x.,x-],...,x.,...,x ..) ij J J for a)G CP(LM, FM) and Xi e LM- (See §3.1 forthedefinition of the multiplication.) Ifwe put CDR(LM, FM)= {a) e C(LM, FM) w isFM-multilinear}, then CDR(LM, FM) is a differential graded subalgebra, isomorphic to ftM the de Rham differential graded algebra. Hence, by thede Rham theorem, we have H DR(LM F M aH *(M R) ' where H T)D(LM F M^ denotes the cohomology of C DR(LM F ^ a n d H *(M R) the singular cohomology of M. Next, consider CA(LM, FM)= {wG C(LM, FM) O J is support preserving}, which is in fact a differential graded subalgebra of C(LM, FM)• A theorem of Losik ([22]) asserts that Received by theeditors December 11, 1979. 1
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