Finite-dimensional Lie algebras of formal vector fields and characteristic classes of homogeneous foliations

D. B. Fuchs

Izvestiya. Mathematics , Tome 10 (1976) no. 1, pp. 55-62

Résumé

In [5], I. M. Gel'fand and the author computed the cohomology of the Lie algebra $W_n$ of formal vector fields in $n$-dimensional space. The present article is devoted to the study of homomorphisms $H^*(W_n;\mathbf R)\to H^*(\mathfrak g;\mathbf R)$ induced by imbeddings of finite-dimensional subalgebras in $W_n$. We show that there exist elements of $H^*(W_n;\mathbf R)$ which are annihilated by any such homomorphism. On the other hand, we show that the image of the cohomology homomorphism induced by the well-known embedding $\mathfrak{sl}(n+1,\mathbf R)\to W_n$ has dimension $2^{n-1}+1$. The results are applied to characteristic classes of foliations. Bibliography: 9 titles.