In the present paper we determine for each parallelizable smooth compact manifold $M$ the second cohomology spaces of the Lie algebra ${𝒱}_M$ of smooth vector fields on $M$ with values in the module $\overline{\Omega }{}_M^p={\Omega }_M^p/d{\Omega }_M^{p-1}$. The case of $p=1$ is of particular interest since the gauge algebra of functions on $M$ with values in a finite-dimensional simple Lie algebra has the universal central extension with center ${\overline{\Omega }}_M^1$, generalizing affine Kac-Moody algebras. The second cohomology $H^2({𝒱}_M,{\overline{\Omega }}_M^1)$ classifies twists of the semidirect product of ${𝒱}_M$ with the...

Hermann and Thurston proved that the group of diffeomorphisms with compact support of a smooth manifold M which are isotopic to the identity is a perfect group. We consider the case where M has a geometric structure. In this paper we shall survey on the recent results of the first homology of the diffeomorphism groups which preserve a smooth G-action or a foliated structure on M. We also work in Lipschitz category.

In this paper we study the real secondary classes of transversely holomorphic foliations. We define a homomorphism from the space $H^{*}({\mathrm{WO}}_{2q})$ of the real secondary classes to the space $H^{*}({\mathrm{WU}}_q)$ of the complex secondary classes that corresponds to forgetting the transverse holomorphic structure. By using this homomorphism we show, for example, the decomposition of the Godbillon-Vey class into the imaginary part of the Bott class and the first Chern class of the complex normal bundle of the foliation. We show also...