We give several sufficients conditions for a 2-cycle of $B$ Diff$(S^1{)}_d$ (resp. $B$ Diff${}_K(\mathbf{R}{)}_d$) represented by a foliated $S^1$-(resp. $\mathbf{R}$-) bundle over a 2-torus to be homologous to zero. Such a 2-cycle is determined by two commuting diffeomorphisms $f$, $g$ of $S^1$ (resp. $\mathbf{R}$). If $f$, $g$ have fixed points, we construct decompositions: $f=\pi f_i$, $g=\pi g_i$, where the interiors of Supp$\left(f_i\right)\cup$ Supp$\left(g_i\right)$ are disjoint, and $f_i$ and $g_i$ belong either to $\{h_i^n;n\in \mathbf{Z}\}$ ($h_i\in$ Diff${}^{\infty }$) or to a one-parameter subgroup generated by a $C^1$-vectorfield ${\xi }_i$. Under some conditions on the norms...

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...

The continuous cohomology theory of the Lie algebra $L\left(M\right)$ of complex analytic vector fields on an open Riemann surface $M$ is studied. We show that the cohomology group with coefficients in the $L\left(M\right)$-module of germs of complex analytic tensor fields on the product space $M^n$ decomposes into the global part derived from the homology of $M$ and the local part coming from the coefficients.