标题： 关于Losik类的微分同胚伪群 显示英文标题

标题： On Losik classes of diffeomorphism pseudogroups

摘要： 设$P$是$n$维光滑流形$M$的局部微分同胚伪群。 遵循Losik的思想，我们将商$M/P$的特征类视为$M/P$上二阶框架丛的de~Rham上同调的元素，这些框架丛来自Gelfand-Fuchs上同调的生成元。 我们给出了Godbillon-Vey-Losik类和第一个Chern-Losik类的显式表达式。 通过约化框架丛，我们构造了定义在$M/P$上的丛，使得 Godbillon-Vey-Losik 类由一个维度为$2n+1$的空间上的体积形式表示，而第一 Chern-Losik 类由一个维度为$2n$的空间上的辛形式表示。 考虑了二维情形的例子。 显示英文摘要

摘要： Let $P$ be a pseudogroup of local diffeomorphisms of an $n$-dimensional smooth manifold $M$. Following Losik we consider characteristic classes of the quotient $M/P$ as elements of the de~Rham cohomology of the second order frame bundles over $M/P$ coming from the generators of the Gelfand-Fuchs cohomology. We provide explicit expressions for the classes that we call Godbillon-Vey-Losik class and the first Chern-Losik class. Reducing the frame bundles we construct bundles over $M/P$ such that the Godbillon-Vey-Losik class is represented by a volume form on a space of dimension $2n+1$, and the first Chern-Losik class is represented by a symplectic form on a space of dimension $2n$. Examples in dimension 2 are considered.