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<p>Let <i>P</i> be a pseudogroup of local diffeomorphisms of an <i>n</i>-dimensional smooth manifold <i>M</i>. Following Losik we consider characteristic classes of the quotient <i>M</i>/<i>P</i> as elements of the de&#xa0;Rham cohomology of the second order frame bundles over <i>M</i>/<i>P</i> 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 <i>M</i>/<i>P</i> such that the Godbillon-Vey-Losik class is represented by a volume form on a space of dimension <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2n+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and the first Chern-Losik class is represented by a symplectic form on a space of dimension 2<i>n</i>. Examples in dimension 2 are considered.</p>

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On Losik Classes of Diffeomorphism Pseudogroups

  • 期刊论文

  • 发表日期：

    2025年11月12日

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  • Yaroslav V. Bazaikin,
  • Yury D. Efremenko,
  • Anton S. Galaev

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 \(2n+1\) 2 n + 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.