Cohomology in algebra, geometry, physicsand statistics

Losik classes for codimension one foliations

Following Losik's approach to Gelfand formal geometry, certain
characteristic classes for codimension one foliations coming from
Gelfand-Fuchs cohomology are considered. Sufficient conditions for
non-triviality in terms of the dynamical properties of generators
of the holonomy groups are found. The non-triviality for the Reeb
foliation is shown; this is in contrast with some classical theorems
on the Godbillon-Vey class, e.g, the Mizutani-Morita-Tsuboi Theorem
about triviality of the Godbillon-Vey class of foliations almost
without holonomy is not true for the classes under consideration. It
is shown that the considered classes are trivial for a large class
of foliations without holonomy. The question of triviality is
related to ergodic theory of dynamical systems on the circle and to
the problem of smooth conjugacy of local diffeomorphisms. Certain
classes are obstructions for the existence of transverse affine and
projective connections.

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