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Higher cohomology for Abelian groups of toral automorphisms

Published online by Cambridge University Press:  19 September 2008

Anatole Katok
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA (e-mail: katok_a@math.psu.edu)
Svetlana Katok
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA (e-mail: katok_s@math.psu.edu)

Abstract

We give a complete description of smooth untwisted cohomology with coefficients in ℝl for ℤk-actions by hyperbolic automorphisms of a torus. For 1 ≤ nk − 1 the nth cohomology trivializes, i.e. every cocycle is cohomologous to a constant cocycle via a smooth coboundary. For n = k a counterpart of the classical Livshitz Theorem holds: the cohomology class of a smooth k-cocycle is determined by periodic data.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1995

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