HIGHER COHOMOLOGY FOR ABELIAN-GROUPS OF TORAL AUTOMORPHISMS

We give a complete description of smooth untwisted cohomology with coefficients in R(e) for Z(k)-actions by hyperbolic automorphisms of a torus. For 1 less than or equal to n less than or equal to k - 1 the nth cohomology trivializes, i.e. every cocycIe 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.