Cohomology of solvable lie algebras and solvmanifolds

The cohomology H*(lambdaomega) (G/Gamma, C) of the de Rham complex Lambda* (G/Gamma) circle times C of a compact solvmanifold G/Gamma with deformed differential d(lambdaomega) = d + lambdaomega, where omega is a closed 1-form, is studied. Such cohomologies naturally arise in Morse-Novikov theory. It is shown that, for any, completely solvable Lie group G containing a cocompact lattice Gamma subset of G, the cohomology H-lambdaomega* (G/Gamma, C) is isomorphic to the cohomology H-lambdaomega* (g) of the tangent Lie algebra g of A the group G with coefficients in the one-dimensional representation Plambdaomega:g--> K defined by P-lambdaomega(xi) = lambdaomega(xi). Moreover, the cohomology H-lambdaomega*,(G/Gamma, C) is nontrivial if and only if -lambda[omega][w] A belongs to a finite subset Omega(g) of H-1 (G/Gamma, C) defined in terms of the Lie algebra g.