Cohomology of solvable Lie algebras and solvmanifolds (Q2387826)

The author studies the de Rham complex of a compact solvmanifold \(M\) with a deformed differential \(d + \omega\) where \(\omega\) is a closed one-form. The cohomology of such a complex naturally appears in the Morse-Novikov theory, i.e. the analog of the classical Morse theory for smooth closed one-forms. It is proved that such a cohomology ring is isomorphic to the cohomology ring of the corresponding solvable Lie algebra with coefficients in the one-dimensional representation \(\rho_\omega(\xi) = \omega(\xi)\) of this algebra. Moreover it is established that this cohomology ring is nontrivial if and only if \(\omega\) belongs to some finite subset of \(H^1(M;\mathbb C)\) defined in terms of the Lie algebra.