Cohomology of solvable lie algebras and solvmanifolds

Abstract

The cohomology Hλω* (G/Γ,ℂ) of the de Rham complex Λ*(G/Γ) ⊗ ℂ of a compact solvmanifold G/Γ with deformed differential dλω = d + λω, where ω 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 Γ ⊂ G, the cohomology Hλω*(G/Γ, ℂ) is isomorphic to the cohomology Hλω*(\(\mathfrak{g}\)) of the tangent Lie algebra \(\mathfrak{g}\) of the group G with coefficients in the one-dimensional representation ρλω : \(\mathfrak{g}\) → \(\mathbb{K}\) defined by ρλω(ξ) = λω(ξ). Moreover, the cohomology Hλω*(G/Γ,ℂ) is nontrivial if and only if -λ[ω] belongs to a finite subset \(\tilde \Omega _\mathfrak{g} \) of H1(G/Γ,ℂ) defined in terms of the Lie algebra \(\mathfrak{g}\).