Cohomology with local coefficients of solvmanifolds and Morse-Novikov theory

Abstract

We study the cohomology $H^*_{λω}(G/Γ, {\mathbb C})$ of the deRham complex $Λ^*(G/Γ)\otimes{\mathbb C}$ of a compact solvmanifold $G/Γ$ with a deformed differential $d_{λω}=d + λω$, where $ω$ is a closed 1-form. This cohomology naturally arises in the Morse-Novikov theory. We show that for a solvable Lie group $G$ with a completely solvable Lie algebra $\mathfrak{g}$ and a cocompact lattice $Γ\subset G$ the cohomology $H^*_{λω}(G/Γ, {\mathbb C})$ coincides with the cohomology $H^*_{λω}(\mathfrak{g})$ of the Lie algebra $\mathfrak{g}$ associated with the one-dimensional representation $ρ_{λω}: \mathfrak{g} \to {\mathbb K}, ρ_{λω}(ξ) = λω(ξ)$. Moreover $H^*_{λω}(G/Γ, {\mathbb C})$ is non-trivial if and only if $-λ[ω]$ belongs to the finite subset $\{0\} \cup \tilde Ω_{\mathfrak{g}}$ in $H^1(G/Γ, {\mathbb C})$ well defined in terms of $\mathfrak{g}$.