Cohomology with local coefficients of solvmanifolds and Morse-Novikov theory

Abstract

We study the cohomology $H^*_{\lambda \omega}(G/\Gamma, {\mathbb C})$ of the deRham complex $\Lambda^*(G/\Gamma)\otimes{\mathbb C}$ of a compact solvmanifold $G/\Gamma$ with a deformed differential $d_{\lambda \omega}=d + \lambda\omega$, where $\omega$ 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 $\Gamma \subset G$ the cohomology $H^*_{\lambda \omega}(G/\Gamma, {\mathbb C})$ coincides with the cohomology $H^*_{\lambda \omega}(\mathfrak{g})$ of the Lie algebra $\mathfrak{g}$ associated with the one-dimensional representation $\rho_{\lambda \omega}: \mathfrak{g} \to {\mathbb K}, \rho_{\lambda \omega}(\xi) = \lambda \omega(\xi)$. Moreover $H^*_{\lambda \omega}(G/\Gamma, {\mathbb C})$ is non-trivial if and only if $-\lambda [\omega]$ belongs to the finite subset $\{0\} \cup \tilde \Omega_{\mathfrak{g}}$ in $H^1(G/\Gamma, {\mathbb C})$ well defined in terms of $\mathfrak{g}$.