Cohomology with local coefficients of solvmanifolds and Morse-Novikov theory

Abstract

We study the cohomology $H_{\lambda \omega }^{\ast }(G/\Gamma ,\mathbb{C})$ of the deRham complex ${\Lambda }^{\ast }(G/\Gamma )\otimes \mathbb{C}$ 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 $H_{\lambda \omega }^{\ast }(G/\Gamma ,\mathbb{C})$ coincides with the cohomology $H_{\lambda \omega }^{\ast }(\mathfrak{g})$ 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 $H_{\lambda \omega }^{\ast }(G/\Gamma ,\mathbb{C})$ 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 $\mathfrak{g}$.

11 pages, Latex