Cohomology with local coefficients of solvmanifolds and Morse-Novikov theory

Details Cohomology with local coefficients of solvmanifolds and Morse-Novikov theory Cohomology with local coefficients of solvmanifolds and Morse-Novikov theory

2002-03-07

arxiv.org/abs/math/0203067v1

Mathematics

Differential Geometry

11 pages, Latex

Scientific paper

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 $\mathfrak{g}$.

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