de Rham and Dolbeault cohomology of solvmanifolds with local systems

Let $G$ be a simply connected solvable Lie group with a lattice $\Gamma$ and the Lie algebra ${\frak{g}}$ and a representation $\rho : G \to GL(V_{\rho})$ whose restriction on the nilradical is unipotent. Consider the flat bundle $E_{\rho}$ given by $\rho$. By using “many” characters $\{ \alpha \}$ of $G$ and “many” flat line bundles $\{ E \alpha \}$ over $G / \Gamma$, we show that an isomorphism\begin{equation*}\bigoplus_{ \{ \alpha \} } H^{\ast}({\frak{g}}, V_{\alpha} \otimes V_{\rho}) \cong \bigoplus_{ \{ E_{\alpha} \} } H^{\ast}(G / \Gamma, E_{\alpha} \otimes E_{\rho})\end{equation*}holds. This isomorphism is a generalization of the well-known fact: “If $G$ is nilpotent and $\rho$ is unipotent then, the isomorphism $H^{\ast} ({\frak{g}}, V_{\rho}) \cong H^{\ast} (G / \Gamma , E_{\rho})$ holds.” By this result, we construct an explicit finite-dimensional cochain complex which compute the cohomology $H^{\ast} (G / \Gamma , E_{\rho})$ of solvmanifolds even if the isomorphism $H^{\ast} ({\frak{g}}, V_{\rho}) \cong H^{\ast} (G / \Gamma , E_{\rho})$ does not hold. For Dolbeault cohomology of complex parallelizable solvmanifolds, we also prove an analogue of the above isomorphism result which is a generalization of computations of Dolbeault cohomology of complex parallelizable nilmanifolds. By this isomorphism, we construct an explicit finite-dimensional cochain complex which compute the Dolbeault cohomology of complex parallelizable solvmanifolds.

Keywords

de Rham cohomology, local system, Lie algebra cohomology, Dolbeault cohomology, solvmanifold

2010 Mathematics Subject Classification

Primary 17B30, 17B56, 22E25, 53C30. Secondary 32M10, 55N25, 58A12.

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Accepted 27 January 2014

Published 27 October 2014