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# Mathematical Research Letters

## Volume 21 (2014)

### Number 4

### de Rham and Dolbeault cohomology of solvmanifolds with local systems

Pages: 781 – 805

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n4.a10

#### Author

#### Abstract

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.