Cohomology computations of solvmanifolds with local coefficients

TL;DR

This paper develops an explicit finite-dimensional cochain complex to compute the cohomology of solvmanifolds with local coefficients, extending previous methods by incorporating twisted representations.

Contribution

It introduces a novel explicit cochain complex for cohomology with local coefficients on solvmanifolds, generalizing Lie algebra cohomology techniques.

Findings

Constructed a finite-dimensional cochain complex for cohomology with local coefficients.

Proved the isomorphism between the complex's cohomology and the manifold's cohomology.

Extended cohomology computation methods to include twisted representations.

Abstract

Let $G$ be a simply connected solvable Lie group with a lattice $\Gamma$ and $N$ the nilradical of $G$. For a complex valued representation $\rho: G\to GL(V_{\rho})$ such that the restriction $\rho_{|_{N}}$ is unipotent, as an advanced variation of cohomology computation of solvmanifolds by using Lie algebra cohomology, we construct a explicit finite dimensional cochain complex whose cohomology is isomorphic to the cohomology of $G/\Gamma$ with twisted by $\rho$.