[Paper Review] On the de Rham cohomology of solvmanifolds

Sergio Console, Anna Fino|ArXiv.org|Dec 10, 2009

Geometry and complex manifolds13 references3 citations

TL;DR

This paper provides a new proof of D. Guan's result on the de Rham cohomology of compact solvmanifolds using D. Witte's superrigidity theorems on lattices in solvable Lie groups. It establishes that for any compact solvmanifold $G/\Gamma$, there exists a finite-index subgroup $\tilde{\Gamma}$ and a diffeomorphic solvmanifold $\tilde{G}/\tilde{\Gamma}$ satisfying the Mostow condition, enabling computation of de Rham cohomology via Lie algebra cohomology even when the original $G$ and $\Gamma$ do not satisfy the condition.

ABSTRACT

By using results by D. Witte on the superigidity of lattices in solvable Lie groups we get a different proof of a recent remarkable result obtained by D. Guan on the de Rham cohomology of a compact solvmanifold, i.e. of a quotient of a connected and simply connected solvable Lie group $G$ by a lattice $Γ$. This result can be applied to compute the Betti numbers of a compact solvmanifold $G/Γ$ even in the case that the solvable Lie group $G$ and the lattice $Γ$ do not satisfy the Mostow condition.

Motivation & Objective

• To provide an alternative proof of Guan's result on the de Rham cohomology of compact solvmanifolds using superrigidity theorems.
• To establish the existence of a finite-index subgroup $\tilde{\Gamma}$ and a diffeomorphic solvmanifold $\tilde{G}/\tilde{\Gamma}$ satisfying the Mostow condition.
• To enable the computation of de Rham cohomology $H^{*}_{dR}(G/\Gamma)$ via Lie algebra cohomology $H^{*}(\mathfrak{g})$ even when $G$ and $\Gamma$ do not satisfy the Mostow condition.
• To extend the applicability of cohomological computations to solvmanifolds in the non-Mostow case, particularly in the context of almost abelian Lie groups.

Proposed method

• Utilize D. Witte's superrigidity result on lattices in solvable Lie groups to refine the Malcev splitting construction.
• Construct a new simply connected solvable Lie group $\tilde{G}$ as a semidirect product $T_{cpt} \ltimes G$, where $T_{cpt}$ is a compact torus.
• Identify a finite-index subgroup $\tilde{\Gamma} \subset \Gamma$ such that $\mathcal{A}(Ad_{\tilde{G}}(\tilde{G})) = \mathcal{A}(Ad_{\tilde{G}}(\tilde{\Gamma}))$.
• Apply the Borel density theorem to relate the algebraic closures of adjoint images of $G$ and $\Gamma$ via maximal compact tori.
• Use the fact that $H^{*}_{dR}(G/\tilde{\Gamma}) \cong H^{*}(\tilde{\mathfrak{g}})$ when the Mostow condition holds.
• Derive $H^{*}_{dR}(G/\Gamma)$ as the invariants under the finite group $\Gamma/\tilde{\Gamma}$ acting on $H^{*}_{dR}(G/\tilde{\Gamma})$.

Experimental results

Research questions

• RQ1 Can the de Rham cohomology of a compact solvmanifold $G/\Gamma$ be computed when $G$ and $\Gamma$ do not satisfy the Mostow condition?
• RQ2 Is there a finite covering $\tilde{G}/\tilde{\Gamma}$ of $G/\Gamma$ such that the Mostow condition holds for $\tilde{G}$ and $\tilde{\Gamma}$?
• RQ3 Can Witte's superrigidity theorems be used to provide a new proof of Guan's result on solvmanifold cohomology?
• RQ4 What conditions on the eigenvalues of the adjoint action ensure that $\mathcal{A}(Ad_G(G)) \neq \mathcal{A}(Ad_G(\Gamma))$?
• RQ5 How does the cohomology of a solvmanifold relate to the cohomology of its finite covers when the Mostow condition fails?

Key findings

• For any compact solvmanifold $G/\Gamma$, there exists a finite-index subgroup $\tilde{\Gamma} \subset \Gamma$ and a simply connected solvable Lie group $\tilde{G}$ such that $\mathcal{A}(Ad_{\tilde{G}}(\tilde{G})) = \mathcal{A}(Ad_{\tilde{G}}(\tilde{\Gamma}))$.
• The de Rham cohomology of $G/\Gamma$ is isomorphic to the invariants of $H^{*}_{dR}(G/\tilde{\Gamma})$ under the action of the finite group $\Gamma/\tilde{\Gamma}$.
• The cohomology $H^{*}_{dR}(G/\tilde{\Gamma})$ is isomorphic to the Lie algebra cohomology $H^{*}(\tilde{\mathfrak{g}})$, enabling explicit computation.
• In the case where $i\pi$ is not a rational linear combination of the eigenvalues $\lambda_k$ of the derivation $Z$, the Mostow condition holds and $H^{*}_{dR}(G/\Gamma) \cong H^{*}(\mathfrak{g})$.
• For the hyperelliptic surface example with $\eta = \pi$, $H^{1}_{dR}(G/\Gamma) \cong \text{span}\langle e^3, e^4 \rangle$, matching $H^{1}(\mathfrak{g})$ despite failure of the Mostow condition.
• The method applies to almost abelian Lie groups, extending known results on the first Betti number and cohomology computation.