On Low-Dimensional Solvmanifolds
Christoph Bock
TL;DR
This work analyzes low-dimensional solvmanifolds arising as quotients $G/ ablaamma$ of connected, simply connected solvable Lie groups by lattices, focusing on lattice existence up to dimension six and on formality, symplectic structure, Lefschetz properties, and Kählerity. It leverages minimal model theory and the Chevalley–Eilenberg complex to compute de Rham cohomology via left-invariant forms, and employs Auslander–semisimple splitting and the nilradical structure to understand lattice criteria. The paper provides a detailed classification of solvmanifolds in dimensions 3–6 with respect to formality and Lefschetz-type properties, supplies explicit lattice constructions for many indecomposable algebras (notably almost abelian cases), and presents concrete examples demonstrating that formality, Lefschetz, and Kähler properties can be independent. It highlights the rich interplay between algebraic structure (nilradical, semisimple splitting) and geometric features (symplecticity, contact structures) in solvmanifolds, offering a catalog of new explicit solvmanifolds with varied topological and geometric behaviors.
Abstract
A nilmanifold resp. solvmanifold is a compact homogeneous space of a connected and simply-connected nilpotent resp. solvable Lie group by a lattice, i.e. a discrete co-compact subgroup. There is an easy criterion for nilpotent Lie groups which enables one to decide whether there is a lattice or not. Moreover, it is easy to decide whether a nilmanifold is formal, Kaehlerian or (Hard) Lefschetz. The study of solvmanifolds meets with noticeably greater obstacles than the study of nilmanifolds. Even the construction of solvmanifolds is considerably more difficult than is the case for nilmanifolds. The reason is that there is no simple criterion for the existence of a lattice in a connected and simply-connected solvable Lie group. We consider the question of existence of lattices in solvable Lie groups up to dimension six and examine whether the corresponding solvmanifolds are formal, symplectic, Kaehler or (Hard) Lefschetz.