Purely coclosed $\mathrm{G}_2$-structures on nilmanifolds -- II

TL;DR

The paper completes the classification of 7-dimensional nilpotent Lie algebras admitting left-invariant purely coclosed $\mathrm{G}_2$-structures in the indecomposable 5- and 6-step cases, building on prior work for decomposable and lower-step cases. It relies on a central SU(3)-data construction to produce $\mathrm{G}_2$-forms, applies a set of obstructions (including a new one) to rule out nonexistence, and provides explicit purely coclosed structures for the algebras that admit them. The authors enumerate the 5-step indecomposable algebras that do and do not admit purely coclosed structures, and show that all 6-step indecomposables do, supplementing with explicit forms in tables. A broader conclusion ties coclosed and purely coclosed notions in decomposable and indecomposable regimes, yielding infinite families of purely coclosed examples and clarifying the landscape of real homotopy types for these geometries. The work also provides SageMath worksheets to verify the constructions and obstructions, facilitating further exploration and potential generalizations.

Abstract

This paper completes the classification of seven-dimensional nilpotent Lie groups endowed with a left-invariant purely coclosed $\text{G}_2$-structure, initiated by the first-named author and collaborators. In this previous work, the authors provided the classification of decomposable seven-dimensional nilpotent Lie groups and of the indecomposable ones up to step $4$ of nilpotency. Here, we address the case of indecomposable $5$- and $6$-step nilpotent Lie groups.

Theorems & Definitions (9)

• Theorem
• Theorem 3.1
• Remark 4.1
• Proposition 4.2: Fourth obstruction
• Theorem 5.1
• Theorem 5.2
• Theorem 6.1
• Remark 6.2
• Remark 6.3