Classification of homogeneous almost complex $4$-manifolds with non-degenerate torsion bundle

We investigate the local and global geometry of almost complex $4$-manifolds admitting non-degenerate torsion bundle.

The rigidity of these structures forces a parallelizable $J$-adapted double cover, which imposes severe topological constraints on the underlying manifold.

Exploiting this rigidity, we give a complete classification in the homogeneous setting.

We show that such a manifold is diffeomorphic either to a $4$-dimensional Lie group carrying an almost complex structure with non-degenerate torsion bundle, or to a product $L(4,1)\times\mathbb R$ or $L(4,1)\times\mathbb T$, where $L(4,1)$ is a lens space.

We also determine exactly which real $4$-dimensional Lie algebras admit such a structure.

Constructively, we realize every admissible algebra by an explicit invariant structure, thereby closing the existence question in dimension $4$.

We also relate these structures to certain Engel structures that we call Nijenhuis--Engel, and answer the resulting existence questions in the homogeneous case.