The handlebody conjecture for self-affine 3-manifolds

A self-affine 3-manifold is a self-affine tile in $\mathbb{R}^3$ that is a 3-manifold. A 3-dimensional handlebody is an orientable 3-manifold with boundary containing pairwise disjoint, tamely embedded disks such that cutting along these disks produces a 3-ball.

The handlebody conjecture. Every self-affine 3-manifold is homeomorphic to a handlebody.

This is the three-dimensional case of the general handlebody conjecture. The paper develops effective recognition methods for self-affine 3-manifolds and shows that every 3-dimensional handlebody can be realized as a self-affine manifold, while the converse remains open.