Geometric interpretation of the sorted complemented algebra invariants

Determine whether the sorted complemented algebra invariants introduced for non-compact 2- and 3-manifolds admit a direct geometric interpretation.

Background

The paper constructs sorted complemented algebras as complete invariants for non-compact 2- and 3-manifolds, using interiors of compact and co-compact rational polyhedra. Although these algebras provide a successful classification in the descriptive-set-theoretic sense, their relationship to more conventional geometric invariants is unresolved. A direct geometric interpretation would help connect the algebraic classification with the geometric classification of open manifolds and address the longstanding difficulty of describing open 3-manifolds through geometric data.