Classification complexity of non-compact manifolds in dimensions greater than three

Determine whether the homeomorphism relation on non-compact \(n\)-manifolds is Borel reducible to isomorphism of countable graphs for every \(n>3\).

Background

The paper proves the desired upper bound for non-compact 2- and 3-manifolds, but its method depends on the uniqueness of PL structures in dimensions 2 and 3. In higher dimensions, homeomorphic triangulated manifolds can fail to be PL-homeomorphic, so the triangulation-based invariant is not known to classify topological manifolds. The question asks whether the same descriptive complexity nevertheless holds for all higher-dimensional non-compact manifolds.