Graph-isomorphism lower bound for PL manifolds

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

Background

The paper establishes that PL-homeomorphism of PL manifolds is classifiable by countable structures in every dimension, but it does not establish that the relation has the maximal complexity of countable-graph isomorphism. The obstruction is the failure, in dimensions greater than three, of the argument that transfers arbitrary homeomorphisms to PL-homeomorphisms. The problem asks for the missing lower bound.

References

Is the isomorphism on countable graphs Borel reducible to the PL-homeomorphism relation on PL n-manifolds for n>3?

— Borel classification of simplicial complexes and non-compact $2$- and $3$-manifolds  (2608.16400 - Iannella et al., 17 Aug 2026) in Section 7, subsection “Higher dimensional manifolds,” second Question