Graph-isomorphism lower bound for Heine-Borel simplicial complexes

Determine whether isomorphism of countable graphs is Borel reducible to PL-homeomorphism on Heine-Borel simplicial complexes.

Background

The paper proves an upper bound for PL-homeomorphism of Heine-Borel simplicial complexes by reducing it to isomorphism of countable sorted complemented algebras. It does not prove the converse reduction from countable graph isomorphism, so the exact Borel complexity of this classification relation remains unresolved. This question isolates the lower-bound problem independently of manifold dimension.

References

Is the isomorphism on countable graphs Borel reducible to the PL-homeomorphism on Heine-Borel simplicial complexes?

— 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,” third Question