Borel parametrization of embedded 3-manifolds

Determine whether the set of closed subsets of \(\mathbb{R}^N\) that are 3-manifolds is Borel, for a sufficiently large Euclidean dimension \(N\).

Background

The paper parametrizes manifolds by countable atlases because it is not known that the more natural coding by closed subsets of a universal or Euclidean space produces a Borel class. A positive answer would provide a direct standard Borel parametrization of embedded 3-manifolds, rather than one mediated by atlases. The authors explain that a Borel substitute containing representatives of every 3-manifold up to homeomorphism can nevertheless be constructed.

References

It is not clear to the authors whether {F\in F(\mathbb{R}N)\mid F\text{ is a 3-manifold}} is a Borel set, but it is possible to find M_3\subset F(\mathbb{R}N) such that every 3-manifold is represented in M_3 up to homeomorphism.

— Borel classification of simplicial complexes and non-compact $2$- and $3$-manifolds  (2608.16400 - Iannella et al., 17 Aug 2026) in Section 6, subsection “Manifolds given not as an atlas”