Borel classification of simplicial complexes and non-compact $2$- and $3$-manifolds

Abstract: We generalize the Stone space of ultrafilters on Boolean algebras and prove a generalization of Stone duality which is applicable to locally compact Polish spaces. Using this, we obtain complete invariants for simplicial complexes up to PL-homeomorphism and for non-compact $2$- and $3$-manifolds up to homeomorphism. We prove that the homeomorphism relation on non-compact $2$-manifolds without boundary, the homeomorphism relation on non-compact $3$-manifolds with or without boundary, the homeomorphism relation on open subsets of $\mathbb{R}<sup>2$ and $\mathbb{R}<sup>3$, and conjugacy of Cantor sets in $\mathbb{R}<sup>3$ are classifiable by countable structures. Together with known lower bounds, this implies that these relations are Borel bireducible with isomorphism of countable graphs. We also show that PL-homeomorphism of Heine-Borel simplicial complexes and PL-homeomorphism of PL $n$-manifolds, for every $n$, are classifiable by countable structures.

• The paper establishes complete algebraic invariants for simplicial complexes and non-compact 2- and 3-manifolds, showing their homeomorphism relations are Borel bireducible with countable graph isomorphism.
• The paper develops blurry duality using sorted complemented algebras and rational polyhedral bases, converting locally compact polyhedral spaces into effective countable invariants based on compact rational polyhedra.
• The paper proves Borel triangulation results for 3-manifolds and applies the framework to PL manifolds, open subsets of Euclidean space, and conjugacy of Cantor sets in R³, while identifying higher-dimensional limitations.

This paper establishes complete algebraic invariants for simplicial complexes up to PL-homeomorphism and for non-compact 2- and 3-manifolds up to homeomorphism, and uses them to settle the descriptive-set-theoretic complexity of the corresponding classification problems (2608.16400). The central technical contribution is a generalization of Stone duality — termed "blurry duality" — which replaces Boolean algebras of clopen sets with sorted complemented algebras (SCAs) built from polyhedral bases. The main theorem states that the homeomorphism relations on non-compact 2-manifolds without boundary and on non-compact 3-manifolds with or without boundary are classifiable by countable structures; combined with known lower bounds, these relations are Borel bireducible with isomorphism of countable graphs.

Main results

Denote by $\cong_2$, $\cong_3$, and $\cong_3^\partial$ the homeomorphism relations on non-compact 2-manifolds without boundary, non-compact 3-manifolds without boundary, and non-compact 3-manifolds with boundary, respectively. The paper proves:

• $\cong_2$, $\cong_3$, and $\cong_3^\partial$ are classifiable by countable structures, hence Borel bireducible with graph isomorphism.
• Simplicial complexes up to PL-homeomorphism are classifiable by countable structures, as are PL $n$-manifolds in every dimension $n$ (both Heine-Borel complexes and PL manifolds presented via locally PL-compatible atlases).
• Homeomorphism of open subsets of $\mathbb{R}^2$ and $\mathbb{R}^3$ is classifiable by countable structures.
• Conjugacy of Cantor sets in $\cong_3$0 reduces to isomorphism of countable structures, answering Question 5.5 of Gartside–Kovan-Bakan.

A notable consequence is that, despite the sharp contrast between the classical classification theories in dimensions 2 and 3 — Goldman's 1971 algebraic classification of open surfaces versus the absence of any comparable theory for open 3-manifolds, where Whitehead-type examples obstruct geometric decompositions — the corresponding homeomorphism relations have identical Borel complexity. Moreover, since these relations sit at the level of graph isomorphism, they are strictly simpler than homeomorphism of compact Polish spaces or equivalence of wild knots, both of which lie strictly above graph isomorphism.

Blurry duality

Classical Stone duality reconstructs a compact totally disconnected space from its Boolean algebra of clopen sets via ultrafilters. The paper adapts this to locally compact Polish spaces carrying a polyhedral basis, where two obstructions arise: a homeomorphism need not preserve a chosen countable basis, and the Stone space of an ultrafilter construction is always totally disconnected.

The first obstruction is handled by restricting attention to rational polyhedra. A complex $\cong_3$1 has a canonical dense set $\cong_3$2 of points with rational barycentric coordinates, and a subdivision is a $\cong_3$3-subdivision if all transition maps carry rational points to rational points. The key structural result is that PL-homeomorphic complexes are always $\cong_3$4-PL-homeomorphic: given any continuous function $\cong_3$5, a PL-homeomorphism can be approximated within $\cong_3$6 by one that preserves all rational polyhedra. This relies on iterated stellar subdivisions and a common-subdivision theorem for infinite triangulations from the authors' companion work.

The second obstruction leads to the definition of sorted complemented algebras: partial orders $\cong_3$7 satisfying complement-like axioms but not necessarily admitting joins or meets, equipped with a distinguished downward-closed sort $\cong_3$8 of "compact" elements. Filters are generalized to blurry filters, which drop the ultrafilter requirement that $\cong_3$9 or $\cong_3^\partial$0 belong to the filter, replacing it with a weaker condition. For a countable complemented locally compact Polish (CCLCP) basis space $\cong_3^\partial$1, the map $\cong_3^\partial$2 produces an SCA whose elements index basic open sets, ordered by containment of closures. The blurry duality theorem shows that $\cong_3^\partial$3 is equivalent, as a basis space, to the space of blurry $\cong_3^\partial$4-filters on $\cong_3^\partial$5; consequently two CCLCP basis spaces are equivalent exactly when their associated SCAs are isomorphic. In particular, the isomorphism type of the restricted order $\cong_3^\partial$6 already determines that of $\cong_3^\partial$7 — so the invariant can be simplified to a bare partial order of compact rational polyhedra ordered by closure-containment-in-interior.

From invariants to Borel reductions

The purely algebraic classification is then made effective. The paper constructs standard Borel spaces of simplicial complexes (as sequences of embeddings of standard simplexes into the Urysohn space), of manifolds (via atlases satisfying local-finiteness and Heine-Borel conditions), and of SCAs, and verifies that the invariant assignment is Borel throughout. Two technically demanding ingredients deserve emphasis.

First, the space of PL-embeddings between finite complexes is shown to be $\cong_3^\partial$8 inside the space of all embeddings, by writing it as a countable union of compact sets of bilipschitz simplicial maps over stellar subdivisions; this makes the relation "$\cong_3^\partial$9 is PL" Borel when quantifying over atlases.

Second, and described by the authors as the hardest part, is a Borel triangulation theorem for 3-manifolds: there is a Borel map assigning to each atlas-coded 3-manifold a genuine triangulation. The proof adapts Moise–Bing's triangulation arguments, converting existence statements into selections of Borel functions via the Kechris countable-section selection theorem. A Borel version of Bing's extension lemma produces, for each chart, a $\cong_2$0-PL approximation of the transition data that extends previously constructed pieces, and an exhaustion by compact submanifolds with controlled overlaps ensures the gluing converges. For 2-manifolds without boundary, the authors instead import the Borel triangulation of Bergfalk–Smythe and translate between their parametrization and the atlas framework. For 3-manifolds with boundary, the boundary is triangulated as a surface, extended over a collar, and then over the whole manifold.

Once triangulations are available in a Borel way, the reduction chain is short: manifold → compatible triangulation → SCA → countable structure, each step Borel, with correctness following from the Moise–Bing Hauptvermutung for dimensions 2 and 3.

Applications and lower bounds

Two further classes are handled by reduction. Open subsets of $\cong_2$1 ($\cong_2$2) admit Borel Whitney-decomposition atlases that are automatically PL-compatible, reducing their homeomorphism relation to the PL case. Cantor set conjugacy in $\cong_2$3 reduces to homeomorphism of complements, using the fact that Cantor sets are locally non-separating, so ambient homeomorphism of complements implies conjugacy.

On the lower-bound side, graph isomorphism reduces to homeomorphism of open subsets of $\cong_2$4 for all $\cong_2$5: one codes a compact zero-dimensional space into a Cantor subset of a line, removes it, and uses extension theorems for homeomorphisms of totally disconnected subsets. Combining upper and lower bounds yields Borel bireducibility with $\cong_2$6 for all the relations listed above, and hence also with isomorphism of torsion-free abelian groups via Paolini–Shelah.

Limitations and open questions

The method depends essentially on uniqueness of PL structure, which fails above dimension 3: there exist homeomorphic but not PL-homeomorphic triangulated non-compact $\cong_2$7-manifolds for $\cong_2$8, so the SCA of a triangulation may distinguish homeomorphic manifolds. Consequently the paper does not determine the complexity of $\cong_2$9 for $\cong_3$0; the matching upper bound is pursued independently by Gompf–Panagiotopoulos by different methods. Two further questions remain open: whether graph isomorphism Borel reduces to PL-homeomorphism of PL $\cong_3$1-manifolds for $\cong_3$2 (the upper bound holds in all dimensions, but the lower bound is unestablished beyond the cases handled here), and the analogous question for Heine-Borel simplicial complexes. The authors also leave open whether the space of closed subsets of $\cong_3$3 that are manifolds is Borel, working around this via atlas parametrizations. Finally, no direct geometric interpretation of the SCA invariant is known; Maillot's observation that open 3-manifolds lack even a conjectural geometric classification remains valid, though the results remove any descriptive-set-theoretic obstruction to such a classification existing.

Conclusion

The paper supplies a uniform Stone-type duality for locally compact polyhedral spaces and derives from it the first classification-by-countable-structures results for non-compact 3-manifolds, together with complete algebraic invariants stated as partial orders of rational compact polyhedra. The equivalence of $\cong_3$4 and $\cong_3$5 with graph isomorphism places manifold classification at the canonical threshold of descriptive complexity, and the Borel triangulation theorem for 3-manifolds is a substantive strengthening of the classical Moise–Bing theory. The natural continuation — extending the upper bound past dimension 3 and interpreting the SCA invariant geometrically — is left as explicit open problems.