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The Borel conjecture for closed aspherical four-manifolds (homeomorphism form)

A connected manifold is aspherical if its universal cover is contractible, so its fundamental group determines its homotopy type. In a letter to Serre of May 2, 1953, Borel asked whether compact manifolds that are classifying spaces for the same group are homeomorphic. The topological Borel conjecture asks, in every dimension, that every homotopy equivalence between closed aspherical manifolds be homotopic to a homeomorphism; the weaker homeomorphism-existence form asks only that homotopy-equivalent closed aspherical manifolds be homeomorphic. It holds in dimensions at most three, and in dimensions at least five for large classes of groups (Farrell-Jones, Bartels-Lück, including hyperbolic and CAT(0) groups). In dimension four it was known only with extra hypotheses such as a good fundamental group in Freedman's sense. Are homotopy-equivalent closed aspherical four-manifolds always homeomorphic?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Topology of manifolds; aspherical four-manifolds
Posed by
Armand Borel
Year posed
1953
Years open
73y
Solved
2026-10-04
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
60 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there are closed connected aspherical topological four-manifolds MM and NN, with the same word-hyperbolic fundamental group, that are homotopy equivalent but not homeomorphic. Theorem 1.2: some such MM has a self-homotopy equivalence not homotopic to any homeomorphism, so the prescribed-class form also fails. The examples come from Davis reflection applied to a Poincaré chamber whose odd cyclic covers have no manifold model. This disproves the Borel conjecture as an all-dimension statement and in dimension four specifically. Not shown: anything in dimensions five or higher, or smooth-category statements; it also shows that Khan's rigidity up to s-cobordism cannot be upgraded to homeomorphism for such groups (curator's reading of the introduction).

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript depends on two other release manuscripts, A marked tensor obstruction to four-dimensional disk embedding and A PD4 group without an aspherical manifold model (both September 24, 2026, family 305).

Verification

No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 were read against Borel's question and the homeomorphism-existence formulation; the paper claims a counterexample in dimension four with word-hyperbolic fundamental group. The proof was not refereed. No Lean formalization exists for this family (no lean/docs/320.md at the pinned commit, nothing in lean/formalization.yaml). The proof imports results from two unreviewed manuscripts of the same release (the marked tensor obstruction and the PD4 group without manifold model, family 305), whose own claims include that the free group is not good in Freedman's sense; this entry stands or falls with them. The result is in the topological category; it does not touch the conjecture in dimensions five and higher.

Sources

Changelog1 change

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