Counterexample to the Borel conjecture in dimension four
An OpenAI manuscript constructs two closed aspherical topological four-manifolds that are homotopy equivalent but not homeomorphic, which would disprove the four-dimensional case of the Borel conjecture even in the weak form Borel first raised in a 1953 letter to Serre.
- Lab
- OpenAI
- Model
- Unreleased internal OpenAI model, not named in the release
- Field
- Mathematics
- Date
- 2026-10-06
- Problem posed
- 1953 · open 73 yrs
Sources
Original work
Announcement
What was found
The Borel conjecture asks that homotopy equivalences between closed aspherical manifolds be homotopic to homeomorphisms; the weaker homeomorphism-existence form asks only that homotopy-equivalent ones be homeomorphic. It holds in dimensions up to three, and in dimensions five and above it is known for large classes of fundamental groups, including word-hyperbolic groups by Bartels and Lück. The manuscript gives a pair , with a common word-hyperbolic fundamental group, plus a self-homotopy equivalence of not homotopic to any homeomorphism. The construction applies Davis's reflection trick to a four-dimensional Poincaré chamber whose cyclic covers have a manifold filling in degree two but none in odd degree, glues the degree-two filling in two different ways, and separates the results by showing the outer automorphism group of the fundamental group is finite and computing a normalizer inside it. It is a topological statement, distinct from the 2026 smooth examples of Davis, Hayden, Huang, Ruberman and Sunukjian, which are homeomorphic but not diffeomorphic.
Novelty check
Read the introduction and bibliography at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb on 2026-10-12. The history it gives matches the standard account: Hsiang and Wall and Farrell and Jones in high dimensions, Bartels and Lück's theorem with a good-fundamental-group hypothesis in dimension four, and Khan's rigidity up to topological s-cobordism for torsion-free hyperbolic groups. No earlier counterexample in dimension four was found in the registry or in that literature. The family is a disproof, not a proof: the manuscript map and the paper both say so.
Caveats and known objections
The argument rests on two companion manuscripts in the same release, 'A marked tensor obstruction to four-dimensional disk embedding' and 'A PD4 group without an aspherical manifold model', which claim respectively that the unrestricted four-dimensional disk-embedding conjecture fails (so that the free group on two generators is not good in Freedman and Quinn's sense) and that Wall's conjecture fails. Either would be a major result in four-manifold topology on its own, neither has been checked outside OpenAI, and this counterexample inherits whatever they contain. No Lean formalization accompanies the manuscript or its companions. Nobody outside OpenAI is on record as having read it. Autonomy is graded ai-led rather than autonomous because humans chose the problems and filtered the output; the README places this family under its single-agent procedure, an account that rests on OpenAI alone.
The open problem
316143
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Also recorded at
vibemathedborel-conjecture-dimension-four
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Nobody outside the lab has checked this yet.
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History of this entry
- CorrectedRecorded the vibemathed record for this result and the MathDB record for the conjecture as registrations. No grade moved.
- AddedEntered the registry graded Claimed and AI-led.
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Community discussion
Graded claimed for verification and ai-led for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Counterexample to the Borel conjecture in dimension four. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-10-06-borel-conjecture-four-manifoldsBibTeX
@misc{whataifound-openai-2026-manifolds,
title = {Counterexample to the Borel conjecture in dimension four},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by OpenAI. Verification: Claimed. Autonomy: AI-led.},
url = {https://whataifound.org/finding/2026-10-06-borel-conjecture-four-manifolds}
}