View source for The Borel conjecture in dimension four

The '''Borel conjecture''' predicts that a closed aspherical manifold is determined up to homeomorphism by its fundamental group: homotopy-equivalent closed aspherical manifolds should be homeomorphic, and, in the stronger common formulation, every homotopy equivalence between them should be homotopic to a homeomorphism. The question goes back to Borel's letter to Serre of 1953. A manuscript in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 4 October 2026, claims counterexamples to both formulations in dimension four, with a word-hyperbolic fundamental group, in the topological category. The family has no Lean formalization in the repository's catalogue. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.

== Background ==
* As recounted in the introduction of the manuscript:
** Rigidity holds in dimensions at most two, and for closed orientable aspherical three-manifolds by geometrization. In dimensions at least five, Hsiang and Wall proved it for homotopy tori, Farrell and Jones for closed negatively curved manifolds, and Bartels and Lück proved the stronger formulation for a class of groups containing word-hyperbolic groups and groups acting geometrically on finite-dimensional CAT(0) spaces.
** In dimension four these rigidity theorems need further input: the four-dimensional application of Bartels and Lück assumes that the fundamental group is good in the sense of Freedman, and Khan proved rigidity up to topological ''s''-cobordism for torsion-free word-hyperbolic fundamental groups.
** Davis, Hayden, Huang, Ruberman and Sunukjian recently gave closed aspherical smooth four-manifolds that are homeomorphic but not diffeomorphic; the manuscript notes that its own conclusion concerns homeomorphism in the topological category.

== The original result ==
* '''Claim.''' Theorem 1.1 of ''Nonhomeomorphic closed aspherical four-manifolds with the same homotopy type'' states that there exist closed connected aspherical topological four-manifolds ''M'' and ''N'' that are homotopy equivalent but not homeomorphic, with a common word-hyperbolic fundamental group. Theorem 1.2 states that the same ''M'' has a self-homotopy equivalence that is not homotopic to any homeomorphism. Together they claim counterexamples to the homeomorphism-existence formulation and to the homotopy-class formulation of the Borel conjecture in dimension four.
* ''Method, as described:'' the manuscript builds a four-dimensional Poincaré chamber whose boundary admits a marked filling in its double cyclic cover but in no odd cyclic cover of degree at least three, the obstruction being a parity argument for assignments of red and blue plumbing caps around a cycle. Davis's reflection construction turns fillings into closed aspherical models, and a cyclic-cover criterion, which composes a lifted homeomorphism with a deck translation and needs no finite-order assumption, rules out homeomorphisms realizing a given self-homotopy equivalence. The two manifolds are distinguished by which symmetries homeomorphisms can realize, through a Sylow argument in a finite group.
* '''Dependence on other manuscripts.''' The manuscript states that it imports the marked algebraicity theorem of ''A marked tensor obstruction to four-dimensional disk embedding'' and the controlled filling implication and support construction of ''A PD4 group without an aspherical manifold model'', two manuscripts of the same collection grouped in its result 305 on four-dimensional disk embedding and Wall's conjecture.
* '''Source''' (openai/math repository, commit <code>7ef5c0c</code>): [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/Nonhomeomorphic-closed-aspherical-four-manifolds-with-the-same-homotopy-type-October-4-2026 Nonhomeomorphic closed aspherical four-manifolds with the same homotopy type], dated 4 October 2026.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 30 October 2026)
|-
| 2026-10-07 || [https://lowrank.github.io/latent-seminar/OpenAI-Math/320-borel-four-manifolds-cyclic-covers-realized-symmetries ''Borel rigidity in dimension four: cyclic covers and realized symmetries''], an article of the Latent Seminar blog at lowrank.github.io || Reading (exposition) || Explains homotopy type and manifold structure, the chamber with a parity obstruction, how reflection converts fillings into closed models, the cyclic-cover criterion and the two models with different realized symmetries, with proof outlines on the even filling, the exact capacity count behind the odd-cover contradiction, the manifold quotient from bounded displacement and the distinction of unmarked manifolds. || Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs.
|-
| 2026-10-16
| J. Jardine, Lattice24, referee review [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_282.json rank_282] in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| Reports no gap. Its one minor note calls the reliance on the two companion manuscripts a load-bearing dependency rather than a demonstrated gap: the step from a closed four-manifold model of the reflection to a marked filling is extracted from the proof of one companion rather than from a stated theorem, the odd-cover argument imports results of the other unchanged, and together these imply results far beyond the previous state of the art, such as that free groups are not good, so that the paper stands or falls with them.
| Part of an audit that its README describes as complete for the collection's manuscripts without a Lean formalization. The review is published under CC BY 4.0.
|-
| 2026-10-18 || ''AI Proved What?'', episode 320 [https://aiprovedwhat.com/episodes/320-borel-four/ Same group, not the same?], remade version of 23 October 2026 || Reading (exposition) || Video explainer of the claim and its history. The episode states that nothing smooth, no other dimension, no explicit manifold and no invariant such as an Euler characteristic is claimed, and that the paper stands or falls with two companion papers of result 305. It adds, as its own reading rather than the paper's claim, that with a 2012 theorem of Khan the two manifolds would also be ''s''-cobordant but not homeomorphic. || Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, 60 pages, dated October 4, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: no formalization for this item. The page states that it found no specialist assessment, no reported error and no competing claim.
|}

== Notes ==
* The background summarizes the manuscript's own account of earlier work.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]