View source for The Singer conjecture in dimension four

'''The Singer conjecture''' predicts that the ''L''<sup>2</sup>-Betti numbers of the universal cover of a closed aspherical manifold vanish outside the middle degree. In dimension four it reduces to vanishing in degree one and implies that the Euler characteristic of a closed aspherical four-manifold is nonnegative. A manuscript in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 25 September 2026, claims a proof in dimension four, also for finite aspherical Poincaré complexes. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introduction of the manuscript:
** Atiyah's ''L''<sup>2</sup> index theorem identifies the Euler characteristic of a closed manifold with the alternating sum of the ''L''<sup>2</sup>-Betti numbers of its universal cover, and Dodziuk connected the analytic and combinatorial descriptions. Dodziuk records Singer's proposal to obtain the sign of the Euler characteristic from the vanishing of ''L''<sup>2</sup>-harmonic forms; the manuscript cites the aspherical form of the conjecture from Davis and Okun and from Kirstein, Kremer and Lück. In dimension three the conjecture follows from work of Lott and Lück together with geometrization.
** Earlier four-dimensional cases: right-angled Coxeter groups whose nerves are flag triangulations of the three-sphere (Davis and Okun); groups of type VF with a hierarchy and Coxeter groups whose nerves triangulate the three-sphere (Okun and Schreve); closed aspherical Kähler surfaces, through Gromov's work; closed aspherical complex surfaces with residually finite fundamental group, and those outside class VII<sub>0</sub><sup>+</sup> (Albanese, Di Cerbo and Lombardi); and Gromov–Thurston branched covers (Avramidi, Okun and Schreve).
** The extension to finite Poincaré complexes was formulated in the earlier literature, which the manuscript cites as a remark of Lück.

== The original result ==
* '''Claim.''' Theorem 1.1 of ''The Singer conjecture in dimension four'' states that for every finite connected aspherical integral Poincaré complex ''Q'' of formal dimension four, with arbitrary orientation character, the ''L''<sup>2</sup>-Betti numbers ''b''<sub>''p''</sub><sup>(2)</sup> of its universal cover vanish for every ''p'' ≠ 2. Corollary 1.2 states the same for every closed connected aspherical topological four-manifold, with no orientability, smoothness or geometric structure assumed.
* '''Stated consequences.''' The Euler characteristic of such a complex equals its middle ''L''<sup>2</sup>-Betti number and is therefore nonnegative, and it vanishes exactly when all ''L''<sup>2</sup>-Betti numbers vanish. For an oriented manifold the ''L''<sup>2</sup>-signature theorem also gives χ ≥ |σ|, and normalized Betti numbers along suitable towers of finite covers converge to these limits.
* ''Method, as described in the outline:'' assuming a positive first ''L''<sup>2</sup>-Betti number, a bounded function of finite Dirichlet energy on a Cayley graph and its translates give a compactification of the group whose boundary is a nondegenerate continuum. The group acts on it as a convergence group; the boundary has dimension at most one and controlled first Čech cohomology. A tree of global cut points, a continuum without cut points, component labels with annular depth and a path-pushing argument show local connectedness, which contradicts the earlier properties.
* '''Source''' (openai/math repository, commit <code>7ef5c0c</code>): [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/The-Singer-conjecture-in-dimension-four-September-25-2026 The Singer conjecture in dimension four], dated 25 September 2026, 45 pages.
* '''Formalization.''' The collection lists no Lean scope note for this family.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 29 October 2026)
|-
| 2026-10-07 || [https://lowrank.github.io/latent-seminar/OpenAI-Math/315-singer-four-manifolds-finite-energy-boundary-paths ''The four-dimensional Singer conjecture: finite energy and boundary paths''], an article of the Latent Seminar blog at lowrank.github.io || Reading (exposition) || Explains the reduction to vanishing in degree one, the compact boundary built from finite-energy functions, its dimension and cohomology, the cut-point reduction, annular depth with a threshold independent of the path, and the final continuum contradiction with its consequences. || Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs.
|-
| 2026-10-11 || [https://kokunoyumeto.github.io/open-math-courses-public/courses/continua-arcs-and-cut-points/ ''Continua, arcs and cut points''], a course on Open Mathematics Courses ([https://github.com/kokunoyumeto/open-math-courses-public kokunoyumeto/open-math-courses-public]) || Reading (exposition) || Four lessons with complete proofs on locally connected continua: connected open sets are path connected, every path contains an arc, first Čech cohomology of a compact metric space by finite covers, and the theorem that a Peano continuum of dimension at most one with vanishing first Čech cohomology is a dendrite with cut points, which the course describes as the last step of the manuscript's proof. || The lessons state that they were written by Claude Opus 5.5 (Anthropic) in October 2026 and self-checked by the writing AI; the course lists its status as author self-checked.
|-
| 2026-10-11 || [https://kokunoyumeto.github.io/open-math-courses-public/courses/convergence-groups/ ''Convergence groups''], a course on Open Mathematics Courses ([https://github.com/kokunoyumeto/open-math-courses-public kokunoyumeto/open-math-courses-public]) || Reading (exposition) || Eight lessons on convergence groups on compact metric spaces: the classification of elements, quotients and minimal actions, the dendrite quotient of a continuum and conditions under which it is not a point, the tree of cut points, and the fixed-point property for trees of finitely generated one-ended groups, which the course states these results assume. || The lessons state that they were written by Claude Opus 5.5 (Anthropic) in October 2026 and self-checked by the writing AI; the course lists its status as author self-checked.
|-
| 2026-10-11 || [https://kokunoyumeto.github.io/open-math-courses-public/courses/the-finite-energy-boundary-of-a-group/ ''The finite-energy boundary of a group''], a course on Open Mathematics Courses ([https://github.com/kokunoyumeto/open-math-courses-public kokunoyumeto/open-math-courses-public]) || Reading (exposition) || Ten lessons on the part of the manuscript's proof that takes place on the group: finite-energy functions on the Cayley complex of a finitely presented group, the compact boundary they give a one-ended group, its convergence action, its dimension and first Čech cohomology, a continuum without cut points and the proof that it is locally connected, ending with the statement that under named hypotheses every finite-energy gradient is a limit of gradients of finitely supported functions. || The lessons state that they were written by Claude Opus 5.5 (Anthropic) in October 2026 and self-checked by the writing AI; the course lists its status as author self-checked.
|-
| 2026-10-18 || ''AI Proved What?'', episode 315 [https://aiprovedwhat.com/episodes/315-singer-four/ Never negative?], remade version of 23 October 2026 || Reading (exposition) || Video explainer of the claim that for every closed aspherical topological four-manifold, and more generally every finite aspherical Poincaré complex of formal dimension four, including nonorientable ones, the ''L''<sup>2</sup>-Betti numbers vanish outside degree two, so that the Euler characteristic is never negative and, for oriented manifolds, at least the absolute value of the signature. The episode states that only degree one is hard, and that nothing is claimed about other dimensions, the signature bound for Poincaré complexes, or the middle ''L''<sup>2</sup>-Betti number beyond its equalling the Euler characteristic. || 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, 45 pages, dated September 25, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: no formalization of the theorem, a 235-line file checks part of one lemma. The page also states that the paper is unrefereed and unchanged since the release, that the Lean file concerns an elementary fact about finite-energy gradients on graphs, and that in the first 12 days the site found no expert verdict, reported error or 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]]