A complex structure on the six-sphere
In August 2026 Levent Alpöge put a 108-page manuscript, signed "Claude and Levent Alpöge", on his site, constructing a complex structure on the six-dimensional sphere S^6, which would answer Hopf’s problem in the affirmative. The earliest archived capture of the file is 23 August. In September a mathematician reproduced the proof in a separate paper and two more papers build on it; no journal has refereed it.
Why it matters
Hopf’s question has long had both positive and negative claims, which, according to Engel, were found not to be fully sound. Here the claim comes with a separate exposition by a mathematician who writes that he verified the content himself. That is not refereeing, but it makes this claim’s state different from "claimed" and "disputed".
State of the claim. Claimed: the manuscript alpo.ge/s6.pdf "The (3, 4, ∞) modular family of 2-tori, completed at its three special points, is a complex structure on S^6", by Claude and Levent Alpöge, 108 pages, undated; it was changed after it first appeared (the file size differs between archive captures). The earliest Internet Archive capture is 23 August, 22:22 UTC; the manuscript names no day of publication, so this is a bound. The pass read its first three pages (a summary of the main steps), not the rest. Checked by people: Philip Engel, "Complex structures on S^6" (dated 13 September inside; an earlier version first archived 27 August), gives "a self-contained proof" of a one-parameter family of compact complex threefolds diffeomorphic to S^6 and writes that the mathematical content "was independently verified by the author, who takes full responsibility"; he used ChatGPT for exploration and drafting. Zhangchi Chen (arXiv 2609.26706, 22 September) proves the Oka property for members of the family assuming Alpöge’s identification of the underlying manifold with S^6; Jeff Viaclovsky (arXiv 2609.33785, 27 September) constructs a two-parameter family of complex structures on S^6 "not biholomorphic to the original Alpöge–Claude examples". No document read reports a formalisation in Lean. No dispute was found in what was read; the construction contradicts a published result that the algebraic dimension of such a structure must be zero, and the manuscript explains why that argument does not apply (Engel calls the result nearly thirty years old). The model’s role. Engel: the manuscript was "produced by Claude under the direction of Levent Alpöge"; "the precise nature of the interaction between Alpöge and Claude is not, at present, documented". What the record does not claim: that Hopf’s problem is accepted as solved, that the pass checked the mathematics, or the day the manuscript was published.