The Hopf problem: a complex structure on the 6-sphere
Problem statement
Notes: The Hopf problem asks whether S⁶ admits a complex structure. An affirmative construction has been claimed (Alpöge 2026); its correctness is not assumed by this benchmark. Two holes: ChartedSpace data into ℂ³ (the plain function type Fin 3 → ℂ; the model norm is immaterial) on the topological 6-sphere, and the IsManifold 𝓘(ℂ, Fin 3 → ℂ) ω proof that the atlas is holomorphic. No compatibility with the standard smooth atlas is demanded; by Smale's h-cobordism theorem and Θ₆ = 0 (Kervaire–Milnor) this is equivalent to a complex structure on the standard smooth S⁶. Known constraints: S² and S⁶ are the only spheres with almost complex structures (Borel–Serre); no complex structure on S⁶ is orthogonal for the round metric (LeBrun 1987).
Source: H. Hopf, 'Zur Topologie der komplexen Mannigfaltigkeiten', Studies and Essays Presented to R. Courant, Interscience, 1948. Survey: I. Agricola, G. Bazzoni, O. Goertsches, P. Konstantis, S. Rollenske, 'On the history of the Hopf problem', Differ. Geom. Appl. 57 (2018). https://arxiv.org/abs/1708.01068
Informal solution: Unavailable.
/-- **Hole 1 (data).** An atlas of charts from the 6-sphere — the unit sphere
in `ℝ⁷`, with its subspace topology — into `ℂ³`, taken as the plain function
type `Fin 3 → ℂ`: all norms on a finite-dimensional space are equivalent, so
the choice of model norm is immaterial to the statement. -/
instance declaration uses `sorry`instChartedSpaceS6 :
ChartedSpace (Fin 3 → ℂ)
(Metric.sphere (0 : EuclideanSpace ℝ (Fin 7)) 1) :=
sorry/-- **Hole 2 (proof).** The atlas of hole 1 is holomorphic: its transition
functions are `ℂ`-analytic on their open domains. -/
instance declaration uses `sorry`instIsManifoldS6 :
IsManifold 𝓘(ℂ, Fin 3 → ℂ) ω
(Metric.sphere (0 : EuclideanSpace ℝ (Fin 7)) 1) :=
sorry