FormalConjectures.Mathoverflow.«1973»

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import FormalConjecturesUtil

Mathoverflow 1973

Does the 6-sphere $S^6$ admit the structure of a complex manifold?

Reference: mathoverflow/1973 asked by user Fetchinson0234

open scoped Manifoldnamespace Mathoverflow1973

The unit n-sphere, defined as Metric.sphere 0 1 in EuclideanSpace ℝ (Fin (n + 1)).

abbrev unitSphere (n : ℕ) : Set (EuclideanSpace ℝ (Fin (n + 1))) := Metric.sphere 0 1

Does the 6-sphere admit a complex structure, i.e. an atlas of holomorphically compatible charts relating it to EuclideanSpace ℂ (Fin 3)?

@[category research open, AMS 32]
theorem declaration uses 'sorry'mathoverflow_1973 :
    answer(sorry) ↔ ∃ atlas : ChartedSpace (EuclideanSpace ℂ (Fin 3)) (unitSphere 6),
      IsManifold 𝓘(ℂ, EuclideanSpace ℂ (Fin 3)) 1 (unitSphere 6) := by⊢ True ↔ ∃ atlas, IsManifold 𝓘(ℂ, EuclideanSpace ℂ (Fin 3)) 1 ↑(unitSphere 6)
  sorryAll goals completed! 🐙


end Mathoverflow1973