feat: add the Hopf S⁶ complex-structure problem - #557
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Two-hole open problem (the benchmark's first): ChartedSpace data into C^3 on the topological 6-sphere plus the IsManifold omega proof that the atlas is holomorphic. Declarations are top-level (no namespace): the workspace generator currently re-emits a LeanEval.* namespace chain with a dangling 'open LeanEval' in Submission.lean. Validated locally: module build (two expected sorry warnings), validate-manifest (structure-only + per-module inventory), and a full build of the generated workspace (Challenge/Submission/Solution). Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
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| instance instChartedSpaceS6 : | ||
| ChartedSpace (EuclideanSpace ℂ (Fin 3)) | ||
| (Metric.sphere (0 : EuclideanSpace ℝ (Fin 7)) 1) := | ||
| sorry | ||
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| /-- **Hole 2 (proof).** The atlas of hole 1 is holomorphic: its transition | ||
| functions are `ℂ`-analytic on their open domains. -/ | ||
| @[eval_problem] | ||
| instance instIsManifoldS6 : | ||
| IsManifold 𝓘(ℂ, EuclideanSpace ℂ (Fin 3)) ω | ||
| (Metric.sphere (0 : EuclideanSpace ℝ (Fin 7)) 1) := |
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You could replace EuclideanSpace ℂ (Fin 3) by Fin 3 → ℂ and the statement will still be equivalent (all norms are equivalent on a finite dimensional space). You could even replace EuclideanSpace ℝ (Fin 7) by Fin 7 → ℝ and although the resulting sphere is no longer smooth, it's still homeomorphic to the Euclidean sphere ...
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Adds the Hopf problem — does
S⁶admit a complex structure?Note that we do not require a proof that this structure is diffeomorphic to the standard real smooth structure on S⁶, since it is a standard fact that S⁶ admits only one real smooth structure.
There is an ongoing formalization of Alpöge's claimed construction (deancureton/sphere-six-complex).
Formalized with the assistance of Claude (Anthropic), reviewed by me.