sphere-topology: SimplyConnectedSpace (OnePoint ℂ) and genus X = 0 for simply connected X #
API summary (see docs/design/sphere-topology.md), namespace RS.SphereTopology unless noted.
Builds on projective-line (CC5) and paths-and-integrals (CC6); no van Kampen, no universal cover,
no covering-space monodromy anywhere — the two headline results are assembled from more
elementary mathlib pieces (contractibility/local-path-connectedness/open-embedding transport) plus
paths-and-integrals' own loop-perturbation and homotopy-invariance lemmas.
Topology (
SimplyConnectedP1.lean):isSimplyConnected_compl_infty/isSimplyConnected_compl_coeZero(the two polar caps ofOnePoint ℂ's atlas, each ≃ₜℂ, hence simply connected),instance : PathConnectedSpace (OnePoint ℂ), the headlineinstance simplyConnectedSpace_onePoint : SimplyConnectedSpace (OnePoint ℂ)(via paths-and-integrals'RS.Loop.exists_homotopic_avoiding, basepoint-split on∞vs↑0),simplyConnectedSpace_of_homeoOnePoint(generic homeomorphism transfer), andinstance simplyConnectedSpace_sphere : SimplyConnectedSpace (Metric.sphere ... 1)(the literal challenge sphere model, viaRS.P1.homeoSphere).(Development note: this unit was built while
RS.Loop.exists_homotopic_avoiding(Jacobian/Path/Perturb.lean) still carried one documented upstream placeholder step, design risk R4 of paths-and-integrals; that step has since been resolved upstream — seedocs/build-log.md, "Perturb.lean FIXED" — so no downstream change was needed here.)Analysis — the genus-0 engine (
GlobalPrimitive.lean, standing surface variables perCONVENTIONS.md):contMDiff_and_mdifferential_eq_of_isPrimitiveAlongMap_id(a primitive ofηalongid : X → Xis a genuine holomorphic global primitive withmdifferential = η— reusable verbatim by the monodromy unit, see that file's docstring),exists_isPrimitiveAlongMap_id(constructs the global primitive on simply connectedXviapathIntegralfrom a base point),form1_eq_zero_of_simplyConnectedSpace(Forster 10.5: every holomorphic 1-form vanishes on a simply connected compact surface), theSubsingleton (Form1 X)instance, and the headlinegenus_eq_zero_of_simplyConnectedSpace : genus X = 0.Assembly (
Headline.lean):genus_eq_zero_of_homeo_sphere— the exact backward-headline signatureNonempty (X ≃ₜ Metric.sphere (0 : EuclideanSpace ℝ (Fin 3)) 1) → genus X = 0, consumed bygenus-zero-headlinealongside Riemann–Roch's forward half.
Downstream consumers: genus-zero-headline (consumes genus_eq_zero_of_homeo_sphere directly);
monodromy (may reuse contMDiff_and_mdifferential_eq_of_isPrimitiveAlongMap_id verbatim, or
re-derive the same conclusion from its own chain-continuation IsPrimitiveAlongMap id η F univ
fact — both routes agree, no dependency is required either way).