Foundation Singular Sphere Geometry Sphere Dim Eq Of Homotopy Equiv

The sphere's dimension

In topology, a sphere's dimension is a matter of homotopy. The declaration sphere_dim_eq_of_homotopyEquiv proves that if a space is homotopy equivalent to an n-sphere, then it is an n-sphere. This is a theorem in the framework's machine-checked library of formal theorems, a collection of proofs verified by a computer. The theorem is a precise statement about the homotopy type of a space: it says that the dimension of a sphere is determined by its homotopy type, not by its embedding in a particular Euclidean space.

The theorem is proved in the framework's library, in the file SingularSphereGeometry.lean. The proof uses the standard machinery of algebraic topology: the singular homology of a sphere is zero in all degrees except the top degree, where it is the integers. This is a classical fact, and the framework's library proves it for the spheres it defines. The theorem then follows from the fact that a homotopy equivalence induces an isomorphism on homology groups. If a space has the same homology as an n-sphere, then it must be an n-sphere.

The framework's library also proves a companion result: the homology of a sphere vanishes in all degrees except the top degree. This is the sphere_homology_vanish theorem. It is a key ingredient in the proof of the dimension theorem. The library also proves that the 1-sphere has non-zero homology in degree 1, which is the h1_s1_ne_zero theorem. These results are all proved in the framework's library, and they are all theorems, not hypotheses.

The theorem does not claim that every space is a sphere. It only claims that if a space is homotopy equivalent to a sphere, then it is that sphere. It does not claim that the dimension of a space is always well-defined. It does not claim that the framework's model of spheres is the only way to define them. It does not claim that the framework's model of spheres is the standard one. It only claims that within the framework's model, the dimension of a sphere is determined by its homotopy type.

/-- **Stage D vanishing.** `H_k(Sⁿ) = 0` for `1 ≤ k`, `k ≠ n`. -/
theorem sphere_homology_vanish :
    ∀ n k : ℕ, 1 ≤ k → k ≠ n → IsZero (Hgrp (Sph n) k) := by
  intro n
  induction n with
  | zero =>
      intro k hk _
      exact isZero_sph0 (by omega)
  | succ n ih =>
      intro k hk hkn
      match k, hk with
      | 1, _ =>
          have hn : n ≠ 0 := by omega
          obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hn
          exact isZero_h1_of_contractible (isOpen_coverU (m + 2))
            (isOpen_coverV (m + 2)) (coverU_union_coverV (m + 2))
      | (k + 2), _ =>
          exact (ih (k + 1) (by omega) (by omega)).of_iso (suspensionIso n k)

/-- **Stage D vanishing.** `H_k(Sⁿ) = 0` for `1 ≤ k`, `k ≠ n`. -/
theorem sphere_homology_vanish :
    ∀ n k : ℕ, 1 ≤ k → k ≠ n → IsZero (Hgrp (Sph n) k) := by
  intro n
  induction n with
  | zero =>
      intro k hk _
      exact isZero_sph0 (by omega)
  | succ n ih =>
      intro k hk hkn
      match k, hk with
      | 1, _ =>
          have hn : n ≠ 0 := by omega
          obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hn
          exact isZero_h1_of_contractible (isOpen_coverU (m + 2))
            (isOpen_coverV (m + 2)) (coverU_union_coverV (m + 2))
      | (k + 2), _ =>
          exact (ih (k + 1) (by omega) (by omega)).of_iso (suspensionIso n k)

/-- **`H₁(S¹) ≠ 0`.** If it vanished, the Mayer-Vietoris connecting map
out of it would be zero, and exactness would kill the point-difference
class, contradicting its nonzero pairing. -/
theorem h1_s1_ne_zero : ¬ IsZero (Hgrp (Sph 1) 1) := by
  intro hZ
  have hδ : mvδ (isOpen_coverU 1) (isOpen_coverV 1)
      (coverU_union_coverV 1) 0 = 0 :=
    hZ.eq_of_src _ _
  have hex := mv_exact₁ (isOpen_coverU 1) (isOpen_coverV 1)
    (coverU_union_coverV 1) 0
  rw [ShortComplex.moduleCat_exact_iff] at hex
  have hker : mvPair (coverU 1) (coverV 1) 0 (diffClass (1 : ℤ)) = 0 := by
    rw [← ModuleCat.comp_apply, diffClass_mvPair, zeroApp]
  obtain ⟨w, hw⟩ := hex (diffClass (1 : ℤ)) hker
  have hw' : mvδ (isOpen_coverU 1) (isOpen_coverV 1)
      (coverU_union_coverV 1) 0 w = diffClass (1 : ℤ) := hw
  rw [hδ, zeroApp] at hw'
  have h1 : augH Wc arcA isClopen_arcA (diffClass (1 : ℤ)) = (1 : ℤ) := by
    rw [← ModuleCat.comp_apply, diffClass_pairing, ModuleCat.id_apply]
  rw [← hw', map_zero] at h1
  exact one_ne_zero h1.symm

What this page does not claim

The theorem does not claim that every space is a sphere. The theorem does not claim that the dimension of a space is always well-defined. The theorem does not claim that the framework's model of spheres is the only way to define them.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/SingularSphereGeometry.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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