Foundation Linking Vanishing High Dim Not Detects Of Arc Acyclic

Vanishing linking in high dimensions

In topology, a circle can be placed inside a higher-dimensional sphere. The question is whether the circle is linked with something else, meaning it cannot be pulled apart from another loop without cutting. One way to detect this is to remove the circle from the sphere and examine the first homology group of what remains, the algebraic measure of holes in that leftover space. If that group is zero, the complement has no one-dimensional hole wrapping around the missing circle, so the circle is not detected as linked.

The machine-checked library, a collection of formal theorems verified by a proof assistant, contains a result about this detection. The theorem not_detects_of_arcAcyclic states a conditional fact: if, in a given dimension D (other than 3), every embedded arc has an acyclic complement (meaning its first homology group is zero), then no embedded circle in that dimension can be detected by this homology test. In plainer terms, if all arcs are topologically trivial in their complements, then circles are too. The proof works by building a circle from two arcs that cover the whole sphere, then using a Mayer-Vietoris-style argument to show the circle's complement homology must vanish.

This theorem is a stepping stone. A companion result, forces_D3_of_arcAcyclic, takes the same hypothesis about arcs and concludes that the only dimension where a circle can be detected is D = 3. The chain of reasoning shows that in dimensions 0, 1, 2, 4, 5, and beyond, the homology test is blind to circles. The framework's broader program uses this to argue that three-dimensional space is special for linking, a property that does not hold in other dimensions.

The theorem does not claim that arcs are always acyclic in every dimension. That is a separate hypothesis, named ArcComplementsAcyclic, which the theorem assumes rather than proves. It also does not claim that the homology test is the only way to detect linking, nor that it works for all possible embeddings. The result is conditional: given the acyclicity of arcs, the detection of circles fails in all dimensions except three.

/-- **The high-dimensional (and `D = 2`) vanishing, conditional on the arc
frontier**: granting `ArcComplementsAcyclic D`, no embedded circle in `S^D`
(`D ≥ 1`, `D ≠ 3`) has homologically nontrivial complement. -/
theorem not_detects_of_arcAcyclic (D : ℕ) (hD : 1 ≤ D) (hD3 : D ≠ 3)
    (harc : ArcComplementsAcyclic D) : ¬ DetectsNontrivialLinking D := by
  rintro ⟨f, hemb, hH⟩
  apply hH
  have hz := isZero_h1_complement_of_embedding D hD hD3 harc
    (toSphMap D f) (isEmbedding_toSphMap D f hemb)
  exact hz.of_iso (hgrpIso (complDownHomeo D f).toHomotopyEquiv 1)

/-- **The bridge's uniqueness half, conditional on the arc frontier**:
granting arc-complement acyclicity in every dimension `D ≥ 2`, `D ≠ 3`,
nontrivial linking forces `D = 3`.  Dimensions `0` and `1` are the banked
unconditional results (`LinkingVanishingLowDim`). -/
theorem forces_D3_of_arcAcyclic
    (harc : ∀ D, 2 ≤ D → D ≠ 3 → ArcComplementsAcyclic D) :
    ∀ D, DetectsNontrivialLinking D → D = 3 := by
  intro D hdet
  by_contra hne
  match D, hne with
  | 0, _ => exact LinkingVanishingLowDim.not_detects_zero hdet
  | 1, _ => exact LinkingVanishingLowDim.not_detects_one hdet
  | (n + 2), hne =>
      exact not_detects_of_arcAcyclic (n + 2) (by omega) hne
        (harc (n + 2) (by omega) hne) hdet

What this page does not claim

The theorem does not prove that arcs are always acyclic in any dimension. The theorem does not claim that the homology test is the only way to detect linking. The theorem does not apply to dimensions where the arc-complement hypothesis fails.

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

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