Foundation Arc Complement Acyclic Class Of Eq Zero Iff

The zero-class criterion

In algebraic topology, a cycle is a chain whose boundary is zero, and a boundary is a chain that is itself the boundary of one dimension higher. The classical question is whether every cycle is a boundary, which measures the holes in a space. The declaration classOf_eq_zero_iff answers a sharper version: a cycle represents the zero class in homology exactly when it is a boundary. The statement is a biconditional, proved for any chain complex K, any dimension n, and any cycle z: classOf K n z hz = 0 ↔ ∃ w, z = K.d (n+2) (n+1) w. In words, the homology class of a cycle vanishes if and only if the cycle is the boundary of some higher-dimensional chain.

The proof is a direct calculation in the framework's machine-checked library of formal theorems. It unfolds the definition of classOf, which sends a cycle to its equivalence class modulo boundaries, and then applies the standard fact that a quotient is zero exactly when its numerator lies in the kernel. The lemma cls_eq_zero_iff specializes this to the framework's recognition spaces, where the chain complex is built from spheres and their complements. The result is a structural fact about homology, not a claim about any particular space: it holds for every chain complex in the library.

What the declaration does not claim is equally precise. It does not assert that every cycle is a boundary; that would require the homology group to be trivial, which is false for spheres. It does not identify which higher-dimensional chain w bounds a given cycle; it only states that one exists. And it does not connect the zero class to any physical recognition event, cost, or ledger operation. The lemma is a tool for later theorems about arc complements and acyclicity, such as arcComplementsAcyclic, but it carries no geometric or physical interpretation by itself.

/-- **The vanishing criterion.** The class of a cycle is zero iff the cycle
is a boundary. -/
lemma classOf_eq_zero_iff (n : ℕ) (z : ↥(K.X (n + 1))) (hz : K.d (n + 1) n z = 0) :
    classOf K n z hz = 0 ↔ ∃ w : ↥(K.X (n + 2)), z = K.d (n + 2) (n + 1) w := by
  have h1 : ∀ x : ↥((K.sc' (n + 2) (n + 1) n).homology),
      CategoryTheory.ShortComplex.homologyMap (scIso K n).inv x = 0 ↔ x = 0 := fun x =>
    hom_apply_eq_zero_iff (CategoryTheory.ShortComplex.homologyMapIso (scIso K n)).symm x
  have h2 : ∀ q, (K.sc' (n + 2) (n + 1) n).moduleCatLeftHomologyData.homologyIso.inv q = 0 ↔
      q = 0 := fun q =>
    hom_apply_eq_zero_iff (K.sc' (n + 2) (n + 1) n).moduleCatLeftHomologyData.homologyIso.symm q
  unfold classOf
  rw [h1, h2, Submodule.Quotient.mk_eq_zero, LinearMap.mem_range]
  constructor
  · rintro ⟨w, hw⟩
    exact ⟨w, (congrArg Subtype.val hw).symm⟩
  · rintro ⟨w, hw⟩
    exact ⟨w, Subtype.ext hw.symm⟩

What this page does not claim

This does not claim that every cycle is a boundary in any space. This does not claim that the zero class corresponds to a physical recognition event. This does not identify the specific bounding chain when one exists.

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

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Derived articles

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