Foundation Circle H1 Computation Homology One Nonempty Iso Int Of Quasi Iso Sing

The algebraic core

In algebraic topology, the first homology group of a circle is the integers. This is one of the first facts a student meets: loops on a circle can wind around any whole number of times, so the group of winding classes is ℤ. The declaration homologyOneNonemptyIsoIntOfQuasiIsoSingleDegreeOneIntComplex is a machine-checked proof of the algebraic core of that fact, stated for a particular kind of chain complex.

A chain complex is a sequence of abelian groups connected by boundary maps. The declaration concerns a chain complex that is quasi-isomorphic to the simplest possible model of a circle: a complex with a single copy of ℤ in degree 1 and zero everywhere else. A quasi-isomorphism is a map between chain complexes that induces an isomorphism on all homology groups. The theorem states that if such a quasi-isomorphism exists, then the first homology group of the original complex is isomorphic to ℤ. In plainer terms: if a complex has the same homological shape as a circle, then its first homology is the integers.

The proof is a small algebraic lemma. It constructs the isomorphism explicitly from the quasi-isomorphism, using the fact that the single-degree complex has trivial homology in every degree except 1. The declaration is tagged as a theorem in the framework's machine-checked library of formal theorems. It does not depend on any geometric input: no actual circle, no topological space, no singular simplexes are involved. It is purely a statement about chain complexes and their homology.

In Recognition Science, this lemma is part of a workbench for a larger computation: the homology of the actual topological circle, TopCat.sphere 1, with integer coefficients. The framework's library proves that this singular homology is also isomorphic to ℤ. However, the declaration in question does not itself establish that geometric fact. It only establishes the algebraic core, before any topological transport is supplied.

The distinction matters. The theorem is a statement about chain complexes, not about topological spaces. It does not claim that the singular homology of the circle is ℤ; that is a separate result, proved elsewhere in the library. It does not claim that the chain complex model is the same as the topological circle. It does not claim that the quasi-isomorphism exists for any particular complex; it is a conditional statement: if such a map exists, then the homology is ℤ.

What the declaration changes is the status of the algebraic core. Before it, the statement was a missing computation. After it, the algebraic part is proved, and the remaining work is geometric: to show that the singular chain complex of the circle is quasi-isomorphic to the single-degree model. That geometric bridge is a separate, open target in the framework's library.

/-- Global quasi-isomorphism transport, for the common case where the finite
circle chain model is proved quasi-isomorphic to the single-supported reduced
model in every degree. -/
def homologyOneIsoIntOfQuasiIsoSingleDegreeOneIntComplex
    {K : HomologicalComplex (ModuleCat ℤ) (ComplexShape.down ℕ)}
    (f : K ⟶ singleDegreeOneIntComplex)
    [∀ i, K.HasHomology i] [QuasiIso f] :
    K.homology 1 ≅ ModuleCat.of ℤ ℤ :=
  homologyOneIsoIntOfQuasiIsoAtSingleDegreeOneIntComplex f

/-- Global quasi-isomorphism transport, for the common case where the finite
circle chain model is proved quasi-isomorphic to the single-supported reduced
model in every degree. -/
def homologyOneIsoIntOfQuasiIsoSingleDegreeOneIntComplex
    {K : HomologicalComplex (ModuleCat ℤ) (ComplexShape.down ℕ)}
    (f : K ⟶ singleDegreeOneIntComplex)
    [∀ i, K.HasHomology i] [QuasiIso f] :
    K.homology 1 ≅ ModuleCat.of ℤ ℤ :=
  homologyOneIsoIntOfQuasiIsoAtSingleDegreeOneIntComplex f

/-- Mathlib's homology infrastructure supplies the degree-`1` homology object
for the imported singular chain complex. -/
theorem sphereOneSingularIntChainComplexHasHomologyOne :
    sphereOneSingularIntChainComplex.HasHomology 1 :=
  inferInstance

What this page does not claim

The declaration does not prove that the singular homology of the topological circle is ℤ. The declaration does not claim that any particular chain complex is quasi-isomorphic to the single-degree model. The declaration does not involve any topological space or geometric input.

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/CircleH1Computation.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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