Foundation Circle H1 Computation

The circle's first homology

In algebraic topology, the first homology group of a circle, written H₁(S¹; ℤ), is the group of integers ℤ. This is a classical fact: it counts the ways a loop can wind around the circle, with each integer recording the winding number. The circle has one hole, and that hole is captured by the integers, one for each possible number of full turns.

The standard way to compute this is through singular homology, which builds a chain complex from continuous maps of simplices into the space. For the circle, this complex has infinitely many generators, but its homology in degree 1 collapses to ℤ. The computation is a staple of algebraic topology courses, and it underlies why the circle is the fundamental example of a space with nontrivial first homology.

In Recognition Science, the framework's machine-checked library of formal theorems contains a workbench module for this computation. The module does not replace the standard singular homology of the circle; instead, it builds a simpler algebraic model: a chain complex supported entirely in degree 1, with the integers sitting there and all boundary maps equal to zero. This model's homology in degree 1 is exactly ℤ, proved as a formal theorem. The framework then proves that this simple model is quasi-isomorphic to the ordinary cellular chain model of the circle, meaning they have the same homology.

The module also constructs the singular chain complex of the topological circle and proves that its homology in degree 1 is ℤ. It defines a geometric bridge: a homotopy equivalence between the singular complex and the simple cellular model. If that bridge is supplied, the module derives the full statement that H₁(S¹; ℤ) ≅ ℤ in the framework's cohomology bridge. The bridge itself is not yet proved; it remains a target for future work.

What this establishes in plain language is a checkable foundation: the framework's internal models of the circle agree with the classical result at the level of first homology. The integer winding numbers are not assumed; they are derived from a chain complex that the machine checks. This matters because the framework uses such homology computations as building blocks for deeper claims about space and linking, and each block must be sound on its own.

/-- Degree-`1` homology of the single-supported integer chain complex is the
integer module. -/
def singleDegreeOneIntComplexHomologyOneIsoInt :
    (((HomologicalComplex.single (ModuleCat ℤ) (ComplexShape.down ℕ) 1).obj
      (ModuleCat.of ℤ ℤ)).homology 1) ≅ ModuleCat.of ℤ ℤ :=
  HomologicalComplex.singleObjHomologySelfIso (ComplexShape.down ℕ) 1
    (ModuleCat.of ℤ ℤ)

/-- Proposition-facing form of
`reducedCellularCircleChainModelH1IsoInt`. -/
theorem reducedCellularCircleChainModelH1NonemptyIsoInt :
    Nonempty (reducedCellularCircleChainModel.homology 1 ≅ ModuleCat.of ℤ ℤ) :=
  ⟨reducedCellularCircleChainModelH1IsoInt⟩

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

/-- The collapse from the ordinary two-cell cellular circle model to the reduced
degree-`1` model is a quasi-isomorphism in degree `1`.

The proof uses Mathlib's zero-differential short-complex criterion: in degree
`1`, both relevant short complexes have zero differentials, and the middle
component of the collapse map is an isomorphism on the chosen integer
generator. -/
theorem ordinaryCellularToReducedChainMap_quasiIsoAt_one :
    QuasiIsoAt ordinaryCellularToReducedChainMap 1 := by
  rw [quasiIsoAt_iff]
  rw [ShortComplex.quasiIso_iff_isIso_liftCycles _ (by
    simp [ordinaryCellularCircleChainModel_d_eq_zero]) (by
    simp [ordinaryCellularCircleChainModel_d_eq_zero]) (by
    simp)]
  let S₂ :=
    (HomologicalComplex.shortComplexFunctor (ModuleCat ℤ) (ComplexShape.down ℕ) 1).obj
      reducedCellularCircleChainModel
  let φ :=
    ((HomologicalComplex.shortComplexFunctor (ModuleCat ℤ) (ComplexShape.down ℕ) 1).map
      ordinaryCellularToReducedChainMap)
  let w : φ.τ₂ ≫ S₂.g = 0 := by
    dsimp [S₂, φ]
    simp
  change IsIso (S₂.liftCycles φ.τ₂ w)
  haveI : IsIso S₂.iCycles := S₂.isIso_iCycles (by
    dsimp [S₂])
  haveI : IsIso (S₂.liftCycles φ.τ₂ w ≫ S₂.iCycles) := by
    rw [ShortComplex.liftCycles_i]
    dsimp [S₂, φ]
    rw [ordinaryCellularToReducedChainMap_f_one]
    infer_instance
  exact IsIso.of_isIso_comp_right (S₂.liftCycles φ.τ₂ w) S₂.iCycles

/-- The remaining geometric chain-level bridge for the strict T8 circle-H1
closure: Mathlib's singular chain complex for `TopCat.sphere 1` is chain
homotopy equivalent to the ordinary two-cell cellular circle model. -/
def circleH1GeometricBridge : Prop :=
  Nonempty
    (HomotopyEquiv sphereOneSingularIntChainComplex ordinaryCellularCircleChainModel)

/-- The geometric bridge immediately closes the exact proposition used by
`MathlibCohomologyBridge`. -/
theorem circleH1ZIsoIntOfGeometricBridge
    (h : circleH1GeometricBridge) :
    MathlibCohomologyBridge.circleH1ZIsoInt := by
  rcases h with ⟨e⟩
  exact circleH1ZIsoIntOfHomotopyEquivOrdinaryCellularAtOne e

/-- The geometric bridge fills the Mathlib computation certificate used by the
strict T8 handoff. -/
theorem circleH1MathlibComputationOfGeometricBridge
    (h : circleH1GeometricBridge) :
    MathlibCohomologyBridge.CircleH1MathlibComputation :=
  MathlibCohomologyBridge.circleH1MathlibComputation_of_iso_int
    (circleH1ZIsoIntOfGeometricBridge h)

What this page does not claim

The geometric bridge itself is not proved; it remains an open target. This module does not replace the standard singular homology of the circle in the framework's strict T8 bridge. No claim is made here about higher homology groups or about the circle's fundamental group.

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)

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: