Foundation Mathlib Cohomology Bridge Mathlib Circle Linking Backend Nonempty Iff

The linking backend equivalence

In algebraic topology, the first cohomology group of a circle with integer coefficients is a standard object: it classifies circle bundles and is isomorphic to the integers. The declaration mathlibCircleLinkingBackend_nonempty_iff_circleH1ZNonzero states an equivalence between two properties in the framework's machine-checked library of formal theorems. The first property is that a structure called MathlibCircleLinkingBackend exists, which is a formal object encoding the claim that the framework's linking relation supports a dimension. The second property is that the cohomology group circleH1Z is non-zero, meaning it is not the trivial group. The theorem proves these two properties are equivalent: one holds if and only if the other holds.

To make this concrete, consider what each side means. The cohomology group circleH1Z is defined as the singular homology functor applied to the 1-sphere, with integer coefficients. Its non-triviality is a basic fact: the circle has a one-dimensional hole, so its first homology is the integers, not zero. The linking backend is a formal structure that, given a dimension, asserts that the dimension supports a non-trivial linking relation, and it requires this to hold exactly when the dimension is 3 and the cohomology group is non-zero. The equivalence theorem links these two: the existence of such a backend is exactly the statement that the cohomology group is non-zero.

This equivalence is a bridge within the framework's library. It shows that the abstract condition of having a linking backend is not an extra assumption but is equivalent to a concrete, checkable topological fact. The theorem is proved in the library by constructing a backend from the non-zero cohomology group, and conversely extracting the non-zero cohomology group from any backend. This is a formal, machine-checked result, not a paper derivation. It is a step toward replacing a concrete encoding of the circle's cohomology with a computation that uses the standard singular homology API.

What the theorem does not claim is equally important. It does not prove that the cohomology group of the circle is non-zero; that is a separate fact, assumed or supplied by another theorem. It does not prove that three-dimensional space is forced; that conclusion would require additional steps beyond this equivalence. It only establishes the logical equivalence between the two stated conditions. The theorem is a bridge, not a destination: it connects a structural condition to a topological one, and leaves the physical interpretation to other parts of the framework.

/-- The remaining backend object is equivalent to the single concrete Mathlib
homology computation `circleH1ZNonzero`. -/
theorem mathlibCircleLinkingBackend_nonempty_iff_circleH1ZNonzero :
    Nonempty MathlibCircleLinkingBackend ↔ circleH1ZNonzero := by
  constructor
  · rintro ⟨B⟩
    exact B.circle_h1_nonzero
  · intro hH1
    exact ⟨mathlibCircleLinkingBackend_from_circleH1ZNonzero hH1⟩

/-- The concrete Mathlib object that must eventually be computed:
first singular homology of the topological circle with integer coefficients. -/
noncomputable abbrev circleH1Z : ModuleCat ℤ :=
  ((AlgebraicTopology.singularHomologyFunctor (ModuleCat ℤ) 1).obj
    (ModuleCat.of ℤ ℤ)).obj (TopCat.sphere 1)

/-- The backend required to replace the current concrete S¹ cohomology encoding.

The key field is `linking_from_circle_h1`: a future backend must prove the
nonvanishing of the concrete Mathlib circle homology object and route linking
through that computation. -/
structure MathlibCircleLinkingBackend where
  supportsLinking : DimensionForcing.Dimension → Prop
  uses_singular_homology_api : MathlibSingularHomologyAPIAvailable
  circle_h1_nonzero : circleH1ZNonzero
  linking_from_circle_h1 :
    ∀ D : DimensionForcing.Dimension,
      supportsLinking D ↔ D = 3 ∧ circleH1ZNonzero

What this page does not claim

This equivalence does not prove that the circle's cohomology group is non-zero. This equivalence does not by itself force three spatial dimensions. This theorem does not establish the Alexander-duality bridge needed for the full computation.

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/MathlibCohomologyBridge.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: