Foundation Singular Pair Pair Ses Degreewise Short Exact

What the exactness lemma says

In algebraic topology, a short exact sequence is a three-term chain of maps where the image of each map equals the kernel of the next, a precise way of saying that the middle object is built from the outer two. The Recognition Science declaration pairSES_degreewise_shortExact proves, in the framework's machine-checked library of formal theorems, that for any injective continuous map between topological spaces, the associated singular chain complex (a discrete record of continuous maps from standard simplices into the space) gives a short exact sequence at every degree, meaning at every dimension of simplex. The result is a formal theorem: it is proved from the axioms of the framework's logic with no unproved assumptions.

The proof proceeds by first showing that an injective continuous map induces an injective map on singular n-simplices, then building a retraction that splits the chain map, and finally assembling the short exact sequence from the kernel and cokernel. The lemma is the degreewise version of a stronger statement, pairSES_shortExact, which establishes the same exactness for the entire chain complex at once. The degreewise version evaluates the complex at a fixed dimension n and confirms that the three-term sequence of abelian groups is exact there. This is a standard and expected property in homological algebra; the contribution here is that it is fully formalized and machine-checked.

What the lemma does not claim is that the sequence splits, that the spaces involved are manifolds or have any special structure beyond being topological spaces, or that the homology groups are finitely generated or computable. It also does not claim that the map between spaces is surjective, only injective. The result is purely about the algebraic structure of singular homology for a pair of spaces, not about any particular geometric or physical consequence. It is a foundational building block, not an end in itself.

/-- Degreewise form of the short exact sequence: in every degree `n`,
`0 ⟶ C_n(A) ⟶ C_n(X) ⟶ C_n(X, A) ⟶ 0` is a short exact sequence of
`ℤ`-modules. -/
lemma pairSES_degreewise_shortExact (f : A ⟶ X) (hf : Function.Injective f.hom)
    (n : ℕ) :
    ((pairSES f).map
      (HomologicalComplex.eval (ModuleCat.{0} ℤ) (ComplexShape.down ℕ) n)).ShortExact :=
  (pairSES_shortExact f hf).map_of_exact _

/-- An injective continuous map induces an injective map on singular
`n`-simplices (postcomposition with an injective map is injective). -/
lemma toSSet_map_app_injective (f : A ⟶ X) (hf : Function.Injective f.hom) (n : ℕ) :
    Function.Injective ((TopCat.toSSet.map f).app (op ⦋n⦌)) := by
  intro a b hab
  have h1 : f.hom.comp (A.toSSetObjEquiv (op ⦋n⦌) a) =
      f.hom.comp (A.toSSetObjEquiv (op ⦋n⦌) b) := by
    rw [← toSSetObjEquiv_map f a, ← toSSetObjEquiv_map f b, hab]
  have h2 : A.toSSetObjEquiv (op ⦋n⦌) a = A.toSSetObjEquiv (op ⦋n⦌) b := by
    ext t
    apply hf
    simpa only [ContinuousMap.comp_apply] using ContinuousMap.congr_fun h1 t
  exact (A.toSSetObjEquiv (op ⦋n⦌)).injective h2

/-- For an injective continuous map, the sequence
`0 ⟶ C_*(A) ⟶ C_*(X) ⟶ C_*(X, A) ⟶ 0` is short exact. -/
lemma pairSES_shortExact (f : A ⟶ X) (hf : Function.Injective f.hom) :
    (pairSES f).ShortExact where
  exact := ShortComplex.exact_of_g_is_cokernel _ (cokernelIsCokernel (sChainMap f))
  mono_f := sChainMap_mono f hf
  epi_g := by
    show Epi (cokernel.π (sChainMap f))
    infer_instance

What this page does not claim

The sequence splits, meaning the middle homology group is not claimed to be a direct sum of the outer two. The spaces involved are manifolds or have any special structure beyond being topological spaces. The homology groups are finitely generated or computable in practice.

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/SingularPair.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

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