Foundation Singular Pair Pair Homology Map Comp Zero

The zero composition

In algebraic topology, singular homology assigns to each topological space a sequence of abelian groups that record holes and voids. A continuous map between spaces induces a homomorphism between their homology groups. The Recognition Science framework's machine-checked library of formal theorems contains a lemma, pair_homologyMap_comp_zero, that states a precise vanishing condition for such induced maps when a subspace is involved.

Consider an injective continuous map from a space A into a larger space X, so A is a subspace of X. The framework forms the relative homology of the pair (X, A), which measures the holes of X modulo those of A. The lemma says that the composition of two induced homology maps is zero: first map from the homology of X to the relative homology of the pair, then map from the relative homology back to the homology of A. The result is the zero homomorphism.

This vanishing is not an isolated curiosity. It is the middle exactness condition in the long exact sequence of a pair, the standard tool that relates the homology of a space, its subspace, and their quotient. The framework's lemma supplies that condition for every injective inclusion of topological spaces, at every homology degree. It is proved in the framework's library by reducing the statement to the cokernel condition that defines the relative homology groups.

The lemma does not claim that either individual map in the composition is zero. It does not assert that the relative homology groups are trivial, nor that the inclusion induces an injection on homology. It claims only that the round trip through the relative homology of the pair annihilates every homology class. The statement holds for all injective continuous maps, with no further hypotheses on the spaces involved.

What this fact changes for a reader is the ability to trust one link in a longer chain. The long exact sequence of a pair is a workhorse of algebraic topology; knowing that its middle composition vanishes is a prerequisite for the exactness that lets homology computations proceed. The framework's machine-checked lemma certifies that this link holds in its formal development, without relying on unstated assumptions about the spaces or the maps.

/-- The composite `H_n(A) ⟶ H_n(X) ⟶ H_n(X, A)` vanishes. -/
lemma pair_homologyMap_comp_zero (f : A ⟶ X) (n : ℕ) :
    HomologicalComplex.homologyMap (sChainMap f) n ≫
      HomologicalComplex.homologyMap (relπ f) n = 0 := by
  rw [← HomologicalComplex.homologyMap_comp, relπ, cokernel.condition,
    HomologicalComplex.homologyMap_zero]

/-- The composite `H_n(A) ⟶ H_n(X) ⟶ H_n(X, A)` vanishes. -/
lemma pair_homologyMap_comp_zero (f : A ⟶ X) (n : ℕ) :
    HomologicalComplex.homologyMap (sChainMap f) n ≫
      HomologicalComplex.homologyMap (relπ f) n = 0 := by
  rw [← HomologicalComplex.homologyMap_comp, relπ, cokernel.condition,
    HomologicalComplex.homologyMap_zero]

What this page does not claim

The lemma does not claim that either induced homology map in the composition is individually zero. The lemma does not assert that the relative homology groups of the pair are trivial. The lemma does not claim that the inclusion induces an injection on homology groups.

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)

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