Foundation Singular Mayer Vietoris Is Iso Homology Map Of Elementwise

The elementwise isomorphism

The declaration establishes a precise bridge between local and global shape data. In algebraic topology, homology groups are algebraic invariants that count holes of various dimensions in a topological space. The Mayer-Vietoris theorem is a classical tool for computing these groups by breaking a space into two overlapping pieces. The declaration isIso_homologyMap_of_elementwise proves a version of this theorem for singular homology, which uses continuous maps from standard simplices to detect holes. The proof is machine-checked, meaning every step is verified by a computer program that checks logical correctness.

The theorem states that if two open subsets U and V cover the whole space X, then the homology groups of U, V, and their intersection fit together into an exact sequence that determines the homology of X. The declaration specifically proves that the map from the homology of the union to the direct sum of the homologies of the pieces is an isomorphism. This means the local hole data from the two pieces, combined with the data from their overlap, completely determines the global hole structure. The proof works by constructing a chain complex of "small" simplices, those whose image lies entirely in U or entirely in V, and showing this subcomplex captures all the homology.

The declaration does not claim that the homology of a space can always be computed by this method without additional conditions. The theorem requires the two subsets to be open and to cover the space. It also does not claim anything about spaces that are not covered by two such subsets, nor does it address homotopy groups or other algebraic invariants. The result is a formal statement about singular homology groups with integer coefficients, not a general statement about all topological invariants.

In Recognition Science, this result is a foundational tool. The framework models physical structure through discrete recognition events, and this theorem provides a way to assemble local algebraic data into global structure. The machine-checked proof ensures the framework's mathematical core is sound, allowing it to build on this result without relying on unverified assumptions.

lemma mvSESdeg_exact (n : ℕ) : (mvSESdeg U V n).Exact := by
  rw [ShortComplex.moduleCat_exact_iff]
  intro z hz
  have hz' : uInc U V n
      ((biprod.fst : Cgrp (TopCat.of U) n ⊞ Cgrp (TopCat.of V) n ⟶ _) z) +
      vInc U V n
      ((biprod.snd : Cgrp (TopCat.of U) n ⊞ Cgrp (TopCat.of V) n ⟶ _) z) = 0 := by
    rw [← descApp]
    exact hz
  obtain ⟨x, hxU, hxV⟩ := mv_middle_exact U V n hz'
  refine ⟨x, ?_⟩
  apply biprod_elem_ext
  · rw [show (mvSESdeg U V n).f x = biprod.lift (chainMap (mvInclU U V) n)
      (-(chainMap (mvInclV U V) n)) x from rfl, fst_liftApp]
    exact hxU
  · rw [show (mvSESdeg U V n).f x = biprod.lift (chainMap (mvInclU U V) n)
      (-(chainMap (mvInclV U V) n)) x from rfl, snd_liftApp, negApp, hxV,
      neg_neg]

lemma mvSESdeg_exact (n : ℕ) : (mvSESdeg U V n).Exact := by
  rw [ShortComplex.moduleCat_exact_iff]
  intro z hz
  have hz' : uInc U V n
      ((biprod.fst : Cgrp (TopCat.of U) n ⊞ Cgrp (TopCat.of V) n ⟶ _) z) +
      vInc U V n
      ((biprod.snd : Cgrp (TopCat.of U) n ⊞ Cgrp (TopCat.of V) n ⟶ _) z) = 0 := by
    rw [← descApp]
    exact hz
  obtain ⟨x, hxU, hxV⟩ := mv_middle_exact U V n hz'
  refine ⟨x, ?_⟩
  apply biprod_elem_ext
  · rw [show (mvSESdeg U V n).f x = biprod.lift (chainMap (mvInclU U V) n)
      (-(chainMap (mvInclV U V) n)) x from rfl, fst_liftApp]
    exact hxU
  · rw [show (mvSESdeg U V n).f x = biprod.lift (chainMap (mvInclU U V) n)
      (-(chainMap (mvInclV U V) n)) x from rfl, snd_liftApp, negApp, hxV,
      neg_neg]

/-- **The degree-`0` tail**: `H_0(U) ⊞ H_0(V) ⟶ H_0(X)` is surjective; the
Mayer-Vietoris sequence ends `⋯ ⟶ H_0(U) ⊞ H_0(V) ⟶ H_0(X) ⟶ 0`. -/
theorem mvSum_epi_zero (hU : IsOpen U) (hV : IsOpen V)
    (hUV : U ∪ V = Set.univ) : Epi (mvSum U V 0) := by
  haveI h1 : Epi (HomologicalComplex.homologyMap (mvβ U V) 0) := by
    refine epi_homologyMap_chain_zero (mvβ U V) ?_
    intro y
    have hepi : Epi ((mvβ U V).f 0) := (mvSES_degreewise_shortExact U V 0).epi_g
    have hsurj : Function.Surjective ((mvβ U V).f 0) :=
      (ModuleCat.epi_iff_surjective _).mp hepi
    obtain ⟨x, hx⟩ := hsurj y
    refine ⟨x, 0, ?_⟩
    rw [map_zero, add_zero, hx]
  haveI h2 : IsIso (HomologicalComplex.homologyMap (smallι U V) 0) :=
    smallι_isIso_homologyMap hU hV hUV 0
  rw [mvSum_eq]
  infer_instance

What this page does not claim

The declaration does not claim that homology can be computed without the open cover condition. It does not claim anything about homotopy groups or other algebraic invariants beyond singular homology with integer coefficients. The theorem does not address spaces that are not covered by two open subsets.

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/SingularMayerVietoris.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.

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