Foundation Linking Vanishing High Dim Is Zero H1 Union Compl

The union lemma

In topology, a central question is whether a space has holes that a loop can catch. The first homology group, written H₁, is the algebraic object that counts such holes. A space whose H₁ is zero has no loops that cannot be shrunk to a point. The theorem isZero_h1_unionCompl, proved in the machine-checked library of formal theorems, states a condition under which the union of two regions also has this property.

Specifically, let W be a topological space, and let KP and KM be two closed subsets of W. The theorem assumes three things: the intersection of KP and KM has zero second homology, the complement of KP has zero first homology, and the complement of KM has zero first homology. Under these hypotheses, the theorem concludes that the complement of the union KP ∪ KM also has zero first homology. In plainer words, if two closed regions individually leave no holes in the space around them, and their overlap is similarly well-behaved one dimension up, then the space outside both regions together is also hole-free in the loop-catching sense.

This lemma is a building block in a larger argument about when a circle can be embedded in a sphere without creating a hole in the complement. The framework uses this to show that in dimensions other than three, such an embedding always leaves a trivial complement, a fact that helps force the dimension of space to be three. The theorem itself is a formal statement about topological spaces and homology, not about physical space directly; the physical interpretation is a separate step.

What the theorem does not claim is equally important. It does not assert that the union of the two sets is itself simply connected, only that the complement of the union has zero first homology. It does not apply to arbitrary subsets; the closedness condition on KP and KM is essential. And it does not, by itself, establish anything about the dimension of physical space; that conclusion requires additional theorems about embeddings and acyclic complements, which are proved elsewhere in the library.

/-- **Abstract two-arc Mayer-Vietoris**: two closed sets `KP, KM` in a space
`W`, with `H₂(W \ (KP ∩ KM)) = 0` and `H₁`-acyclic complements, have
`H₁`-acyclic union complement. -/
theorem isZero_h1_unionCompl {W : TopCat.{0}} (KP KM : Set ↥W)
    (hKPc : IsClosed KP) (hKMc : IsClosed KM)
    (hX2 : IsZero (Hgrp (TopCat.of ((KP ∩ KM)ᶜ : Set ↥W)) 2))
    (hP1 : IsZero (Hgrp (TopCat.of {y : ↥W // y ∉ KP}) 1))
    (hM1 : IsZero (Hgrp (TopCat.of {y : ↥W // y ∉ KM}) 1)) :
    IsZero (Hgrp (TopCat.of {y : ↥W // y ∉ KP ∪ KM}) 1) := by
  -- the MV cover of `W \ (KP ∩ KM)` by the complements of the two arcs
  have hUopen : IsOpen
      ({x : ↥(TopCat.of ((KP ∩ KM)ᶜ : Set ↥W)) | x.1 ∉ KP} :
        Set (TopCat.of ((KP ∩ KM)ᶜ : Set ↥W))) :=
    hKPc.isOpen_compl.preimage continuous_subtype_val
  have hVopen : IsOpen
      ({x : ↥(TopCat.of ((KP ∩ KM)ᶜ : Set ↥W)) | x.1 ∉ KM} :
        Set (TopCat.of ((KP ∩ KM)ᶜ : Set ↥W))) :=
    hKMc.isOpen_compl.preimage continuous_subtype_val
  have hUVcover :
      ({x : ↥(TopCat.of ((KP ∩ KM)ᶜ : Set ↥W)) | x.1 ∉ KP} :
        Set (TopCat.of ((KP ∩ KM)ᶜ : Set ↥W))) ∪
      {x : ↥(TopCat.of ((KP ∩ KM)ᶜ : Set ↥W)) | x.1 ∉ KM} = Set.univ := by
    rw [Set.eq_univ_iff_forall]
    intro x
    by_cases hxP : x.1 ∈ KP
    · right
      intro hxM
      exact x.2 ⟨hxP, hxM⟩
    · left
      exact hxP
  -- H₁ of the pieces, flattened to the arc complements
  have hU1 : IsZero (Hgrp (TopCat.of
      ({x : ↥(TopCat.of ((KP ∩ KM)ᶜ : Set ↥W)) | x.1 ∉ KP} :
        Set (TopCat.of ((KP ∩ KM)ᶜ : Set ↥W)))) 1) :=
    hP1.of_iso (hgrpIso
      (flattenComplHomeo (KP ∩ KM) KP Set.inter_subset_left).toHomotopyEquiv 1)
  have hV1 : IsZero (Hgrp (TopCat.of
      ({x : ↥(TopCat.of ((KP ∩ KM)ᶜ : Set ↥W)) | x.1 ∉ KM} :
        Set (TopCat.of ((KP ∩ KM)ᶜ : Set ↥W)))) 1) :=
    hM1.of_iso (hgrpIso
      (flattenComplHomeo (KP ∩ KM) KM Set.inter_subset_right).toHomotopyEquiv 1)
  -- MV middle vanishing, then flatten the intersection
  have hmid := isZero_h1_inter hUopen hVopen hUVcover hX2 hU1 hV1
  have hUVeq :
      (({x : ↥(TopCat.of ((KP ∩ KM)ᶜ : Set ↥W)) | x.1 ∉ KP} :
        Set (TopCat.of ((KP ∩ KM)ᶜ : Set ↥W))) ∩
      {x : ↥(TopCat.of ((KP ∩ KM)ᶜ : Set ↥W)) | x.1 ∉ KM}) =
      {x : ↥(TopCat.of ((KP ∩ KM)ᶜ : Set ↥W)) | x.1 ∉ KP ∪ KM} := by
    ext x
    rw [Set.mem_inter_iff, Set.mem_setOf_eq, Set.mem_setOf_eq,
      Set.mem_setOf_eq]
    constructor
    · rintro ⟨h1, h2⟩ (h | h)
      · exact h1 h
      · exact h2 h
    · intro h
      exact ⟨fun hP => h (Set.mem_union_left _ hP),
        fun hM => h (Set.mem_union_right _ hM)⟩
  rw [hUVeq] at hmid
  exact hmid.of_iso (hgrpIso
    (flattenComplHomeo (KP ∩ KM) (KP ∪ KM)
      (Set.inter_subset_left.trans
        Set.subset_union_left)).symm.toHomotopyEquiv 1)

What this page does not claim

The theorem does not claim that the union of the two closed sets is simply connected. The theorem does not apply to subsets that are not closed. The theorem does not by itself establish the dimensionality of physical space.

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/LinkingVanishingHighDim.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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