PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Ordinary homology theories have mayer vietoris for cw covers

Statement

Let $X = U \cup V$ be a cover by CW subcomplexes and $W = U \cap V$ . Every ordinary homology theory has a natural exact sequence $\dots \to h_{n} (W) \overset{(i_{*}, - j_{*})}{\to} h_{n} (U) \oplus h_{n} (V) \overset{a_{*} + b_{*}}{\to} h_{n} (X) \overset{Δ}{\to} h_{n - 1} (W) \to \dots,$ where all four maps $i, j, a, b$ are inclusions. The same sequence holds for a CW pair $(X, C)$ covered by $(U, C \cap U)$ and $(V, C \cap V)$ , with the corresponding relative groups.

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

Ordinary unreduced theories on CW pairs and reduced ordinary theories on based CW spaces with vertex basepoints determine one another, naturally and compatibly with morphisms and coefficients. For $A \neq \emptyset$ the correspondence gives $h_{n} (X, A) ≅ {\tilde{h}}_{n} (X / A)$ ; for $A = \emptyset$ it gives $h_{n} (X) ≅ {\tilde{h}}_{n} (X_{+})$ , where $X_{+} = X ⊔ {*}$ . Under this correspondence, pair boundaries are cofiber boundaries followed by inverse suspension, and arbitrary disjoint-sum additivity corresponds to arbitrary wedge additivity. For a CW triple $B \subset A \subset X$ there is a natural exact sequence $\dots \to h_{n} (A, B) \to h_{n} (X, B) \to h_{n} (X, A) \to h_{n - 1} (A, B) \to \dots$ , whose last map is the pair boundary followed by $h_{n - 1} (A) \to h_{n - 1} (A, B)$ . (Unreduced pair and reduced quotient axioms are equivalent on cw pairs)

Proof

1.1

Let $q_{U} : h_{n} (U) \to h_{n} (U, W)$ and $q_{V} : h_{n} (X) \to h_{n} (X, V)$ be the pair maps, and let $e : h_{n} (U, W) \to h_{n} (X, V)$ be the excision isomorphism. F1 supplies these pair sequences and their naturality, including $\partial_{V} e = j_{*} \partial_{U}$ . Define $Δ = \partial_{U} e^{- 1} q_{V}$ . The three successive composites in the asserted sequence vanish by these identities and pair exactness.

2.1

If $Δ x = 0$ , write $z = e^{- 1} q_{V} x$ . Then $\partial_{U} z = 0$ , so $z = q_{U} u$ for some $u \in h_{n} (U)$ . Thus $q_{V} (x - a_{*} u) = 0$ , giving $x - a_{*} u = b_{*} v$ for some $v \in h_{n} (V)$ . This proves exactness at $h_{n} (X)$ .

F1step 1.1

2.2

If $a_{*} u + b_{*} v = 0$ , then $e q_{U} u = 0$ , so $u = i_{*} w$ . Now $b_{*} (j_{*} w + v) = 0$ , hence $j_{*} w + v = \partial_{V} t$ for some $t \in h_{n + 1} (X, V)$ . Write $t = e z$ . Then $j_{*} w + v = j_{*} \partial_{U} z$ . Set $w^{'} = w - \partial_{U} z$ . Pair exactness gives $i_{*} w^{'} = u$ and the displayed equation gives $- j_{*} w^{'} = v$ . This proves exactness at the direct sum.

F1step 1.1

2.3

If $i_{*} w = j_{*} w = 0$ , choose $z \in h_{n + 1} (U, W)$ with $\partial_{U} z = w$ . Then $\partial_{V} e z = j_{*} w = 0$ , so $e z = q_{V} x$ for some $x \in h_{n + 1} (X)$ . Consequently $Δ x = w$ , proving exactness at $h_{n} (W)$ . All the maps defining $Δ$ are natural, including the inverse of the natural isomorphism $e$ , so this is a natural sequence.

F1step 1.1

3.1

For a subcomplex $C$ , work in the based quotient $X_{+} / C_{+}$ with its cover by the images of $U_{+}$ and $V_{+}$ . These are the based quotients by $(C \cap U)_{+}$ and $(C \cap V)_{+}$ ; their intersection is $W_{+} / (C \cap W)_{+}$ . Use the reduced version of the same chase, or subtract the split basepoint sequence from the unreduced one. The quotient identification in F1 converts every term to the asserted relative term. Empty members, empty intersections, and $C = X$ give the corresponding zero terms without changing the chase.

F1step 2.1step 2.2step 2.3∎

Depends on

Unreduced pair and reduced quotient axioms are equivalent on cw pairs

Used by

Additivity and compact cell support control the infinite cw colimit Lemma

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hatcher, Algebraic Topology, Axioms for Homology, Mayer–Vietoris derivation p.162 (standard reference, not scraped)
Miller, Algebraic Topology I lecture notes, Lemma 11.6, pp.27–28 (standard reference, not scraped)
May, A Concise Course in Algebraic Topology, 14§5, first Mayer–Vietoris theorem pp.112–113 (standard reference, not scraped)