TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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Singular homology satisfies homotopy exactness and excision

Statement

For every fixed abelian group $G$ , singular homology $H_{n} (-, -; G)$ , extended by zero in negative degrees, satisfies homotopy invariance, pair exactness, naturality of the connecting maps, and CW excision in Unreduced homology theory on cw pairs.

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

A CW pair is $(X, A)$ with $A$ a CW subcomplex of $X$ , as in def-skeleta-cw-subcomplex-and-relative-cw-complex. Morphisms are all continuous maps of pairs, not just cellular maps. An ordinary unreduced homology theory assigns covariant functors $h_{n}$ from CW pairs to abelian groups, for every $n \in Z$ , and natural homomorphisms $\partial : h_{n} (X, A) \to h_{n - 1} (A)$ , where $h_{n} (X) = h_{n} (X, \emptyset)$ , satisfying: - Homotopic maps of pairs induce equal homomorphisms. - The inclusion maps and $\partial$ form an exact sequence $\dots \to h_{n} (A) \to h_{n} (X) \to h_{n} (X, A) \to h_{n - 1} (A) \to \dots$ . - For CW subcomplexes $U, V$ of $X = U \cup V$ , inclusion induces $h_{n} (U, U \cap V) ≅ h_{n} (X, V)$ . - For a point $*$ , $h_{n} (*) = 0$ when $n \neq 0$ ; write $G = h_{0} (*)$ . - For every set-indexed family of CW pairs, including the empty family, the inclusions induce $⨁_{α} h_{n} (X_{α}, A_{α}) ≅ h_{n} (⨆_{α} X_{α}, ⨆_{α} A_{α})$ . Thus $h_{n} (\emptyset) = 0$ . No finite-dimensionality or finite-cell restriction is implicit. (Unreduced homology theory on cw pairs)

[F2]

If $f, g : X \to Y$ are homotopic continuous maps, then for every $n \geq 0$ and every abelian group $G$ the induced homomorphisms on singular homology agree: $H_{n} (f_{#}) = H_{n} (g_{#}) : H_{n}^{s i n g} (X; G) \to H_{n}^{s i n g} (Y; G) .$ (Homotopic maps induce the same map on singular homology)

[F3]

For $A \subseteq X$ there is an exact sequence $\dots \to H_{n} (A; G) \to H_{n} (X; G) \to H_{n} (X, A; G) \overset{δ}{\to} H_{n - 1} (A; G) \to H_{n - 1} (X; G) \to \dots .$ (Long exact sequence of a pair)

[F4]

A map of pairs $f : (X, A) \to (Y, B)$ induces a commuting morphism from the long exact sequence of $(X, A)$ to that of $(Y, B)$ , including the connecting maps. (Naturality of the pair long exact sequence)

[F5]

If $Z \subseteq X$ and $\overline{Z} \subseteq {int}_{X} (A)$ , then inclusion $(X - Z, A - Z) \to (X, A)$ induces isomorphisms $H_{n} (X - Z, A - Z; G) ≅ H_{n} (X, A; G)$ for every $n$ . (Excision for singular homology)

[F6]

If $(X, A)$ is a relative CW complex, then $A ↪ X$ has the homotopy extension property; in particular it is a cofibration. (Relative CW inclusions are cofibrations)

Proof

1.1F3F4

The singular pair sequence is exact and its connecting maps commute with every map of pairs, by F3 and F4. These assertions hold for arbitrary subspaces and hence for CW pairs.

1.2F2given

For a homotopy of pairs, the prism chain homotopy on $C_{*} (X; G)$ preserves $C_{*} (A; G)$ : every prism simplex over a simplex in $A$ stays in the target subspace. It therefore descends to relative chain quotients. The identity $f_{#} - g_{#} = \partial P + P \partial$ gives relative homotopy invariance, with the same absolute conclusion as F2. Negative degrees are zero.

1.3F6algebra

Put $W = U \cap V$ and form $D = U \cup_{W} (W \times [0, 1]) \cup_{W} V$ , attaching $W \times {0}$ to $U$ and $W \times {1}$ to $V$ . Let $c : D \to X = U \cup V$ collapse each cylinder fibre to $W$ . Since $(U, W)$ is a CW pair, [F6] extends the homotopy $W \times [0, 1] \to D$ , $(w, t) \mapsto (w, t)$ , starting from the inclusion $j_{U} : U \to D$ , to a homotopy $H_{U} : U \times [0, 1] \to D$ with $H_{U} (w, t) = (w, t)$ . Its end map sends $W$ to the copy in $V$ , so it glues with $j_{V} : V \to D$ to give $h : X \to D$ that fixes $V$ . The maps $c H_{U} (-, t)$ on $U$ and the identity on $V$ glue to a homotopy $c h ≃ {id}_{X}$ fixed on $V$ , since $c H_{U} (w, t) = w$ . Conversely, use $H_{U} (-, t)$ on $U$ , $(w, s) \mapsto (w, s + t (1 - s))$ on the cylinder, and the identity on $V$ . These formulas agree at both attaching ends and give a homotopy ${id}_{D} ≃ h c$ fixed on $V$ . Thus $c$ is a homotopy equivalence of pairs $(D, V) ≃ (X, V)$ , with explicit relative homotopies.

2.1F5F3F4step 1.2step 1.3

In $D$ set $O = U \cup (W \times [0, 2 / 3))$ and $N = V \cup (W \times (1 / 3, 1])$ . They are open and cover $D$ . The closure of $D ∖ O$ is contained in $N$ . Excision therefore identifies $H_{n} (O, O \cap N; G)$ with $H_{n} (D, N; G)$ . Retraction of $N$ to $V$ and the pair exact sequence identify the latter with $H_{n} (D, V; G)$ . The collapse $O \to U$ restricts on $O \cap N = W \times (1 / 3, 2 / 3)$ to projection onto $W$ . Both absolute maps are homotopy equivalences, so their commuting pair exact sequences give an isomorphism $H_{n} (O, O \cap N; G) \to H_{n} (U, W; G)$ by the exact-sequence injectivity and surjectivity chase. No inverse map of these pairs is needed. All maps commute with collapse to $(U \cup V, V)$ , so the resulting isomorphism is the homomorphism of the original inclusion.

3.1F1step 1.1step 1.2step 2.1∎

These are exactly the four structural requirements in F1. If either subcomplex or their intersection is empty the cylinder construction has the corresponding empty pieces; excision and the relative chain quotients still give the same conclusion.

Depends on

Used by

Dependency tree · two levels

29 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

Miller, Algebraic Topology I lecture notes, Definition 11.1 and ensuing verification, pp.25–26 (standard reference, not scraped)
Hatcher, Algebraic Topology, Axioms for Homology, pp.160–162 (standard reference, not scraped)