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

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Relative homology of the standard handle pair

Statement

For any abelian group $G$ , integers $0 \leq k \leq n$ , and $i \geq 0$ , the standard handle pair has $H_{i} (D^{k} \times D^{n - k}, S^{k - 1} \times D^{n - k}; G) ≅ G$ if $i = k$ and zero otherwise. Here $D^{0}$ is a point and $S^{- 1} = \emptyset$ .

Facts & Assumptions

[F1]

K handle core cocore attaching region and belt sphere: For integers $0 \leq k \leq n$ , the standard $n$ -dimensional $k$ -handle is $D^{k} \times D^{n - k}$ . Its core is $D^{k} \times {0}$ , its cocore is ${0} \times D^{n - k}$ , its attaching region is $S^{k - 1} \times D^{n - k}$ , and its attaching sphere is $S^{k - 1} \times {0}$ . The outgoing region is $D^{k} \times S^{n - k - 1}$ and the belt sphere is ${0} \times S^{n - k - 1}$ . Here $D^{j}$ is the closed unit disk, $D^{0}$ is a point, and $S^{- 1} = \emptyset$ . For $n = 0$ both boundary regions are empty.

[F2]

The singular chain homotopy formula: Let $H : X \times I \to Y$ be a homotopy from $f$ to $g$ . Then the prism operator $P_{H}$ of def-prism-operator-for-a-homotopy satisfies $g_{#} - f_{#} = \partial P_{H} + P_{H} \partial$ as homomorphisms $C_{n} (X; G) \to C_{n} (Y; G)$ for every $n \geq 1$ and every abelian group $G$ . In degree $0$ , the same identity reduces to $g_{#, 0} - f_{#, 0} = \partial P_{H} : C_{0} (X; G) \to C_{0} (Y; G) .$

[F3]

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

[F4]

Homology of spheres: For $n \geq 1$ , ${\tilde{H}}_{k} (S^{n}; G)$ is $G$ for $k = n$ and $0$ otherwise. For $S^{0}$ , ${\tilde{H}}_{0} (S^{0}; G) ≅ G$ and all other reduced groups vanish. Thus $H_{0} (S^{n}; G) ≅ G$ for $n \geq 1$ , whereas $H_{0} (S^{0}; G) ≅ G \oplus G$ .

[F5]

Contractible nonempty spaces have the homology of a point: If $X$ is a nonempty contractible topological space, then for every $n \geq 0$ and every abelian group $G$ , $H_{n}^{s i n g} (X; G) ≅ H_{n}^{s i n g} (*; G),$ where $*$ denotes a one-point space.

Proof

Given: The objects and hypotheses in the statement.

1.1F1algebra

The standard pair contracts its second disk factor by $(u, v) \mapsto (u, (1 - t) v)$ , with projection to and inclusion of $(D^{k}, S^{k - 1})$ as inverse maps up to a homotopy of pairs. The attaching subspace is preserved even when empty.

2.1F2step 1.1

The prism formula descends to relative chains: prisms of simplices in the subspace remain in the subspace, so their classes vanish in the quotient chain complex. Thus the two maps in the previous step induce inverse homology maps, in degree zero as well as positive degrees.

3.1F5step 2.1algebra

If $k = 0$ , the reduced pair is $(*, \emptyset)$ , with homology $G$ in degree zero and zero otherwise. Explicitly the chain complex of a point has one copy of $G$ in each degree, and its boundary is the identity in positive even degrees and zero in odd degrees, so this computation includes arbitrary $G$ . Every nonempty disk has that same homology by contractibility.

4.1F3F4step 3.1algebra∎

For $k \geq 2$ , the pair exact sequence and sphere homology give the only nonzero relative group in degree $k$ , isomorphic to $G$ ; in degrees zero and one the map $H_{0} (S^{k - 1}; G) \to H_{0} (D^{k}; G)$ is the identity on $G$ . For $k = 1$ , it is $(g, h) \mapsto g + h$ from $G \oplus G$ to $G$ , whose kernel is ${(g, - g)}$ and whose cokernel is zero. Hence $H_{1} (D^{1}, S^{0}; G) ≅ G$ and $H_{0} = 0$ . This proves all cases, including $G = 0$ .

Depends on

Used by

Dependency tree · two levels

17 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

Nicolaescu, An Invitation to Morse Theory (standard reference, not scraped)
Audin–Damian, Morse Theory and Floer Homology (standard reference, not scraped)