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Attaching handles of index at least q preserves homology below q-1

Statement

Assume ${A C}_{ω}$ , let $F$ be a field, and let $q$ be an integer. Let $W^{'}$ be obtained from a smooth manifold $W$ with boundary by successively attaching finitely many rounded handles of indices at least $q$ (Attaching a smooth handle with corner rounding). Then $H_{i} (W^{'}, W; F) = 0$ for every $i \leq q - 1$ . Consequently the inclusion $W \to W^{'}$ induces isomorphisms $H_{i} (W; F) \to H_{i} (W^{'}; F)$ for $i \leq q - 2$ and a surjection for $i = q - 1$ .

Facts & Assumptions

Given: A field $F$ , a smooth manifold $W$ with boundary, handles attached successively to obtain $W^{'} \supseteq W$ with indices $ℓ_{1}, \dots, ℓ_{r} \geq q$ , and the intermediate manifolds $W_{0} = W \subseteq W_{1} \subseteq \dots \subseteq W_{r} = W^{'}$ .

[F1]

If $N^{'} = N \cup_{φ} h^{k}$ is obtained by attaching a rounded $k$ -handle to a smooth manifold $N$ with boundary, then $H_{i} (N^{'}, N; F) = 0$ for $i \neq k$ and $H_{k} (N^{'}, N; F) ≅ F$ , with the core class as generator (One handle changes relative homology in one degree only, part (a)).

[F2]

For spaces $B \subseteq A \subseteq X$ and every abelian group $G$ there is a long exact sequence $\dots \to H_{n} (A, B; G) \to H_{n} (X, B; G) \to H_{n} (X, A; G) \overset{δ}{\to} H_{n - 1} (A, B; G) \to \dots,$ with the first two maps induced by inclusions (Long exact sequence of a triple in singular homology).

[F3]

For $A \subseteq X$ the pair sequence $\dots \to H_{n} (A; G) \to H_{n} (X; G) \to H_{n} (X, A; G) \overset{δ}{\to} H_{n - 1} (A; G) \to \dots$ is exact, with the first two maps induced by inclusions (Long exact sequence of a pair).

[L1]

For $A = X$ the relative group $H_{n} (X, X; G)$ vanishes for every $n$ (Relative singular homology).

Proof

technique · induction-on-handles

1.1given

The intermediate manifolds $W_{j}$ are smooth manifolds with boundary, since attaching a rounded handle to a smooth manifold with boundary produces one, and $W_{j}$ is obtained from $W_{j - 1}$ by attaching one handle of index $ℓ_{j} \geq q$ for each $j$ .

1.2L1given

Claim, by induction on $j$ , that $H_{i} (W_{j}, W; F) = 0$ for every $i \leq q - 1$ . For $j = 0$ we have $W_{0} = W$ and $H_{i} (W, W; F) = 0$ by [L1].

2.1F1step 1.1

Induction step: by [F1] applied to the attachment $W_{j - 1} \to W_{j}$ , the group $H_{i} (W_{j}, W_{j - 1}; F)$ is $F$ for $i = ℓ_{j}$ and zero otherwise; since $ℓ_{j} \geq q$ , it vanishes for every $i \leq q - 1$ .

3.1F2step 1.2step 2.1

The long exact sequence of the triple $(W_{j}, W_{j - 1}, W)$ from [F2] reads $\dots \to H_{i} (W_{j - 1}, W; F) \to H_{i} (W_{j}, W; F) \to H_{i} (W_{j}, W_{j - 1}; F) \to H_{i - 1} (W_{j - 1}, W; F) \to \dots .$ For $i \leq q - 1$ the first and last terms vanish by the induction hypothesis of step 1.2 and the middle term vanishes by step 2.1; exactness at the middle node then forces $H_{i} (W_{j}, W; F) = 0$ . This completes the induction.

4.1step 3.1

Taking $j = r$ gives $H_{i} (W^{'}, W; F) = 0$ for every $i \leq q - 1$ .

5.1F3step 4.1∎

The pair sequence contains $H_{i + 1} (W', W; F) \to H_{i} (W; F) \to H_{i} (W'; F) \to H_{i} (W', W; F)$ . For $i \leq q - 2$ both relative groups vanish by step 4.1, giving an isomorphism. For $i = q - 1$ only the right relative group is known to vanish, giving a surjection; its kernel is the image of the connecting map from $H_{q} (W', W; F)$ . Thus injection in this degree is not asserted.

Remarks

The case $q = 0$ . Every index is at least $0$ , so the vanishing statement is about $i \leq - 1$ , where all homology vanishes; the isomorphism statements are about degrees $\leq - 2$ and $- 1$ and are vacuous. The lemma is used only for $q \geq 1$ .
Use. This is the handle analogue of the skeletal stabilization step in the computation of cellular homology: handles attached above degree $j$ leave the homology below $j$ unchanged, which is what lets the handle chain complex of an index-ordered presentation compute $H_{*} (M; F)$ .

Depends on

Used by

The handle chain complex computes singular homology Proposition

Dependency tree · two levels

31 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

Allen Hatcher, Algebraic Topology, Sections 2.1-2.2 (relative homology and long exact sequences) (standard reference, not scraped)
Ralph L. Cohen, Bundles, Manifolds, and Homotopy, Chapter 12 Section 5, printed pp. 489-493 (PDF pp. 501-505) (standard reference, not scraped)