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The dual handle retraction onto the cocore, with the outgoing region carried onto the belt sphere

Statement

For the standard $n$ -dimensional $k$ -handle $H = D^{k} \times D^{n - k}$ (K handle core cocore attaching region and belt sphere, Euclidean spheres and closed balls as subspaces of $R^{n}$ ) with core $K = D^{k} \times {0}$ , cocore $C = {0} \times D^{n - k}$ , attaching region $S^{k - 1} \times D^{n - k}$ , attaching sphere $S^{k - 1} \times {0}$ , outgoing region $R = D^{k} \times S^{n - k - 1}$ and belt sphere $B = {0} \times S^{n - k - 1}$ :

(a) The formula $H_{s} (x, y) : = (x, (1 - s) y)$ defines a strong deformation retraction of $H$ onto the core $K$ (Retractions and deformation retracts, with a deformation retraction required to fix the retract pointwise) that maps the attaching region into itself and maps it onto the attaching sphere at $s = 1$ .

(b) The formula $G_{s} (x, y) : = ((1 - s) x, y)$ defines a strong deformation retraction of $H$ onto the cocore $C$ that maps the outgoing region $R$ into itself and maps it onto the belt sphere $B$ at $s = 1$ .

(c) $R ∖ B$ strongly deformation retracts onto $S^{k - 1} \times S^{n - k - 1}$ by the radial map $(x, y) \mapsto (x / ∣ x ∣, y)$ , and this retraction fixes $S^{k - 1} \times S^{n - k - 1}$ pointwise.

Facts & Assumptions

Given: Integers $0 \leq k \leq n$ and the standard handle $H = D^{k} \times D^{n - k}$ with its core, cocore, attaching region, outgoing region and belt sphere.

[F1]

The standard $n$ -dimensional $k$ -handle is $D^{k} \times D^{n - k}$ , with core $D^{k} \times {0}$ , cocore ${0} \times D^{n - k}$ , attaching region $S^{k - 1} \times D^{n - k}$ , attaching sphere $S^{k - 1} \times {0}$ , outgoing region $D^{k} \times S^{n - k - 1}$ and belt sphere ${0} \times S^{n - k - 1}$ ; $D^{0}$ is a point and $S^{- 1} = \emptyset$ (K handle core cocore attaching region and belt sphere).

[L1]

$D^{j}$ is the Euclidean closed unit ball and $S^{j - 1}$ its boundary sphere, carrying the subspace topology of $R^{j}$ (Euclidean spheres and closed balls as subspaces of $R^{n}$ ).

[F2]

A strong deformation retraction of $X$ onto $A \subseteq X$ is a retraction $r : X \to A$ together with a homotopy $H : {id}_{X} ≃_{A} i \circ r$ from the identity to $i \circ r$ that fixes $A$ pointwise (Retractions and deformation retracts, with a deformation retraction required to fix the retract pointwise).

[F3]

The model handle is glued by a smooth embedding of the attaching region that extends over a neighbourhood of the disk factor, with the framing part of the data; there is no corner to round when $k = 0$ or $k = n$ (Attaching a smooth handle with corner rounding).

Proof

technique · explicit formulas

1.1F1F2givenconstruct

For $(x, y) \in H$ and $s \in [0, 1]$ put $H_{s} (x, y) : = (x, (1 - s) y)$ . This map is continuous, $H_{0} = {id}_{H}$ , and $H_{1} (x, y) = (x, 0) \in K$ ; moreover $H_{s} (x, 0) = (x, 0)$ for all $s$ , so $K$ is fixed pointwise. Hence $H_{s}$ is a strong deformation retraction of $H$ onto $K$ in the sense of [F2]. Its restriction to the attaching region satisfies $H_{s} (S^{k - 1} \times D^{n - k}) = S^{k - 1} \times (1 - s) D^{n - k} \subseteq S^{k - 1} \times D^{n - k}$ , and at $s = 1$ the image is $S^{k - 1} \times {0}$ , the attaching sphere.

1.2F1F2givenconstruct

For $(x, y) \in H$ and $s \in [0, 1]$ put $G_{s} (x, y) : = ((1 - s) x, y)$ . This map is continuous, $G_{0} = {id}_{H}$ , and $G_{1} (x, y) = (0, y) \in C$ ; moreover $G_{s} (0, y) = (0, y)$ for all $s$ , so $C$ is fixed pointwise and $G_{s}$ is a strong deformation retraction of $H$ onto the cocore $C$ by [F2]. Its restriction to the outgoing region satisfies $G_{s} (D^{k} \times S^{n - k - 1}) = (1 - s) D^{k} \times S^{n - k - 1} \subseteq D^{k} \times S^{n - k - 1} = R$ , and at $s = 1$ the image is ${0} \times S^{n - k - 1} = B$ , the belt sphere.

1.3F1L1F2givenconstruct

The complement of the belt sphere in the outgoing region is $R ∖ B = {(x, y) \in D^{k} \times S^{n - k - 1} : x \neq 0}$ . Define $K_{s} (x, y) : = ((1 - s) x + s x / ∣ x ∣, y)$ on $(R ∖ B) \times [0, 1]$ ; this is well defined and continuous because $x \neq 0$ on the domain and $y \in S^{n - k - 1}$ is unchanged, and by [L1] the norm $∣ x ∣$ is the Euclidean norm. We have $K_{0} = id$ , $K_{1} (x, y) = (x / ∣ x ∣, y) \in S^{k - 1} \times S^{n - k - 1}$ , and $K_{s}$ fixes every point of $S^{k - 1} \times S^{n - k - 1}$ pointwise, because $x / ∣ x ∣ = x$ there. Hence $K_{s}$ is a strong deformation retraction of $R ∖ B$ onto $S^{k - 1} \times S^{n - k - 1}$ in the sense of [F2].

2.1F1F3step 1.1step 1.2step 1.3∎

The three formulas are explicit and continuous for all $0 \leq k \leq n$ , including the endpoint cases $k = 0$ and $k = n$ : at $k = 0$ the attaching region $S^{- 1} \times D^{n}$ is empty, the core is a point, and $R ∖ B = \emptyset$ because $B = R$ ; at $k = n$ the outgoing region and belt sphere are empty while the cocore is a point. Thus (a), (b) and (c) hold as stated, and the standard model is the one glued by [F3].

Remarks

Relation to Wall's retraction. Statement (a) is Wall's handle retraction onto the core and attaching region in the disk-factor direction (Wall, Figure 5.6), and (b) is its dual in the complementary disk-factor direction; (c) is the punctured-disk retraction $D^{k} ∖ {0} \to S^{k - 1}$ written in the outgoing coordinates.
Use. In Milnor's proof of Lemma 7.2 the local computation is exactly (b) together with (c): the handle retracts to its cocore while the complement of the belt sphere in the outgoing region is pushed back onto the attaching boundary $S^{k - 1} \times S^{n - k - 1}$ .
Choice. All three homotopies are explicit formulas, so no choice principle is used.

Depends on

Used by

Handle boundary coefficients are attaching-belt intersection numbers Lemma

Dependency tree · two levels

11 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

John Milnor, Lectures on the h-Cobordism Theorem (notes by L. Siebenmann and J. Sondow), Section 6 and Section 7, PDF pp. 87-93 (standard reference, not scraped)
C. T. C. Wall, Differential Topology, Sections 5.1-5.4, printed pp. 129-148 (PDF pp. 137-151) (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)