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The outgoing boundary of a handle attachment trades the disk factors

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $N$ be a smooth $n$ -manifold with boundary, let $0 \leq k \leq n$ , and let $N^{'} = N \cup_{ψ} h^{k}$ be obtained by attaching the standard $k$ -handle $D^{k} \times D^{n - k}$ along an embedding $ψ : S^{k - 1} \times D^{n - k} \to \partial N$ that extends over a neighbourhood of the disk factor, with corners rounded. Then:

(i) $\partial (D^{k} \times D^{n - k}) = (S^{k - 1} \times D^{n - k}) \cup (D^{k} \times S^{n - k - 1})$ with intersection $S^{k - 1} \times S^{n - k - 1}$ ;

(ii) the boundary of $N^{'}$ is obtained from $\partial N$ by trading the open attaching region for the outgoing region, $\partial N^{'} ≅ (\partial N ∖ ψ (S^{k - 1} \times int D^{n - k})) \cup_{ψ ∣_{S^{k - 1} \times S^{n - k - 1}}} (D^{k} \times S^{n - k - 1}),$ the identification on the overlap being $ψ$ ;

(iii) the belt sphere ${0} \times S^{n - k - 1}$ is a closed embedded submanifold of $\partial N^{'}$ , and its normal bundle in $\partial N^{'}$ is identified with the bundle of $D^{k}$ -factor directions.

Endpoint cases: for $k = 0$ the attaching region is empty and $\partial N^{'} = \partial N ⊔ S^{n - 1}$ ; for $k = n$ the outgoing region is empty and the attaching region is the sphere $S^{n - 1} \times D^{0}$ , which the handle $D^{n} \times D^{0}$ caps.

Facts & Assumptions

Given: a smooth $n$ -manifold $N$ with boundary, an integer $0 \leq k \leq n$ , an embedding $ψ : S^{k - 1} \times D^{n - k} \to \partial N$ extending over a neighbourhood of the disk factor, and the attached manifold $N^{'} = N \cup_{ψ} h^{k}$ with rounded corners.

[F1]

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

[F2]

Attaching a smooth handle with corner rounding: The handle is attached by gluing $D^{k} \times D^{n - k}$ to $N$ along the attaching region, identifying $z$ with $ψ (z)$ ; the framing is part of the data, the seam receives product charts from collars, and the compact codimension-two corner is rounded by a compatible profile. There is no corner to round when $k = 0$ or $k = n$ .

[F3]

Smooth handle attachment is independent of corner rounding up to diffeomorphism: For fixed attaching and product-collar data, two compatible roundings are related by a diffeomorphism equal to the identity outside the collar.

[F4]

Collar neighborhood theorem: every smooth manifold with boundary has a smooth collar (Smooth collars of a manifold boundary), so $\partial N$ has a neighbourhood identified with $\partial N \times [0, 1)$ .

[F5]

The double has a well-defined smooth structure: gluing a manifold with boundary to itself along its boundary using a collar produces a boundaryless smooth manifold whose structure is well defined up to a diffeomorphism fixing the seam pointwise; this is the model for the seam charts used in an attachment.

Proof

Given: the objects and hypotheses of the statement.

1.1F1algebra

For the product $D^{k} \times D^{n - k}$ the boundary is the union of the two products with the boundary of one factor, $\partial (D^{k} \times D^{n - k}) = \partial D^{k} \times D^{n - k} \cup D^{k} \times \partial D^{n - k}$ , overlapping exactly in $\partial D^{k} \times \partial D^{n - k}$ ; writing $\partial D^{k} = S^{k - 1}$ and $\partial D^{n - k} = S^{n - k - 1}$ gives claim (i). The degenerate cases are included: for $k = 0$ the first term is $S^{- 1} \times D^{n} = \emptyset$ and the second is $D^{0} \times S^{n - 1} = S^{n - 1}$ .

1.2F2F4F5given

The glued manifold $N^{'}$ is covered by the interior of $N ∖ ψ (S^{k - 1} \times int D^{n - k})$ , the interior of the handle, and a collar neighbourhood of the seam supplied by [F4], in which the two pieces are presented as half-spaces meeting along the seam; the seam has the product model recorded in [F5], so the union is a smooth manifold with boundary.

2.1F1F2F3step 1.1step 1.2

A point of $N^{'}$ is a boundary point exactly when it lies in $\partial N$ outside the open attaching region or in the outgoing region $D^{k} \times S^{n - k - 1}$ of the handle: points of the attaching region and of the seam that lie over its interior are interior points of $N^{'}$ by the collar model of step 1.2, and the remaining boundary points of the handle are precisely its outgoing region by claim (i). The two parts meet exactly along $ψ (S^{k - 1} \times S^{n - k - 1})$ , where $ψ$ and the boundary identification of the handle agree. This proves claim (ii); the rounding enters only through the smooth structure of the seam, and changing it changes the result at most by a diffeomorphism equal to the identity outside the collar by [F3].

3.1F1F2step 2.1algebra

The belt sphere ${0} \times S^{n - k - 1}$ lies in the outgoing region $D^{k} \times S^{n - k - 1} \subseteq \partial N^{'}$ and is closed there because $S^{n - k - 1}$ is closed in $D^{k} \times S^{n - k - 1}$ . Near a point $(0, y)$ the outgoing region is an open subset of $D^{k} \times S^{n - k - 1}$ with the product smooth structure, and the tangent directions of ${0} \times S^{n - k - 1}$ are the $S^{n - k - 1}$ -directions, so the complementary normal directions inside $\partial N^{'}$ are the $D^{k}$ -factor directions; the product trivialization identifies this normal bundle with the trivial bundle of rank $k$ . This is claim (iii), and it makes the belt sphere a closed embedded submanifold of $\partial N^{'}$ in the sense of Embedded smooth submanifolds with boundary.

4.1F1F2step 1.1∎

Endpoint cases. For $k = 0$ the attaching region is $S^{- 1} \times D^{n} = \emptyset$ , the handle is the disk $D^{n}$ attached along the empty set, and the formula of claim (ii) reduces to $\partial N^{'} = \partial N ⊔ \partial D^{n} = \partial N ⊔ S^{n - 1}$ . For $k = n$ the outgoing region is $D^{n} \times S^{- 1} = \emptyset$ , the attaching region is $S^{n - 1} \times D^{0} = S^{n - 1}$ , and the handle $D^{n} \times D^{0} = D^{n}$ caps the attaching sphere, so the formula removes $ψ (S^{n - 1} \times {0})$ from $\partial N$ and glues nothing; no rounding is needed in either case by [F2].

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Sources

Wolfgang Lück, A Basic Introduction to Surgery Theory (lecture notes, Münster, 27 October 2004) (standard reference, not scraped)
Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, 2002; electronic copy) (standard reference, not scraped)
C. T. C. Wall, Differential Topology (Cambridge Studies in Advanced Mathematics 156, Cambridge University Press 2016) (standard reference, not scraped)