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The relative handle decomposition of a cylinder

Example

Assume ${A C}_{ω}$ . For a compact smooth manifold $M$ without boundary, the cylinder $W = M \times [0, 1]$ with faces $M_{0} = M \times {0}$ and $M_{1} = M \times {1}$ has the empty handle decomposition relative to $M_{0}$ : no handles are attached, and $W$ is the collar $M_{0} \times [0, 1]$ with the projection $π$ as adapted critical-point-free Morse function.

Facts & Assumptions

Given: A compact smooth manifold $M$ without boundary, ${A C}_{ω}$ , and the cylinder $W = M \times [0, 1]$ with the faces $M_{0} = M \times {0}$ and $M_{1} = M \times {1}$ .

[F1]

Smooth cobordism triad for Morse theory: A smooth cobordism triad $(W; M_{0}, M_{1})$ is a compact smooth manifold with boundary $W$ together with, for $n \geq 1$ , closed embedded $(n - 1)$ -submanifolds $M_{0}, M_{1} \subseteq \partial W$ with $\partial W = M_{0} ⊔ M_{1}$ and fixed collars; for $n = 0$ both faces and collar domains are empty, with their unique collar maps; either face may be empty and no orientation is needed.

[F2]

Handle decomposition relative to the incoming boundary: A finite handle decomposition of $(W; M_{0}, M_{1})$ relative to $M_{0}$ is a finite ordered list of indices with attaching embeddings such that $W$ is diffeomorphic, relative to $M_{0}$ , to the manifold obtained from the collar $M_{0} \times [0, ε]$ by successively attaching the handles with corners rounded. The empty list is allowed and presents the collar itself.

[F3]

Product cobordisms have critical-point-free presentations: Assume ${A C}_{ω}$ . For a compact smooth manifold $M$ without boundary the projection $π : M \times [0, 1] \to [0, 1]$ is an adapted Morse function with no critical points, $W$ has the empty handle decomposition relative to $M_{0}$ , and $W$ is diffeomorphic to the collar $M_{0} \times [0, 1]$ ; the hypothesis requires $\partial M = \emptyset$ .

[F4]

Smooth manifolds and their smooth charts applies to the boundaryless factor $M$ and its faces. The cylinder $W = M \times [0, 1]$ is a manifold with boundary in the category of [F1], with product boundary charts; its smooth maps and diffeomorphisms are read in Smooth maps between manifolds with boundary.

Verification

technique · direct

1.1F1F4given

The projection $π : W \to [0, 1]$ , $π (x, t) = t$ , is smooth, and its differential is $d t$ , which is nowhere zero; hence $π$ has no critical point and is a Morse function with empty critical set, and the condition of excellence is vacuous. It satisfies $π^{- 1} (0) = M \times {0} = M_{0}$ and $π^{- 1} (1) = M_{1}$ , and it is constant on each face, so it is adapted (with the boundary collar containing no critical point since there are none).

2.1F2F4step 1.1constructalgebra

For every $ε > 0$ , the map $W \to M_{0} \times [0, ε]$ , $(x, t) \mapsto ((x, 0), ε t)$ , is a diffeomorphism of manifolds with boundary, with inverse $((x, 0), s) \mapsto (x, s / ε)$ . It fixes $M_{0}$ pointwise and carries the projection to the rescaled collar coordinate. Thus the initial collar stage already presents the whole cylinder up to the required relative diffeomorphism.

3.1F2F3step 1.1step 2.1algebra∎

By [F2] the empty ordered list is an allowed handle decomposition: it presents the collar $M_{0} \times [0, ε]$ itself. By step 2.1 the cylinder is that collar, so the empty list is a handle decomposition of $W$ relative to $M_{0}$ in which no handle is attached; and by [F3] this is exactly the critical-point-free presentation whose Morse function is the projection.

Depends on

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Sources

C. T. C. Wall, Differential Topology (Cambridge Studies in Advanced Mathematics 156), Sections 5.1-5.4, printed pp. 129-148 (standard reference, not scraped)
John Milnor, Lectures on the h-Cobordism Theorem (notes by L. Siebenmann and J. Sondow), Sections 2-4, printed pp. 10-48 (standard reference, not scraped)