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Product cobordisms have critical-point-free presentations

Statement

Assume ${A C}_{ω}$ . Let $M$ be a compact smooth manifold without boundary, including the empty manifold. For the triad $W = M \times [0, 1]$ , $M_{0} = M \times {0}$ , $M_{1} = M \times {1}$ , the projection $π$ is adapted excellent with no critical points. The cylinder has the empty handle decomposition relative to $M_{0}$ .

Facts & Assumptions

[F1]

Smooth cobordism triad for Morse theory requires $\partial W = M_{0} ⊔ M_{1}$ .

[F2]

Morse function adapted to a cobordism uses completeness on a boundaryless collar extension.

[F3]

Handle decomposition relative to the incoming boundary permits the empty list, presenting the incoming collar.

Proof

Given: The compact boundaryless $M$ and its cylinder.

1.1F1F2givenconstruct

The product is a smooth manifold with exactly the two boundary faces. Its projection has differential $d t \neq 0$ , endpoint fibres exactly $M_{0}, M_{1}$ , and no critical points. The field $X = - \partial_{t}$ is descending, outward at $M_{0}$ and inward at $M_{1}$ ; the complete translation field on $M \times R$ restricts to $X$ . Thus the pair is adapted and excellence is vacuous.

2.1F3step 1.1construct∎

The map $(x, t) \mapsto ((x, 0), t)$ identifies $W$ with the incoming collar, fixing $M_{0}$ . Rescaling the interval gives any positive collar length in [F3], so the empty handle list presents $W$ . The same formulas give the empty presentation when $M = \emptyset$ . A product with $\partial M \neq \emptyset$ has an additional side face and is outside this triad convention.

Depends on

Used by

Surgery trace cobordism Definition
A product cobordism is an h-cobordism Example
Simply connected h-cobordisms have zero Whitehead obstruction Example
The relative handle decomposition of a cylinder Example
p-surgery kills the represented pi-p class below the middle dimension Lemma
Product h-cobordisms have zero Whitehead torsion Lemma
The homology effect of surgery away from the middle dimensions Proposition
Handle decompositions are not canonical Remark
A cobordism with no handles is a product Theorem
The upper boundary of the surgery trace is the surgered manifold Theorem

Dependency tree · two levels

33 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

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)