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A product cobordism is an h-cobordism

Example

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $M_{0}$ be a closed smooth $n$ -manifold with $n \geq 1$ and let $W = M_{0} \times [0, 1]$ with faces $M_{0} \times {0}$ and $M_{0} \times {1}$ . Then $(W; M_{0} \times {0}, M_{0} \times {1})$ is an h-cobordism: the maps $W \to M_{0} \times {i}$ , $(x, t) \mapsto (x, i)$ , are retractions and the linear homotopies $(x, t) \mapsto (x, (1 - s) t + s i)$ are deformation retractions fixing $M_{0} \times {i}$ pointwise, so both inclusions are homotopy equivalences. Its handle decomposition relative to $M_{0} \times {0}$ is empty, its relative homology vanishes in all degrees, and if $M_{0}$ is simply connected with $n \geq 5$ the h-cobordism theorem returns exactly this product.

Facts & Assumptions

Given: Countable choice and a closed smooth $n$ -manifold $M_{0}$ with $n \geq 1$ , its product $W = M_{0} \times [0, 1]$ with the two faces $M_{0} \times {0}$ and $M_{0} \times {1}$ , and the inclusions $ι_{0} : M_{0} \times {0} ↪ W$ , $ι_{1} : M_{0} \times {1} ↪ W$ .

[F1]

A retraction of $X$ onto $A$ is a continuous $r : X \to A$ with $r \circ i = {id}_{A}$ , equivalently $r (a) = a$ on $A$ ; $A$ is a deformation retract when in addition there is a homotopy $H : {id}_{X} ≃_{A} i \circ r$ fixing $A$ pointwise (Retractions and deformation retracts, with a deformation retraction required to fix the retract pointwise).

[F2]

A compact smooth cobordism triad is an h-cobordism when both face inclusions are homotopy equivalences (h-Cobordism).

[F3]

For a compact boundaryless smooth manifold $M$ the product $W = M \times [0, 1]$ has the empty handle decomposition relative to $M \times {0}$ : the projection $W \to [0, 1]$ is an adapted Morse function without critical points and $W$ is diffeomorphic to the collar $M_{0} \times [0, 1]$ , no handle being attached (Product cobordisms have critical-point-free presentations).

Verification

technique · direct

1.1F1given

The projections $r_{i} : W \to M_{0} \times {i}$ , $r_{i} (x, t) = (x, i)$ , are continuous and satisfy $r_{i} (ι_{i} (x)) = r_{i} (x, i) = (x, i) = ι_{i} (x)$ for every $x \in M_{0}$ , so by [F1] each $r_{i}$ is a retraction onto the corresponding face.

1.2F3given

Read the product presentation through [F3]: $W$ has the empty handle list relative to $M_{0} \times {0}$ , and no handle is attached.

2.1F1F2step 1.1

The linear homotopy $H_{s} (x, t) = (x, (1 - s) t + s i)$ is continuous with $H_{0} = {id}_{W}$ and $H_{1} = ι_{i} \circ r_{i}$ , and it satisfies $H_{s} (x, i) = (x, i)$ for all $s$ ; hence by [F1] each face is a deformation retract of $W$ , in particular each inclusion is a homotopy equivalence with homotopy inverse $r_{i}$ . The product is a compact smooth triad with its product collars and dimension $n + 1 \geq 2$ , so by [F2] the triad $(W; M_{0} \times {0}, M_{0} \times {1})$ is an h-cobordism.

3.1step 1.2step 2.1∎

The relative homology now vanishes by Relative homology of an h-cobordism vanishes at both ends, applied to step 2.1. Therefore the product cobordism has an empty presentation and vanishing relative homology; when $M_{0}$ is simply connected of dimension $n \geq 5$ , the later h-cobordism theorem applied to this h-cobordism returns precisely the product it started from, so the product is the trivial model that the theorem's conclusion describes.

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Sources

John Milnor, Lectures on the h-Cobordism Theorem (notes by L. Siebenmann and J. Sondow, Princeton University Press 1965; scanned edition with searchable text layer) (standard reference, not scraped)
Wolfgang Lück, A Basic Introduction to Surgery Theory (ICTP lecture notes, 27 October 2004; complete author text) (standard reference, not scraped)