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High-dimensional simply connected h-cobordant manifolds are diffeomorphic

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $M_{0}, M_{1}$ be closed simply connected smooth $n$ -manifolds with $n \geq 5$ . If $M_{0}$ and $M_{1}$ are h-cobordant, i.e. if there is a compact smooth h-cobordism $(W; M_{0}, M_{1})$ with $\dim W = n + 1$ (h-Cobordism), then $M_{0}$ and $M_{1}$ are diffeomorphic.

Facts & Assumptions

Given: Countable choice and closed simply connected smooth $n$ -manifolds $M_{0}, M_{1}$ with $n \geq 5$ and a compact smooth h-cobordism $(W; M_{0}, M_{1})$ with $\dim W = n + 1$ .

[L1]

The h-cobordism theorem gives a diffeomorphism $F : W \to M_{0} \times [0, 1]$ whose restriction to $M_{0}$ is the identity (The smooth simply connected h-cobordism theorem).

[L2]

A diffeomorphism of smooth manifolds restricts to a diffeomorphism between corresponding boundary components, and the map $M_{0} \to M_{0} \times {1}$ , $x \mapsto (x, 1)$ , is a diffeomorphism of smooth manifolds (Diffeomorphisms and local diffeomorphisms of manifolds, Smooth embeddings).

Proof

technique · direct

1.1L1given

$W$ is connected because its incoming inclusion is a homotopy equivalence from the connected $M_{0}$ ; the inverse homotopies connect every point of $W$ to that face. Thus every hypothesis of [L1], including countable choice, holds: there is a diffeomorphism $F : W \to M_{0} \times [0, 1]$ that is the identity on $M_{0}$ ; since $F$ maps the boundary of $W$ to the boundary of $M_{0} \times [0, 1]$ and already maps $M_{0}$ onto $M_{0} \times {0}$ , it maps the other face $M_{1}$ diffeomorphically onto the complementary face $M_{0} \times {1}$ .

2.1L2step 1.1∎

The restriction $F ∣_{M_{1}} : M_{1} \to M_{0} \times {1}$ is therefore a diffeomorphism, and the second projection $(x, 1) \mapsto x$ is a diffeomorphism $M_{0} \times {1} \to M_{0}$ by [L2]; composing gives a diffeomorphism $M_{1} \to M_{0}$ , so the two h-cobordant manifolds are diffeomorphic.

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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)