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The smooth s-cobordism theorem: a vanishing presentation implies a product

Statement

Assume ${A C}_{ω}$ . Let $(W; M_{0}, M_{1})$ be a nonempty connected oriented smooth h-cobordism of dimension $n + 1 \geq 6$ , with $M_{0}$ a closed connected oriented smooth $n$ -manifold and $π = π_{1} (M_{0})$ . Then $W$ is diffeomorphic to $M_{0} \times [0, 1]$ relative to $M_{0}$ if and only if there exists a finite handle presentation $H$ of $(W, M_{0})$ with $τ_{H} (W, M_{0}) = 0$ . The forward direction uses the empty height-function presentation of the product; the reverse direction is the vanishing-torsion sufficiency theorem. The class is indexed by its presentation, and this equivalence makes no claim that different presentations have equal torsion. The dimension hypothesis is $n \geq 5$ (equivalently $\dim W \geq 6$ ); nothing is asserted in boundary dimension four, and no orientation-free strengthening is claimed.

Facts & Assumptions

Given: A nonempty connected oriented smooth h-cobordism $(W; M_{0}, M_{1})$ of dimension $n + 1 \geq 6$ with $M_{0}$ closed connected oriented and $π = π_{1} (M_{0})$ .

[F1]

If $W$ is a product $M_{0} \times [0, 1]$ , the height function presents it relative to $M_{0} \times {0}$ with no handles, and the based handle complex is the zero complex, so its presentation-indexed torsion vanishes (Product h-cobordisms have zero Whitehead torsion, Presentation-indexed Whitehead torsion of an h-cobordism).

[F2]

Conversely, if some finite handle presentation $H$ of $(W, M_{0})$ has $τ_{H} (W, M_{0}) = 0$ , then $W$ is diffeomorphic to $M_{0} \times [0, 1]$ relative to $M_{0}$ by the vanishing-torsion sufficiency theorem (Vanishing presentation-indexed torsion implies the product cobordism, h-cobordisms admit two-index normal form presentations, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F3]

The presentation-indexed torsion is an element of the Whitehead group $Wh (π)$ attached to the chosen presentation, and no equality of classes from different presentations is asserted anywhere (Presentation-indexed Whitehead torsion of an h-cobordism, The based handle chain complex over the fundamental group ring, K₁ of a ring and the Whitehead group of a discrete group, h-Cobordism).

Proof

1.1F1given

Assume first that $W$ is diffeomorphic to $M_{0} \times [0, 1]$ relative to $M_{0}$ . Then the product's height-function presentation $H_{0}$ has no handles, and by [F1] its based handle complex is the zero complex with vanishing contraction torsion, so $τ_{H_{0}} (W, M_{0}) = 0$ ; this exhibits the required presentation and proves the forward implication.

1.2F2given

Assume conversely that some finite presentation $H$ has $τ_{H} (W, M_{0}) = 0$ . Then by [F2] the h-cobordism is diffeomorphic to $M_{0} \times [0, 1]$ relative to $M_{0}$ , which proves the reverse implication.

2.1F3step 1.1step 1.2∎

Steps 1.1 and 1.2 prove the two implications of the stated equivalence; by [F3] each side refers to a presentation-indexed class, so the theorem neither asserts nor uses equality of classes attached to different presentations, and the dimension hypothesis $n + 1 \geq 6$ , i.e. $n \geq 5$ , is the one under which both directions were established.

Depends on

Used by

Simply connected h-cobordisms have zero Whitehead obstruction Example

Dependency tree · two levels

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Sources

Wolfgang Lück, A Basic Introduction to Surgery Theory (ICTP lecture notes, 27 October 2004; complete author text) (standard reference, not scraped)
Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, electronic edition) (standard reference, not scraped)