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The h-cobordism theorem when the Whitehead group vanishes

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 $M_{0}$ closed connected oriented and let $π = π_{1} (M_{0})$ . If $Wh (π) = 0$ — for instance if $π$ is trivial — then every presentation-indexed class $τ_{H} (W, M_{0})$ vanishes, and $W$ is diffeomorphic to $M_{0} \times [0, 1]$ relative to $M_{0}$ . In particular the classical simply connected h-cobordism theorem is the case $π = 1$ .

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 $π = π_{1} (M_{0})$ , and the hypothesis $Wh (π) = 0$ .

[F1]

Every finite handle presentation of $(W, M_{0})$ has a well-defined contraction torsion of its based handle complex, and hence a presentation-indexed class $τ_{H} (W, M_{0}) \in Wh (π)$ (Presentation-indexed Whitehead torsion of an h-cobordism, K₁ of a ring and the Whitehead group of a discrete group, h-Cobordism).

[F2]

A finite handle presentation of $(W, M_{0})$ exists: the two-index normal-form lemma applies to the oriented data of the statement and produces a handle decomposition of $W$ relative to $M_{0}$ with handles only in degrees $q$ and $q + 1$ for every $2 \leq q \leq n - 2$ , under the countable-choice hypothesis ${A C}_{ω}$ assumed here (h-cobordisms admit two-index normal form presentations, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F3]

A presentation with vanishing presentation-indexed torsion gives a product structure relative to $M_{0}$ , and the vanishing-torsion sufficiency theorem applies to any finite presentation (Vanishing presentation-indexed torsion implies the product cobordism).

[F4]

The trivial group has vanishing Whitehead group: $Z [1] = Z$ ; the determinant is a well-defined surjection $K_{1} (Z) \to {\pm 1}$ because every elementary matrix has determinant $1$ and the classes of the $1 \times 1$ matrices $(\pm 1)$ occur; and it is injective, since a common divisor of the entries of a column of an invertible integer matrix divides the determinant $\pm 1$ , the division algorithm with the Bézout identity reduces such a primitive column to $(\pm 1, 0, \dots, 0)^{T}$ by elementary row additions and swaps, and induction on the size then presents every invertible integer matrix, up to permutation, as elementarily equivalent to a diagonal matrix with entries $\pm 1$ , whose class is a sum of classes $[\pm 1]$ ; hence $K_{1} (Z) ≅ {\pm 1}$ and $Wh (1) = K_{1} (Z) / ⟨ [\pm 1] ⟩ = 0$ (K₁ of a ring and the Whitehead group of a discrete group, Division with remainder in $Z$ : for $a \in Z$ and $b > 0$ there are unique $q, r \in Z$ with $a = q b + r$ and $0 \leq r < b$ , Bézout's identity: for integers $a, b$ not both zero, $\gcd (a, b)$ is the least positive element of ${a x + b y : x, y \in Z}$ ; in particular $a x + b y = \gcd (a, b)$ has an integer solution).

Proof

1.1F1F2given

Take a finite handle presentation $H$ of $(W, M_{0})$ , which exists by [F2]; by [F1] its presentation-indexed class $τ_{H} (W, M_{0})$ is an element of $Wh (π)$ , and by hypothesis $Wh (π) = 0$ , so $τ_{H} (W, M_{0}) = 0$ automatically, without any independence of presentations being needed.

2.1F3step 1.1

Since the chosen presentation has vanishing presentation-indexed torsion, the vanishing-torsion sufficiency theorem of [F3] applies to it and yields that $W$ is diffeomorphic to $M_{0} \times [0, 1]$ relative to $M_{0}$ ; this argument works for every presentation because every presentation's class lies in the zero group.

3.1F4step 2.1∎

The case $π = 1$ is covered because [F4] gives $Wh (1) = 0$ , so the simply connected h-cobordism theorem is the special case of the statement in which the fundamental group is trivial.

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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)