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The smooth simply connected h-cobordism theorem

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $(W; M_{0}, M_{1})$ be an h-cobordism with $\dim W = n + 1 \geq 6$ (equivalently $n \geq 5$ ), where $M_{0}, M_{1}$ are closed simply connected $n$ -manifolds and $W$ is connected (h-Cobordism, Simply connected topological spaces). Then $W$ is diffeomorphic to $M_{0} \times [0, 1]$ relative to $M_{0}$ : there is a diffeomorphism $W \to M_{0} \times [0, 1]$ that is the identity on $M_{0}$ . In particular every h-cobordism over a closed simply connected $n$ -manifold with $n \geq 5$ is trivial. The dimension hypothesis $n \geq 5$ is used in the elimination of one-handles and in the Whitney-trick step; simple connectivity is used in those steps and in the diagonalisation.

Facts & Assumptions

Given: An h-cobordism $(W; M_{0}, M_{1})$ with $\dim W = n + 1 \geq 6$ , closed simply connected faces and connected $W$ ; ${A C}_{ω}$ .

[F1]

Every compact h-cobordism admits an adapted ordered handle presentation relative to $M_{0}$ , with all $k$ -handles attached at one level before all $(k + 1)$ -handles (h-Cobordisms admit adapted ordered handle decompositions).

[F2]

The relative homology of an h-cobordism vanishes in every degree $k \geq 0$ : $H_{k} (W, M_{0}; Z) = 0$ (Relative homology of an h-cobordism vanishes at both ends).

[F3]

Under the dimension and simple connectivity hypotheses, $W$ admits a presentation relative to $M_{0}$ with no handles of index $0$ or $1$ (Zero- and one-handles are eliminated in a simply connected h-cobordism) and, dually, no handles of index $n$ or $n + 1$ (Duality eliminates the top and codimension-one handles).

[F4]

For every admissible $k \in {2, \dots, n - 2}$ the presentation may be concentrated in the two adjacent indices $k, k + 1$ , with the same number $r$ of handles of each index and relative handle complex $0 \to Z^{r} \to Z^{r} \to 0$ (Trading concentrates a simply connected h-cobordism in two adjacent middle indices).

[F5]

The middle-handle intersection matrix of such a two-index presentation is invertible over $Z$ (Acyclicity makes the simply connected middle-handle matrix unimodular), and handle slides, renumbering and reorientation carry it to the identity matrix $I_{r}$ while preserving $W$ relative to $M_{0}$ (Handle slides, renumberings and reorientations reduce a unimodular middle-handle matrix to the identity).

[F6]

In the resulting presentation the Whitney trick upgrades the identity matrix to the geometric single-point configuration (The Whitney trick realizes algebraic middle-handle cancellation geometrically), and then the pairs cancel, leaving an empty presentation (Middle-handle pairs with one geometric intersection cancel).

[F7]

An empty presentation relative to $M_{0}$ is exactly a product: $W$ is diffeomorphic to $M_{0} \times [0, 1]$ relative to $M_{0}$ (A cobordism with no handles is a product).

Proof

technique · direct

1.1F1F2given

By [F1] put the h-cobordism in self-indexed form relative to $M_{0}$ , so that its handles are attached in order of increasing index and $H_{k} (W, M_{0}; Z) = 0$ for all $k$ by [F2]; the relative handle chain complex of this presentation computes $H_{*} (W, M_{0}; Z)$ .

2.1F3step 1.1

Use the simultaneous elimination supplied by the duality lemma in [F3]: its low-eliminate/reverse/low-eliminate/reverse procedure yields one presentation relative to $M_{0}$ with every index between $2$ and $n - 1$ .

3.1F2F4step 2.1

Since $n \geq 5$ there is an admissible index $k$ with $2 \leq k \leq n - 2$ (for instance $k = 2$ , and in higher dimensions any $k$ with $3 \leq k \leq n - 3$ ), and by [F4] the presentation may be concentrated in the two adjacent indices $k, k + 1$ : handles only of index $k$ and $k + 1$ remain, with the same number $r$ of each, and the relative handle complex is $0 \to Z^{r} \to Z^{r} \to 0$ with differential an isomorphism because the homology $H_{*} (W, M_{0}; Z)$ vanishes by [F2].

4.1F5F6step 3.1

By [F5] the middle-handle intersection matrix $M$ of this presentation is invertible over $Z$ , and handle slides, renumbering and reorientation carry it to the identity $I_{r}$ without changing $W$ relative to $M_{0}$ ; by [F6] the Whitney trick then upgrades the algebraic identity to the geometric single-point configuration, in which each $(k + 1)$ -handle meets the belt sphere of exactly one $k$ -handle once and is disjoint from the others, and the resulting pairs are deleted one after another by handle cancellation.

5.1F7givenstep 4.1∎

When all pairs have been cancelled, the presentation of $W$ relative to $M_{0}$ is empty, and by [F7] an empty presentation is exactly a product: there is a diffeomorphism $W \to M_{0} \times [0, 1]$ whose restriction to $M_{0}$ is the identity. Hence every h-cobordism over a closed simply connected $n$ -manifold with $n \geq 5$ is trivial. This is the classical smooth simply connected h-cobordism theorem of Milnor, Lück and Ranicki.

Depends on

Used by

Dependency tree · two levels

84 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

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)
Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, Oxford University Press 2002; complete electronic copy) (standard reference, not scraped)