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Duality eliminates the top and codimension-one handles

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). In the situation of the previous lemma, $W$ also admits a handle decomposition relative to $M_{0}$ with no handles of index $n$ or $n + 1$ ; combining the two results, $W$ admits a presentation relative to $M_{0}$ in which every handle has index between $2$ and $n - 1$ . Both statements are obtained from the low-index elimination by running it on the reversed triad $(W; M_{1}, M_{0})$ and dualising (h-Cobordism).

Facts & Assumptions

Given: An h-cobordism $(W; M_{0}, M_{1})$ with $\dim W = n + 1$ , $n \geq 5$ , closed simply connected faces and connected $W$ ; ${A C}_{ω}$ .

[F1]

The reversed triad $(W; M_{1}, M_{0})$ of an h-cobordism is again a compact smooth cobordism triad with collared boundary, the same manifold with the face labels exchanged (Smooth cobordism triad for Morse theory), and it is again an h-cobordism because the definition is symmetric in the two faces (h-Cobordism).

[F2]

Zero- and one-handles can be eliminated: a connected h-cobordism of dimension $n + 1$ , $n \geq 5$ , with closed simply connected faces admits a presentation relative to its incoming boundary with no handles of index $0$ or $1$ (Zero- and one-handles are eliminated in a simply connected h-cobordism).

[F3]

Negating an adapted Morse function produces the dual handle decomposition, in which a $k$ -handle of the reversed presentation becomes an $(n + 1 - k)$ -handle of a presentation relative to $M_{0}$ , with attaching and belt spheres interchanged and the index ordered reversed (Handle duality from negating a Morse function, Dual handle decomposition).

[F4]

The low-index procedure removes only $0$ - and $1$ -handles and introduces only $3$ -handles; it preserves any upper index bound at least three (Zero- and one-handles are eliminated in a simply connected h-cobordism, proof step 5.1).

Proof

technique · direct

1.1F1F2given

By [F1] the reversed triad $(W; M_{1}, M_{0})$ is again an h-cobordism with the same dimension, and its faces are the same closed simply connected $n$ -manifolds in the opposite order while $W$ remains connected; hence the low-index elimination [F2] applies to the reversed triad and gives a presentation of $W$ relative to $M_{1}$ with no handles of index $0$ or $1$ .

2.1F3step 1.1

Negate the Morse function of the reversed presentation and dualise by [F3]: a handle of index $k$ in the presentation relative to $M_{1}$ becomes a handle of index $n + 1 - k$ in a presentation relative to $M_{0}$ , with attaching and belt spheres interchanged. Absent indices $0$ and $1$ therefore become absent indices $n + 1$ and $n$ , so the dual presentation relative to $M_{0}$ has no handles of index $n + 1$ or $n$ .

3.1F2F3F4step 2.1given∎

To obtain the two exclusions simultaneously, first eliminate $0, 1$ relative to $M_{0}$ by [F2]. Its reversed presentation has every index at most $n - 1$ . Apply the actual low-index procedure to that presentation relative to $M_{1}$ : by [F4] it deletes $0, 1$ and introduces only $3$ , so its maximum stays at most $n - 1$ , because $n \geq 5$ . Reverse back by [F3]. The resulting original indices lie between $2$ and $n - 1$ ; both exclusions now hold in one presentation. No composition of two independently chosen presentations or face-fixing diffeomorphisms is being assumed.

Depends on

Used by

Dependency tree · two levels

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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)