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A cobordism with no handles is a product

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $(W; M_{0}, M_{1})$ be a compact smooth triad with $\dim W = n + 1$ admitting a handle decomposition relative to $M_{0}$ with empty handle list (Handle decomposition relative to the incoming boundary). Equivalently, $W$ admits an adapted Morse function without critical points (equivalently, $W$ is diffeomorphic to a collar $M_{0} \times [0, ε]$ ) (Morse function adapted to a cobordism). Then $W$ is diffeomorphic to $M_{0} \times [0, 1]$ relative to $M_{0}$ ; that is, there is a diffeomorphism $W \to M_{0} \times [0, 1]$ whose restriction to $M_{0}$ is the identity. Conversely a product cobordism has an empty presentation.

Facts & Assumptions

Given: A compact smooth triad $(W; M_{0}, M_{1})$ with a handle decomposition relative to $M_{0}$ whose handle list is empty; ${A C}_{ω}$ .

[F1]

A finite handle decomposition of a triad $(W; M_{0}, M_{1})$ relative to $M_{0}$ is a finite ordered list of indices together with embeddings of attaching regions such that $W$ is diffeomorphic, relative to $M_{0}$ , to the manifold obtained from the collar $M_{0} \times [0, ε]$ by successively attaching the handles with corners rounded; with an empty list no handle is attached (Handle decomposition relative to the incoming boundary).

[F2]

Every finite handle decomposition of $W$ relative to $M_{0}$ is induced by an adapted excellent Morse function with one critical point per handle of the same index, and conversely every adapted excellent Morse function induces such a decomposition (Morse functions and handle decompositions correspond, Morse function adapted to a cobordism).

[F3]

Under the compact regular closed-band hypothesis the normalized flow crosses the band in controlled time and gives a level-preserving diffeomorphism $T : M_{a} \times [a, b] \to K$ , $T (x, t) = Φ_{t - a} (x)$ , together with a strong deformation retraction of the upper sublevel onto the lower one and a diffeomorphism of the two sublevels (Regular interval diffeomorphism, Normalized gradient crosses a compact regular band in controlled time, Deformation lemma for a critical point free slab, Regular sublevels are diffeomorphic).

[F4]

For a compact boundaryless smooth manifold $M$ the product $M \times [0, 1]$ has the empty handle decomposition relative to $M \times {0}$ and its projection is an adapted Morse function without critical points (Product cobordisms have critical-point-free presentations).

Proof

technique · direct

1.1F1given

By [F1] applied to the empty list, $W$ is diffeomorphic relative to $M_{0}$ to the collar $M_{0} \times [0, ε]$ ; composing with the reparametrization $M_{0} \times [0, ε] \to M_{0} \times [0, 1]$ , $(x, t) \mapsto (x, t / ε)$ , which fixes $M_{0} \times {0}$ pointwise, gives a diffeomorphism $W \to M_{0} \times [0, 1]$ whose restriction to $M_{0}$ is the identity.

1.2F2F4given

Conversely let $W$ be a product cobordism; by [F4] its projection is an adapted Morse function without critical points and $W$ carries the empty handle list relative to $M_{0}$ , which also shows that the empty-presentation formulation and the critical-point-free Morse formulation describe the same triads by [F2].

2.1F2F3step 1.2construct

For the critical-point-free formulation, append signed collars to obtain a boundaryless neighborhood of $W$ . Smoothness up to the boundary gives local extensions of $f$ , and Smooth partitions of unity exist on manifolds patches them to a smooth extension near $W$ , equal to $f$ there. Compactness and $d f \neq 0$ give a smaller neighborhood on which the extension still has nonzero differential. Choose a metric by Every smooth manifold admits a riemannian metric and multiply its normalized ascending gradient by a compactly supported cutoff equal to one near $W$ (A manifold bump for a compact set inside an open set). Its ambient flow $Φ$ is complete by Compactly supported smooth vector fields are complete, and $d f (Φ_{t}^{'}) = 1$ while the trajectory is in $W$ . At $M_{0}$ it enters $W$ and at $M_{1}$ it exits; a first exit before the prescribed value would be at neither boundary fiber, which is impossible. Thus $T (x, t) = Φ_{t} (x)$ maps $M_{0} \times [0, 1]$ onto $W$ , with inverse $y \mapsto (Φ_{- f (y)} (y), f (y))$ . Flow uniqueness and smooth dependence, including the signed collars, make these inverse diffeomorphisms (The fundamental theorem on flows). This proves the boundary version directly; it does not apply a boundaryless closed-band theorem to $W$ without an extension.

3.1F1F2F4step 1.1step 1.2step 2.1∎

Combining step 1.1 with steps 1.2 and 2.1 proves both directions and the claimed equivalence: the empty presentation forces $W ≅ M_{0} \times [0, 1]$ relative to $M_{0}$ , and a product cobordism has an empty presentation. This is Milnor's product theorem for a critical-point-free slab.

Depends on

Used by

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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)