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Dual elimination of top-index handles

Statement

Assume ${A C}_{ω}$ . Let $(W; M_{0}, M_{1})$ be a compact connected triad with $M_{1} \neq \emptyset$ and $\dim W = n$ . Then $W$ admits a handle decomposition relative to $M_{0}$ with no $n$ -handles; if $M_{1}$ is disconnected the dual presentation ends with the corresponding dual $(n - 1)$ -handles coming from the connecting $1$ -handles in the reversed triad, and still has no $n$ -handles. Equivalently, the $n$ -handles of a presentation relative to $M_{0}$ are the duals of the $0$ -handles in the reversed triad, so eliminating those $0$ -handles eliminates these $n$ -handles.

Facts & Assumptions

[F1]

Smooth cobordism triad for Morse theory: the reversed triad of $(W; M_{0}, M_{1})$ is $(W; M_{1}, M_{0})$ , a compact triad with the same collars and the faces exchanged; no orientation is used.

[F2]

Connected cobordisms admit presentations without superfluous zero handles: Assume ${A C}_{ω}$ . A compact connected triad whose incoming boundary is nonempty admits a handle decomposition relative to that boundary with no $0$ -handles; if that incoming boundary has $k$ components the presentation begins with $k - 1$ connecting $1$ -handles; if the incoming boundary is empty, exactly the $0$ -handles needed to create the components remain.

[F3]

Morse functions and handle decompositions correspond: Assume ${A C}_{ω}$ . Every finite handle decomposition of a compact triad relative to its incoming face is induced by an adapted excellent Morse function with one critical point per handle, of the same index.

[F4]

Handle duality from negating a Morse function: Assume ${A C}_{ω}$ . If $f$ is adapted excellent on a compact triad then $1 - f$ is adapted excellent on the reversed triad with indices $n - ind (p)$ at the same critical points, and its handle decomposition relative to the opposite face is the dual of the decomposition of $f$ .

[F5]

Dual handle decomposition: in the dual presentation a $k$ -handle becomes an $(n - k)$ -handle, the order is reversed, and attaching and belt spheres are interchanged.

[F6]

Index n handles cap boundary spheres: an $n$ -handle attaches along its whole boundary sphere $S^{n - 1}$ and caps it; a $0$ -handle attaches along the empty set and creates a component.

[F7]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ : every at most countable family of nonempty sets has a choice function.

[F8]

Under ${A C}_{ω}$ , Every smooth manifold admits a riemannian metric supplies a background metric, Morse lemma supplies the quadratic critical charts, A manifold bump for a compact set inside an open set supplies finite chart cutoffs, and Compactly supported smooth vector fields are complete makes a compactly supported smooth field on a boundaryless carrier complete.

Proof

Given: The compact connected triad $(W; M_{0}, M_{1})$ with $\dim W = n$ and $M_{1} \neq \emptyset$ .

1.1F1F2givenconstruct

The reversed triad $(W; M_{1}, M_{0})$ of [F1] is compact and connected, its incoming face is $M_{1} \neq \emptyset$ , and its outgoing face is $M_{0}$ , which may be empty. By [F2] applied to the reversed triad, $W$ admits a handle decomposition relative to $M_{1}$ with no $0$ -handles; if $M_{1}$ has $k$ components, that presentation begins with $k - 1$ connecting $1$ -handles. Fix this chosen presentation for the dual construction.

2.1F3F4F5F7F8step 1.1construct

Realize the presentation by a Morse function: by [F3] the decomposition of step 1.1 is induced by an adapted excellent Morse function $f$ on the reversed triad $(W; M_{1}, M_{0})$ . To supply the field required by duality, patch the background metric of [F8] to Euclidean metrics in smaller disjoint Morse charts and to product metrics on the realizing function's regular face collars, using finite cutoffs. Its negative gradient has the exact model $(2 u, - 2 v)$ and the required boundary signs. Extend the product collar field across signed face collars, with a cutoff vanishing before their outer ends. The resulting ambient field is compactly supported, hence complete by [F8], and restricts to an adapted field for $f$ . Applying [F4] to that pair, the function $1 - f$ is adapted excellent on the original triad $(W; M_{0}, M_{1})$ , with the same critical points and with indices transformed by $k \mapsto n - k$ , and its handle decomposition relative to $M_{0}$ is the dual of the decomposition of $f$ , in the sense of [F5].

3.1F5step 1.1step 2.1algebra

Since the presentation of step 1.1 has no $0$ -handles, its dual presentation has no $n$ -handles, because a $0$ -handle becomes an $n$ -handle under $k \mapsto n - k$ by [F5]. The connecting $1$ -handles of step 1.1, which join the $k$ components of $M_{1}$ when $M_{1}$ is disconnected, become handles of index $n - 1$ in the dual presentation, again by [F5], and they come last because the order is reversed. Duality bijects the handles and complements their indices, so the resulting presentation of $W$ relative to $M_{0}$ has the same number of $n$ -handles as the reversed presentation has $0$ -handles, namely zero.

4.1F2F5F6step 3.1algebra∎

Equivalently, the $n$ -handles of any presentation relative to $M_{0}$ are the duals of the $0$ -handles of the reversed presentation: a $0$ -handle is an $n$ -disk attached along the empty set [F6], and its dual is an $n$ -handle attached along the whole boundary sphere [F6], so eliminating the $0$ -handles of the reversed presentation by [F2] eliminates exactly the $n$ -handles of the dual presentation relative to $M_{0}$ . This is the dual endpoint elimination; the argument uses the duality, correspondence and elimination suppliers, and through them ${A C}_{ω}$ .

Depends on

Used by

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Sources

C. T. C. Wall, Differential Topology (Cambridge Studies in Advanced Mathematics 156), Sections 5.1-5.4, printed pp. 129-148 (standard reference, not scraped)
John Milnor, Lectures on the h-Cobordism Theorem (notes by L. Siebenmann and J. Sondow), Sections 2-4, printed pp. 10-48 (standard reference, not scraped)