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Middle-handle pairs with one geometric intersection cancel

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $(W; M_{0}, M_{1})$ be a connected simply connected h-cobordism with $\dim W = n + 1 \geq 6$ presented relative to $M_{0}$ with handles only in indices $k, k + 1$ , $2 \leq k \leq n - 2$ , and suppose the attaching sphere of each $(k + 1)$ -handle meets the belt sphere of exactly one $k$ -handle in a single transverse point and is disjoint from the other belt spheres (the configuration produced by the geometric realisation lemma (The Whitney trick realizes algebraic middle-handle cancellation geometrically)). Then $W$ admits a handle decomposition relative to $M_{0}$ with no handles at all: each pair consisting of a $k$ -handle and the $(k + 1)$ -handle meeting its belt sphere once is geometrically cancelling (Geometrically cancelling adjacent handle pair), and the pairs may be deleted one after another, the attaching data of the remaining handles being transported by the relative diffeomorphism.

Facts & Assumptions

Given: A connected simply connected h-cobordism with $\dim W = n + 1 \geq 6$ , presented with handles only in indices $k, k + 1$ , $2 \leq k \leq n - 2$ , with the single-point/disjoint configuration of the realisation lemma; ${A C}_{ω}$ .

[F1]

A consecutive pair is geometrically cancelling when the attaching sphere of the upper handle meets the belt sphere of the lower one transversely in exactly one point (Geometrically cancelling adjacent handle pair).

[F2]

A geometrically cancelling consecutive pair may be deleted by a diffeomorphism relative to the incoming boundary that acts only in a collar of the affected boundary disc and in the two handles, so that the attaching data of all later handles are carried along (Handle cancellation).

[F3]

A handle decomposition relative to $M_{0}$ with empty handle list presents the collar $M_{0} \times [0, ε]$ , and a presentation with no handles left is the empty presentation (Handle decomposition relative to the incoming boundary).

Proof

technique · direct

1.1F1givenalgebra

The two-index relative chain complex is acyclic by The relative handle chain complex computes $H_{*} (W, M_{0})$ and has the intersection matrix as its differential and Relative homology of an h-cobordism vanishes at both ends, so its differential is an isomorphism. Each upper core maps to $\pm$ one lower basis generator by the single-point hypothesis. Surjectivity ensures every lower generator occurs and injectivity ensures none occurs twice. Hence the intersection configuration gives a bijective pairing of the two finite handle families.

2.1F1F2step 1.1construct

Select a matched pair. Reorder the lower handles to put its lower member last and the upper handles to put its upper member first, using Handles of equal index can be attached on one level. They are now consecutive, so [F2] cancels them. Its cancellation model can be supported off the other belt spheres: the selected attaching sphere misses those belts, and a small neighborhood of its attaching data and the chosen lower handle avoids them. Transport the other upper attaching embeddings by the cancellation diffeomorphism; their intersections with the untouched belts stay as prescribed.

3.1F2F3step 2.1∎

Repeat with the remaining finite matched list. Each cancellation removes two handles and preserves the manifold relative to $M_{0}$ . The empty case requires no operation. When no pair remains, [F3] gives the empty presentation, diffeomorphic to $M_{0} \times [0, 1]$ by rescaling its collar coordinate.

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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)