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Elimination lemma: trading a handle for a handle two indices higher

Statement

Assume ${A C}_{ω}$ . Let $W$ be a compact smooth $(n + 1)$ -manifold with a finite index-ordered handle decomposition relative to its incoming face, all indices at least $q$ , where $1 \leq q \leq n - 2$ . Put $N_{j} = \partial_{1} W_{j}$ , and let $N_{q}^{\circ}$ be the common part of $N_{q}$ and $N_{q + 1}$ obtained by deleting the closed attaching tubes of the existing $(q + 1)$ -handles. Fix a $q$ -handle $e$ and a framed attaching embedding $α : S^{q} \times D^{n - q} ↪ N_{q}^{\circ}$ . Suppose:

in $N_{q}$ , $α$ is isotopic through framed attaching embeddings to one whose core meets the belt of $e$ transversely once and misses every other $q$ -handle belt;
in $N_{q + 1}$ , $α$ is isotopic through framed attaching embeddings to a standard trivial embedding, the composite of the standard framed sphere $S^{q} \times D^{n - q} ↪ D^{n}$ with an embedded $n$ -disk in $N_{q + 1}$ .

Then a presentation of the same $W$ relative to its incoming face is obtained by deleting $e$ and adding one $(q + 2)$ -handle, with every other handle index and number unchanged. The diffeomorphism transports later attaching data. Core-sphere nullhomotopy alone does not specify the second framed hypothesis.

Facts & Assumptions

Given: The finite ordered presentation, $e$ , the common-part attaching embedding and its two framed isotopies; countable choice.

[F1]

A standard trivial attachment can be completed to a cancelling pair of adjacent indices in an outgoing disk. Creation of a cancelling handle pair

[F2]

Isotopies of the full attaching regions preserve the attachment and transport later data; stationary reparametrization of the time interval allows the cited endpoint convention. Isotopic attaching embeddings give diffeomorphic handle attachments, Isotopy extension for a compact source with boundary

[F3]

Disjoint equal-index attaching regions may be reordered. Handles of equal index can be attached on one level

[F4]

A consecutive adjacent-index pair with one transverse attaching/belt intersection cancels, transporting later data. Geometrically cancelling adjacent handle pair, Handle cancellation

Proof

technique · direct

1.1F1F2givenconstruct

Temporarily stop the presentation at $W_{q + 1}$ . At the standard trivial embedding of hypothesis 2 introduce a $(q + 1), (q + 2)$ pair by [F1]. Use that framed isotopy and [F2] to move the new lower attachment to $α$ in $N_{q + 1}$ , carrying the new upper attachment along. Restore all higher handles by transporting their attaching data. If a stage is disconnected, perform the construction on its affected connected component and leave the others fixed.

2.1F2F3step 1.1given

Because the entire image of $α$ lies in the common part $N_{q}^{\circ}$ , the new $(q + 1)$ -handle can instead be attached at $N_{q}$ , before all old $(q + 1)$ -handles: their disjoint attaching-region quotient is the same whichever is attached first. Move $e$ to the end of its $q$ -index family by [F3]. Apply the framed isotopy of hypothesis 1 to the new upper attachment in the resulting $N_{q}$ , carrying all later attaching data by [F2]. Now $e$ and that new $(q + 1)$ -handle are consecutive and meet once.

3.1F4step 2.1∎

Cancel this consecutive pair by [F4]. The deleted handles are $e$ and the newly introduced $(q + 1)$ -handle; the new $(q + 2)$ -handle survives, and every old higher handle is carried by the cancellation diffeomorphism. Thus the counts change exactly as asserted, relative to the incoming face. This is the two-isotopy and disjoint-reordering proof of the cited Lück Elimination Lemma, without an isotopy-to-slide decomposition premise.

Depends on

Used by

Dependency tree · two levels

49 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

Wolfgang Lück, A Basic Introduction to Surgery Theory (ICTP lecture notes, 27 October 2004; complete author text) (standard reference, not scraped)
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)