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Group-labelled Whitney tricks realize the diagonalized handle complex

Statement

Assume ${A C}_{ω}$ . Let $(W; M_{0}, M_{1})$ be a nonempty connected smooth h-cobordism of dimension $n + 1 \geq 6$ and let $2 \leq q \leq n - 2$ . Suppose a handle presentation of $(W, M_{0})$ has handles only in degrees $q$ and $q + 1$ , with equal numbers $c$ , and intersection matrix $diag (\pm g_{1}, \dots, \pm g_{c})$ with $g_{i} \in π_{1} (M_{0})$ . Then $W$ admits a handle presentation relative to $M_{0}$ with no handles at all: the attaching embeddings can be isotoped so that the $i$ -th $(q + 1)$ -handle meets exactly the belt sphere of the $i$ -th $q$ -handle, in a single transverse point, and is disjoint from all other belt spheres; the $c$ geometrically cancelling pairs are then removed by the handle cancellation theorem. Consequently $W$ is diffeomorphic to $M_{0} \times [0, 1]$ relative to $M_{0}$ .

Facts & Assumptions

Given: A nonempty connected smooth h-cobordism $(W; M_{0}, M_{1})$ of dimension $n + 1 \geq 6$ , an index $2 \leq q \leq n - 2$ , and a two-index presentation with handles $e_{1}, \dots, e_{c}$ of index $q$ and $f_{1}, \dots, f_{c}$ of index $q + 1$ whose intersection matrix is $diag (\pm g_{1}, \dots, \pm g_{c})$ .

[F1]

For $2 \leq q \leq n - 3$ , the corrected group-labelled homology lemma realizes a unit column by an attaching-sphere isotopy. When $q = 2$ , its incoming fundamental-group injection follows from the h-cobordism condition. The group-labelled homology lemma realizes group-ring handle bases by isotopy, Belt-sphere complements in low handle levels preserve the fundamental group

[F2]

In the based handle complex of a two-index presentation the class of the attaching sphere of the $i$ -th $(q + 1)$ -handle is the $i$ -th column of the intersection matrix, so for a diagonal matrix it is $[φ_{i}] \cdot (\pm g_{i})$ , a unit multiple of the class of the $i$ -th $q$ -handle (The based handle chain complex over the fundamental group ring, The middle-handle intersection matrix of an h-cobordism).

[F3]

Isotoping an attaching embedding changes the presented manifold only up to a diffeomorphism relative to $\partial_{0} W$ , and a pair of a $q$ -handle and a $(q + 1)$ -handle whose attaching sphere meets the belt sphere of the $q$ -handle transversely in exactly one point is geometrically cancelling and can be deleted, with the cancellation diffeomorphism carrying the remaining attaching data along (Isotopic attaching embeddings give diffeomorphic handle attachments, Handle cancellation, Geometrically cancelling adjacent handle pair).

[F4]

A presentation of an h-cobordism relative to $M_{0}$ with no handles exhibits $W$ as diffeomorphic to $M_{0} \times [0, 1]$ relative to $M_{0}$ (A cobordism with no handles is a product, h-Cobordism).

[F5]

At original $q = n - 2$ , the upper handles have index $n - 1$ and become dual $2$ -handles. Their belts are the original attaching spheres, so the reversed h-cobordism belt-complement lemma applies to their union. Handle duality from negating a Morse function, Belt-sphere complements in low handle levels preserve the fundamental group

[F6]

A zero signed group-label sum, or a unit signed label sum with surplus points, contains opposite-sign equal-label pairs. Their two-sheet arc loop is null when labels computed with paths compatible with those arcs agree; this is the orientation-free clause of the label supplier. The signed coefficients are the lifted local core/normal incidence coefficients of [F2], without an assumption of global orientability. A vanishing group-ring coefficient sum pairs off opposite-signed equal labels, The fundamental-group label controls contractibility of the Whitney circle, Arcs joining two points of a connected submanifold avoiding finitely many points

Proof

1.1F1F2F3given

Suppose first $2 \leq q \leq n - 3$ . Each attaching sphere has the unit-column class of [F2]. The incoming fundamental group maps isomorphically to the trace through the $q$ -handles: its later handles have index $q + 1 \geq 3$ and the incoming h-cobordism inclusion is an isomorphism. Thus the $q = 2$ condition of [F1] holds. Apply the homology lemma to each attaching sphere, choosing its disks and tubes disjoint from all other attaching spheres as allowed by $2 + q - n \leq - 1$ . Those spheres and the previously arranged intersections stay fixed. Isotopy transports the attaching framings and [F3] carries the later data without changing $W$ relative to $M_{0}$ .

1.2F5F6given

For $q = n - 2$ , read the same middle level from $M_{1}$ . By [F5] it is the outgoing level of the dual $2$ -handles, whose belt spheres are exactly the original attaching spheres $A_{1}, \dots, A_{c}$ . The reversed h-cobordism supplies the required incoming injection, so their full complement has the same fundamental group as the level. For any surplus intersection pair of $A_{i}$ with an original belt $B_{j} ≅ S^{2}$ , [F6] gives opposite signs and equal labels, hence a null Whitney circle. Choose its arcs to avoid all intersections with the other spheres. Exchange the two sheets and apply the helper's $2$ -sphere construction to the moving $B_{j}$ and fixed $A_{i}$ , with disk interior in the complement of every $A_{l}$ and also avoiding every other $2$ -sphere $B_{l}$ . Equality of labels and opposite signs survive the sheet exchange: the label convention is inverted with common whisker factors, and the orientation interchange factor is $(- 1)^{2 (n - 2)} = 1$ .

2.1F2F3F5F6step 1.2

Let $H_{t}$ be that compactly supported auxiliary ambient isotopy of $B_{j}$ , with the $A_{i}$ comparison fixed. Replace only the attaching sphere $A_{i}$ by $H_{t}^{- 1} (A_{i})$ and keep all original belts fixed. At time one, $H_{1}^{- 1} (A_{i}) \cap B_{j} = H_{1}^{- 1} (A_{i} \cap H_{1} (B_{j}))$ , so precisely the chosen pair disappears. Its tube avoids every other $A_{l}$ and $B_{l}$ , so the other attaching spheres stay disjoint and no other intersection changes. The inverse is an ambient isotopy and transports the attaching frame. Repeat for every surplus pair; the finite signed-label sums leave one point at each diagonal belt and none elsewhere. This is the flipped move in the actual two-index handle context, not a complement assertion about an arbitrary codimension-two sphere.

3.1F3F4step 1.1step 2.1∎

In either case each $q$ -handle and its corresponding $(q + 1)$ -handle is now geometrically cancelling. By [F3] delete the pairs finitely, transporting the remaining attaching data. The empty presentation gives $M_{0} \times [0, 1]$ relative to $M_{0}$ by [F4]. Thus both the original attaching-isotopy conclusion and the full range $2 \leq q \leq n - 2$ are retained.

Depends on

Used by

Vanishing presentation-indexed torsion implies the product cobordism Theorem

Dependency tree · two levels

128 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)
Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, electronic edition) (standard reference, not scraped)