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Handle decomposition relative to the incoming boundary

Definition

Let $(W; M_{0}, M_{1})$ be a smooth cobordism triad with its fixed collars (Smooth cobordism triad for Morse theory). A finite handle decomposition of $(W; M_{0}, M_{1})$ relative to $M_{0}$ is a finite ordered list of indices $k_{1}, \dots, k_{r}$ , together with, for each $i$ , an embedding of the attaching region $S^{k_{i} - 1} \times D^{n - k_{i}}$ of a standard $k_{i}$ -handle (K handle core cocore attaching region and belt sphere) into the outgoing boundary of the manifold built so far (away from the retained $M_{0}$ ), 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 in the given order with corners rounded (Attaching a smooth handle with corner rounding).

The stages of the decomposition are the manifolds $W_{0} : = M_{0} \times [0, ε]$ and $W_{i}$ , the result of attaching the first $i$ handles; the last stage $W_{r}$ is diffeomorphic to $W$ relative to $M_{0}$ . The ordered indices are the indices of the handles; the number of handles of index $k$ is the multiplicity of $k$ in the list.

The endpoint cases are part of the definition, not exceptions.

If $M_{0} = \emptyset$ the initial stage is the empty manifold, and the first handle of any presentation whose later stages are nonempty is a $0$ -handle, which is a disjoint copy of $D^{n}$ attached along $S^{- 1} \times D^{n} = \emptyset$ .
A $k$ -handle with $k = 0$ is a disjoint $n$ -disk; a $k$ -handle with $k = n$ attaches along its whole boundary sphere $S^{n - 1} \times D^{0} = S^{n - 1}$ and has empty outgoing region.
The empty list $r = 0$ is allowed: it presents the collar $M_{0} \times [0, ε]$ itself, and, when $M_{0} = \emptyset$ , the empty manifold.

The definition fixes the geometric data only up to the corner-rounding convention of the cited attachment definition. Existence, uniqueness up to diffeomorphism and the correspondence with Morse functions are not asserted here; they are the content of later items of this page.

Depends on

Used by

Attaching-belt intersection matrix of adjacent-index handles Definition
Dual handle decomposition Definition
Geometrically cancelling adjacent handle pair Definition
The based handle chain complex over the fundamental group ring Definition
A created cancelling pair contributes a (1+t)tᵏ term Example
An elementary cancelling handle pair gives a product cobordism Example
An empty incoming boundary requires zero handles Example
Relative Morse homology of a single-handle cobordism Example
Reordering independent one-handles Example
The relative handle decomposition of a cylinder Example
A handle decomposition gives a relative CW complex Lemma
Compactified unstable manifolds give the Morse--Smale CW decomposition Lemma
Elimination lemma: trading a handle for a handle two indices higher Lemma
Gluing handle Morse models along collars Lemma
Handles of equal index can be attached on one level Lemma
Middle-handle pairs with one geometric intersection cancel Lemma
Open manifolds admit handle filtrations without top-index handles Lemma
Product cobordisms have critical-point-free presentations Lemma
Connected cobordisms admit presentations without superfluous zero handles Proposition
h-Cobordisms admit adapted ordered handle decompositions Proposition
The handle chain complex computes singular homology Proposition
The relative handle chain complex computes H_*(W,M₀) and has the intersection matrix as its differential Proposition
The relative Morse complex of an adapted cobordism Proposition
A cobordism with no handles is a product Theorem
Creation of a cancelling handle pair Theorem
Handle cancellation Theorem
Morse functions and handle decompositions correspond Theorem
Smale–Hirsch for open source manifolds Theorem

Dependency tree · two levels

12 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

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)