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Smooth cobordism triad for Morse theory

Definition

A smooth cobordism triad $(W; M_{0}, M_{1})$ consists of the following data.

A compact smooth $n$ -manifold with boundary $W$ (Topological manifolds with boundary, Smooth charts, atlases, and structures with boundary, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).
For $n \geq 1$ , two closed embedded smooth $(n - 1)$ -submanifolds $M_{0}, M_{1} \subseteq \partial W$ with $\partial W = M_{0} ⊔ M_{1}$ (Interior and boundary of a manifold with boundary, The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold, Embedded smooth submanifolds with boundary).
Fixed collars of both faces in $W$ (Smooth collars of a manifold boundary), stipulated as part of the data. Under ${A C}_{ω}$ , their existence is guaranteed by Collar neighborhood theorem; the definition itself makes no unconditional collar-existence assertion.

For $n = 0$ , the convention is $\partial W = M_{0} = M_{1} = \emptyset$ : a zero-dimensional manifold has discrete point charts and empty boundary. The two collar domains are empty and their unique maps supply the collar data. No manifold of dimension $- 1$ is introduced. Reversal retains the same empty faces.

The two faces are the incoming face $M_{0}$ and the outgoing face $M_{1}$ ; they are disjoint by the decomposition $\partial W = M_{0} ⊔ M_{1}$ fixed by the data. Each face is a union of boundary components and need not be connected. Either face may be empty: $\partial W = \emptyset$ with $M_{0} = M_{1} = \emptyset$ is allowed, and so is $M_{0} = \emptyset$ , $M_{1} = \partial W$ . All smooth maps between manifolds with boundary are the ones of Smooth maps between manifolds with boundary, and all diffeomorphisms below are diffeomorphisms of that category.

The reversed triad of $(W; M_{0}, M_{1})$ is $(W; M_{1}, M_{0})$ : it carries the same manifold $W$ with the two faces exchanged and with the same fixed collars, read with the opposite roles.

No orientation is required, and an orientation, when present, is extra structure: no statement on this page uses one unless it is listed among the hypotheses.

Depends on

Used by

h-Cobordism Definition
Handle decomposition relative to the incoming boundary Definition
Morse function adapted to a cobordism Definition
Surgery trace cobordism Definition
The based handle chain complex over the fundamental group ring Definition
Relative Morse homology of a single-handle cobordism Example
The relative handle decomposition of a cylinder Example
Boundary product function on a collared cobordism Lemma
Compactified unstable manifolds give the Morse--Smale CW decomposition Lemma
Duality eliminates the top and codimension-one handles Lemma
Interior slab handle attachment Lemma
Open manifolds admit handle filtrations without top-index handles Lemma
Product cobordisms have critical-point-free presentations Lemma
Dual elimination of top-index handles Proposition
Relative Morse inequalities for a cobordism Proposition
The relative Morse complex of an adapted cobordism Proposition
Adapted excellent Morse functions exist on compact cobordisms Theorem
Handle duality from negating a Morse function Theorem
Smale–Hirsch for open source manifolds Theorem
The upper boundary of the surgery trace is the surgered manifold Theorem

Dependency tree · two levels

27 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)