Smooth cobordism triad for Morse theory

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 $n\ge 1$, two closed embedded smooth $(n-1)$-submanifolds $M_0,M_1\subseteq \mathrm{\partial }W$ with $\mathrm{\partial }W=M_0\bigsqcup M_1$ (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 ${\mathrm{A}\mathrm{C}}_{\omega }$, their existence is guaranteed by Collar neighborhood theorem; the definition itself makes no unconditional collar-existence assertion.

For $n=0$, the convention is $\mathrm{\partial }W=M_0=M_1=\mathrm{\varnothing }$: 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 $\mathrm{\partial }W=M_0\bigsqcup M_1$ fixed by the data. Each face is a union of boundary components and need not be connected. Either face may be empty: $\mathrm{\partial }W=\mathrm{\varnothing }$ with $M_0=M_1=\mathrm{\varnothing }$ is allowed, and so is $M_0=\mathrm{\varnothing }$, $M_1=\mathrm{\partial }W$. 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.