Morse function adapted to a cobordism

Let $(W;M_0,M_1)$ be a smooth cobordism triad (Smooth cobordism triad for Morse theory) with its fixed collars (Smooth collars of a manifold boundary, Complete vector fields). A smooth function $f:W\to [0,1]$ is adapted to the triad when:

1. $f^{-1}(0)=M_0$ and $f^{-1}(1)=M_1$, and $f$ is constant on each face;
2. every critical point of $f$ is an interior point of $W$ and is nondegenerate, so that $f$ is a Morse function (Morse functions and excellent Morse functions);
3. there is a fixed collar of $\mathrm{\partial }W$ containing no critical point of $f$.

The pair $(f,X)$ is adapted when $f$ is adapted and $X$ is an adapted complete downward gradient-like field for $f$ (Downward gradient-like vector fields for a Morse function) that points outward along $M_0$ and inward along $M_1$ (Inward, outward, and boundary-tangent vectors), so that descending trajectories enter through $M_1$ and can leave only through $M_0$.

Here adapted complete means that $W$ is embedded in a boundaryless smooth collar extension $\widehat{W}$ and $X$ is the restriction of a complete smooth vector field $\widehat{X}$ on $\widehat{W}$ (Complete vector fields). A collar extension is obtained by appending negative collar parameters to the fixed boundary collars. Trajectories in $W$ are the ambient integral curves restricted to the time intervals during which they remain in $W$; they stop at a boundary exit. This does not require $W$ to be invariant under the complete ambient flow. Such invariance would be incompatible with an outward field at a nonempty $M_0$.

The pair is excellent when in addition distinct critical points of $f$ have distinct values.

Convention. The library convention for trajectories is the descending one: $df(X)<0$ off the critical set and $X=(2u,-2v)$ in Morse charts $f=f(p)-\mathrm{\mid }u{\mathrm{\mid }}^2+\mathrm{\mid }v{\mathrm{\mid }}^2$. The upward field of the classical Milnor presentation is $-X$; statements imported from that presentation are translated by this convention before use.

Existence of adapted excellent pairs on a compact collared triad is proved later on this page; nothing beyond the listed properties is asserted here.