Morse cancellation criterion via a unique connecting orbit

Given: A compact collared triad $(W;M_0,M_1)$ with adapted excellent Morse function $f$ and adapted field $X$, regular values $a<b$ with compact slab $K=f^{-1}[a,b]$ containing exactly the critical points $p,q$ of indices $k$ and $k+1$, regular $v\in (f(p),f(q))$, and the spheres $B_p$ (stable sphere of $p$) and $A_q$ (unstable sphere of $q$) in $f^{-1}(v)$ meeting transversely in exactly one point.

Morse function adapted to a cobordism and Downward gradient-like vector fields for a Morse function: adapted means $f^{-1}(0)=M_0$, $f^{-1}(1)=M_1$, critical points interior and nondegenerate, no critical point in a fixed collar; the field is downward gradient-like with $df(X)<0$ off the critical set and $X=(2u,-2v)$ in Morse charts.

Spheres of adjacent critical levels have product neighbourhoods: assume ${\mathrm{A}\mathrm{C}}_{\omega }$; for consecutive critical levels the crossing sets of the trajectories through the local unstable disk of $q$ and the local stable disk of $p$ are compact embedded spheres $A_q,B_p$ in the regular level $f^{-1}(v)$, with product neighbourhoods, and a trajectory from $q$ to $p$ crosses $f^{-1}(v)$ exactly once, at a point of $A_q\cap B_p$, every such point lying on such a trajectory.

A Morse trajectory from one critical point to another fixes the direction of the limiting orbit in the gradient case. Here the trajectories are those of the given downward gradient-like field $X$; the crossing correspondence in [F2] supplies their limits and identifies them up to time translation. No assertion that $X$ equals a particular metric gradient is required.

The fundamental theorem on flows and Regular interval diffeomorphism: integral curves of a smooth field form a smooth local flow, unique through each point; if a closed band $f^{-1}([a,b])$ is compact and critical-point-free, its normalized flow is a level-preserving diffeomorphism $M_a\times [a,b]\to K$.

Local morse sublevel pair is a handle pair: in a small Morse chart of index $k$ the change across the critical value is a rounded index-$k$ handle, with the core $v=0$ and a compact product piece attached along $S^{k-1}\times D^{n-k}$.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): ${\mathrm{A}\mathrm{C}}_{\omega }$ is assumed; it is used through the adapted-field and band suppliers.

Handle cancellation: under ${\mathrm{A}\mathrm{C}}_{\omega }$, a consecutive index-$k$, index-$(k+1)$ pair with one transverse attaching-belt intersection can be deleted relative to the incoming boundary.