Unstable disk is the handle core

One critical point handle attachment: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $f:M\to \mathbb{R}$ be smooth on a boundaryless $n$-manifold and let $a<b$ be regular values. If $f^{-1}([a,b])$ is compact and has exactly one critical point $p$, nondegenerate of index $k$, then $M^b$ is diffeomorphic to $M^a$ with one $k$-handle attached and corners rounded. No orientation or Morse–Smale hypothesis is required.

Adapted descending field near a compact morse band: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Suppose the compact closed band of a smooth function on a boundaryless manifold has only finitely many critical points, all nondegenerate. There is a smooth field $X$ with $df(X)<0$ at every noncritical point of the band and $X=(2u,-2v)$ in smaller disjoint Morse charts $f=f(p)-\mathrm{\mid }u{\mathrm{\mid }}^2+\mathrm{\mid }v{\mathrm{\mid }}^2$. It can be chosen compactly supported on $M$ and hence complete.

Descending flow identifies the local and global attaching regions: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $f^{-1}([a,b])$ be compact, with regular endpoints and exactly one critical point of value $c$. For the local Morse attaching embedding on $M_{c-\epsilon }$, where $a<c-\epsilon <c$, descending flow transports its entire thickening to $M_a$ as an embedded framed attaching region, provided there is no intervening critical value. The regular regions outside the local critical model are identified by collars.