Simultaneous attachment at a morse critical value

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.

Local morse sublevel pair is a handle pair: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. In a sufficiently small Morse chart $f=c-\mathrm{\mid }u{\mathrm{\mid }}^2+\mathrm{\mid }v{\mathrm{\mid }}^2$, with $u\in {\mathbb{R}}^k$ and $v\in {\mathbb{R}}^{n-k}$, the change across $c$ is a rounded index-$k$ handle: a compact product piece attaches along $S^{k-1}\times D^{n-k}$ on $f=c-\epsilon$, its core is $v=0$, and after a local modification the remaining region up to $c+\epsilon$ is a regular collar. The modification agrees with $f$ off a compact subset of the chart. The collar assertion is made inside a compact band having no other critical point.

Local critical-value lowering preserves the upper sublevel: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. In a Morse chart $f=c-\mathrm{\mid }u{\mathrm{\mid }}^2+\mathrm{\mid }v{\mathrm{\mid }}^2$ containing the closed ball $\mathrm{\mid }u{\mathrm{\mid }}^2+\mathrm{\mid }v{\mathrm{\mid }}^2\le 2\epsilon$, choose a smooth $\mu :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ supported in $[0,2\epsilon )$ with $\mu (0)>\epsilon$ and $-1<{\mu }^{\mathrm{\prime }}\le 0$. Set $F=f-\mu (\mathrm{\mid }u{\mathrm{\mid }}^2+2\mathrm{\mid }v{\mathrm{\mid }}^2)$ in the chart and $F=f$ outside. This is smooth, has the same critical points as $f$, lowers $p$ below $c-\epsilon$, and satisfies $\{F\le c+\epsilon \}=\{f\le c+\epsilon \}$. If $f^{-1}([c-\epsilon ,c+\epsilon ])$ is compact with only the critical point $p$, the corresponding closed band of $F$ is compact and regular.

Regular interval diffeomorphism: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. If $a<b$ and the closed band $K=f^{-1}([a,b])$ of a smooth function on a boundaryless manifold is compact and critical-point-free, its normalized flow gives a level-preserving diffeomorphism $T:M_a\times [a,b]\to K$, $T(x,t)={\mathrm{\Phi }}_{t-a}(x)$.

Smooth handle attachment is independent of corner rounding up to diffeomorphism: For fixed attaching and product-collar data, two compatible smooth monotone roundings of a handle attachment are diffeomorphic by an isotopy supported in that collar. The diffeomorphism is the identity outside the collar.