Interior slab handle attachment

Since $f$ has boundary values zero and one while $0<a<b<1$, the band $K=f^{-1}[a,b]$ is a compact subset of $\mathrm{int}W$; in particular $W^a$ contains a collar neighbourhood of $M_0$ and misses a neighbourhood of $M_1$ and the closure of $W^b\setminus W^a$ lies in the compact interior band $K$.

Suppose $K$ contains exactly one critical point $p$ of index $k$, and put $c=f(p)$. Choose $\eta >0$ and a Morse chart with compact closure in $f^{-1}(a,b)$, small enough for [F8] and with $a<c-\eta <c+\eta <b$. The local construction of [F8] attaches the product handle, changes the function only in this chart, and compares the rounded local attachment with the modified lower sublevel by a chart-supported isotopy. Its complementary modified-function band is regular and compact and lies in $\mathrm{int}W$. Use [F9] with support in a relatively compact interior neighbourhood of this band, and absorb the resulting product collar by a smooth increasing collar-interval map equal to the identity at its inner edge. For the original regular outer bands from $a$ to $c-\eta$ and from $c+\eta$ to $b$, use [F9] again, with cutoffs supported in small interior neighbourhoods of those bands, to transport attaching data and absorb the outer collars. Every chart, cutoff, and collar adjustment is thereby supported in a finite union of compact subsets of $\mathrm{int}W$. Each map is the identity near the complement of this union, so it extends smoothly by the identity over a neighbourhood of $M_0$. This constructs $W^b\cong W^a\cup h^k$ fixing $M_0$; it establishes the support property rather than inferring it from the abstract diffeomorphism type in [F1].

The attaching sphere is the flow-transported boundary of the unstable disk: the local model at $p$ is the one considered in [F3], which identifies the local Morse attaching embedding on the lower regular level and transports its thickening to $W^a$ by the descending flow of $X$, with no intervening critical value because $p$ is the only critical point of the band.

For finitely many critical points of common value $c$ and index $k$, choose disjoint Morse charts with compact closure in $f^{-1}(a,b)$ and one $\eta$ valid for all of them. Perform the compact local modifications and handle constructions of [F8] simultaneously. As in [F2], the modified regular complementary band preserves the common upper sublevel and has no remaining critical point. Its cutoff normalized field, the fields on the two original outer regular bands, and every absorbing interval map may be chosen inside the compact interior neighbourhoods used in step 2.1. Thus their comparisons extend by the identity near $M_0$, producing the asserted simultaneous disjoint attachments on $W$. If there are no critical points, only the original regular-band collar is needed.

The disjoint attaching regions give commuting quotient attachments, and [F6] compares their compatible roundings by disjoint collar-supported isotopies. Thus the order is immaterial. For the pair assertion choose a regular $a_{-}<a$, with $a_{-}>0$, below the lower collar adjustments and with no critical value in $[a_{-},a]$. The collar compression of $W^a$ onto $A_0=W^{a_{-}}$ is fixed near $M_0$ and is a deformation retraction; in the attachment model compress its lower-stage collar to the same copy $A_0$. The above comparisons are the identity on $A_0$, so the two lower-stage inclusions agree after these collar homotopies, giving the homotopy-of-pairs assertion. This does not identify the whole original lower boundary pointwise with the attachment seam.