Handles of equal index can be attached on one level

Since $0<c<1$, choose regular $0<a<c<b<1$ sufficiently close to $c$ that the band contains only the critical points at $c$. The added hypothesis excludes all other indices there; compactness and nondegeneracy make these points a finite list $p_1,\dots ,p_m$, all of index $k$. The band is compact and disjoint from $\mathrm{\partial }W$, since the face values are zero and one. It need not avoid the whole fixed critical-point-free boundary collar; it avoids a sufficiently small boundary neighbourhood, which suffices for the local attachment construction. Write $W^a,W^b$ for its lower and upper stages.

Local extension of an arbitrary attaching-region isotopy. Let $A=S^{k-1}\times D^{n-k}$ and let $F_s:A\to Y$, $Y=\mathrm{\partial }N$, be a smooth isotopy of admissible attaching embeddings. If $A$ is empty the assertion is immediate. Otherwise $A$ is compact and has the full dimension of $Y$. The graph map $(s,x)\mapsto (s,F_s(x))$ is an embedding of the compact parameterized attaching region. In a source boundary or time-endpoint chart, joint smoothness means restriction of a smooth local extension to an open coordinate neighborhood; the graph map's derivative has block form $(1,0;{\mathrm{\partial }}_s{F}_s,D_x{F}_s)$ and is invertible because $D_x{F}_s$ has full rank. F11 supplies a smooth inverse on that open neighborhood. There define the time-dependent velocity by $v(s,y)={\mathrm{\partial }}_s{F}_s(F_s^{-1}(y))$. It extends the actual velocity on the graph even at the attaching boundary. A finite cover of that graph, F12 and compactly supported bumps F10 patch these local extensions: every local value on the graph is the same prescribed velocity, so their weighted sum $V(s,y)$ agrees there. Choose the sum supported in a compact subset of $\mathbb{R}\times Y$. No tubular-neighborhood or later isotopy-extension theorem is used.

Changing the attaching embedding by a diffeotopy of the boundary. Let $N$ be a smooth $n$-manifold with boundary, let $f:S^{k-1}\times D^{n-k}\to \mathrm{\partial }N$ be an attaching embedding, and let $H_s$, $s\in [0,1]$, be a smooth isotopy of $\mathrm{\partial }N$ with $H_0=\mathrm{i}\mathrm{d}$ whose every time-$s$ map is a diffeomorphism; write $f_s:=H_s\circ f$. Choose a collar $c:\mathrm{\partial }N\times [0,1)\to N$ by [F7] and a smooth scalar cutoff $\beta :[0,1)\to [0,1]$ with $\beta =1$ on $[0,1\mathrm{/}4]$ and $\beta =0$ on $[1\mathrm{/}2,1)$, obtained by integrating and normalizing a positive bump on $(1\mathrm{/}4,1\mathrm{/}2)$ and taking the complementary integral, and define ${\tilde{H}}_s:N\to N$ by ${\tilde{H}}_s(c(p,t)):=c(H_{s\beta (t)}(p),t)$ on the collar image and ${\tilde{H}}_s:=\mathrm{i}\mathrm{d}$ outside it. Each ${\tilde{H}}_s$ is a diffeomorphism of $N$, with inverse $c(p,t)\mapsto c(H_{s\beta (t)}^{-1}(p),t)$, the map $(s,x)\mapsto {\tilde{H}}_s(x)$ is smooth, and ${\tilde{H}}_s$ is the identity outside $c(\mathrm{\partial }N\times [0,1\mathrm{/}2))$. Then $\mathrm{\Phi }:={\tilde{H}}_1^{-1}$ on $N$ and the identity on the handle $h^k=D^k\times D^{n-k}$ is a diffeomorphism $N{\cup }_{f_1}{h}^k\to N{\cup }_{f_0}{h}^k$: it is well defined and bijective because ${\tilde{H}}_1^{-1}(f_1(z))={\tilde{H}}_1^{-1}(H_1(f(z)))=f(z)=f_0(z)$ for every $z$ in the attaching region; it is smooth across the seam because near the boundary $\beta =1$, so in the product-collar charts of [F5] it acts on the boundary factor by $H_1^{-1}$, turning the $f_1$-identification into the $f_0$-identification while keeping the collar coordinate fixed, and that is the identity expressed in the two charts in which the attachments are glued; and it is the identity outside the collar image. The corner rounding changes both attachments by the collar-supported isotopy of [F4], with which $\mathrm{\Phi }$ composes, so it does not affect the conclusion. Hence attachments along attaching embeddings related by a diffeotopy of the boundary are diffeomorphic relative to the complement of that collar neighbourhood.

Ambient diffeotopy. Choose smooth compactly supported cutoffs $\rho (s)$ and $\chi (y)$, with $\rho =1$ near $[0,1]$ and $\chi =1$ on a neighborhood of the projection of $\mathrm{supp}V$, and put $Z=\rho (s)\chi (y){\mathrm{\partial }}_s+V(s,y)$ on $\mathbb{R}\times Y$. This field is compactly supported and complete by F13. On $\{\chi =1\}$ during the relevant interval its time coordinate has derivative one; its space trajectory cannot leave that set, since $V$ vanishes on a neighborhood of its boundary. Outside that set the space component is stationary. Thus $H_s(y)={\mathrm{pr}}_Y{\mathrm{Fl}}_s^Z(0,y)$ is a smooth diffeomorphism for $0\le s\le 1$, with smooth inverse $y\mapsto {\mathrm{pr}}_Y{\mathrm{Fl}}_{-s}^Z(s,y)$, including points where the space component is stationary. The graph curve $(s,F_s(x))$ solves the same field for every $x$, so uniqueness gives $H_s(F_0(x))=F_s(x)$. This proves the required ambient boundary diffeotopy for arbitrary smooth attaching-region isotopies, with compact support. For finitely many disjoint attaching regions apply the same construction to their compact disjoint union.

Apply [F3] to $f{\mathrm{\mid }}_{\mathrm{int}W}$ and this compact band. For its diffeomorphism comparison use the explicit interior-support construction in [F2], Proof (one-point and simultaneous cases): disjoint compact Morse-chart modifications, cutoff normalized flows on the regular bands, and collar-interval maps fixed at their inner edges. These operations require the Morse function on the interior band; the additional given field in [F2] is used to describe the attaching spheres and is unnecessary for the attachment diffeomorphism type here. All supports are in a compact subset of $\mathrm{int}W$, so the comparison extends by the identity near $M_0$. Thus $W^b$ is obtained from $W^a$ by one disjoint rounded $k$-handle per $p_j$, with the common pushed-in lower copy giving the homotopy-of-pairs comparison. If $m=0$, the same regular collar construction attaches no handles.

Order does not matter. If the attaching regions of two handles in $\mathrm{\partial }{W}^a$ are disjoint, then attaching them in either order gives the same result: the two quotient constructions coincide with the single quotient of $W^a\bigsqcup h_1\bigsqcup h_2$ by the two attaching identifications, and the gluings are supported in disjoint neighbourhoods. Hence successive attachment in any order produces a manifold diffeomorphic to the simultaneous attachment of step 2.2, relative to $W^a$.

Conclusion. The local graph extension, ambient flow and collar-gluing steps show that an arbitrary smooth isotopy of the attaching-region embeddings induces a boundary diffeotopy and a diffeomorphic attached stage, and step 3.1 shows that the attachments may be performed in any order with the same result; therefore handles of equal index at one level may be regarded as attached simultaneously or successively in any order, may be reordered freely, and their attaching embeddings may be changed by such isotopies of the attaching region, the reordering diffeomorphism is relative to the lower sublevel $W^a$, while an embedding isotopy moves the base only inside its boundary collar and fixes the complement of that collar. It need not fix every point of the lower sublevel when the attaching image itself moves. The corner rounding only changes the result by the collar-supported isotopy of [F4], so it does not affect this conclusion. Every step used the collar and handle suppliers with ${\mathrm{A}\mathrm{C}}_{\omega }$, and no full choice principle.