Isotopic attaching embeddings give diffeomorphic handle attachments

Extend the isotopy by ${\phi }_t:={\phi }_0$ for $t\le 0$ and ${\phi }_t:={\phi }_1$ for $t\ge 1$; the constancy hypotheses make this a smooth map $M\times \mathbb{R}\to \mathrm{\partial }W$ whose every time slice is an embedding. Define $\mathrm{\Psi }(x,t):=({\phi }_t(x),t)$ on $M\times \mathbb{R}$; it is injective, since $({\phi }_t(x),t)=({\phi }_s(y),s)$ forces $t=s$ and then $x=y$. Its differential is $d\mathrm{\Psi }(v,s)=(d{\phi }_t(v)+s{\mathrm{\partial }}_t{\phi }_t,s)$, which is injective: $s=0$ from the second component and then $v=0$ from injectivity of $d{\phi }_t$. At points with $x\in \mathrm{int}M$ the source $\mathrm{int}M\times \mathbb{R}$ and the target $\mathrm{\partial }W\times \mathbb{R}$ both have dimension $n$, so $d\mathrm{\Psi }$ is an isomorphism there, and the image $S^{\circ }:=\mathrm{\Psi }(\mathrm{int}M\times \mathbb{R})$ is an open subset of $\mathrm{\partial }W\times \mathbb{R}$.

Apply the compact full-dimensional graph-velocity construction of [F8] to the graph $\mathrm{\Psi }$ of step 1.1. At a source-boundary point, extend the smooth graph map to an open coordinate neighborhood; its derivative is invertible, so [F3] gives the required local inverse and extends the velocity there. The finite partition and bump construction in [F8] gives a smooth compactly supported field $\widehat{Y}(p,t)=(u_t(p),0)$ on $\mathrm{\partial }W\times \mathbb{R}$ with $u_t({\phi }_t(x))={\mathrm{\partial }}_t{\phi }_t(x)$ for every $x\in M$, including $\mathrm{\partial }M$.

Since the isotopy is stationary for $t$ near its endpoints, multiply this extension by a smooth temporal cutoff that is one wherever the prescribed velocity is nonzero and vanishes on smaller endpoint neighborhoods. This preserves its values on the graph and compact support and makes $u_t=0$ near $t=0,1$. Thus the boundary of the disk factor may move; no stationary spatial collar is required.

Writing $(u_t(p),0):=\widehat{Y}(p,t)$ defines a smooth time-dependent vector field $u_t$ on $\mathrm{\partial }W$ whose support over the compact interval $[0,1]$ is a compact subset of $\mathrm{\partial }W$. By [F4] it has a global evolution $H_t:={\mathrm{\Psi }}_{t,0}$, $t\in [0,1]$, with $H_0={\mathrm{id}}_{\mathrm{\partial }W}$; each $H_t$ is a diffeomorphism with inverse ${\mathrm{\Psi }}_{0,t}$, $H_t$ is the identity outside the compact support of the family, and $H_t=\mathrm{id}$ for $t\le \epsilon$ and $H_t=H_1$ for $t\ge 1-\epsilon$ because $u_t$ vanishes there.

Extend the diffeotopy over the interior by a collar deformation. By [F2] fix a collar $c:\mathrm{\partial }W\times [0,\eta )\to W$ and a smooth function $\chi :[0,\eta )\to [0,1]$ with $\chi \equiv 1$ on $[0,\eta \mathrm{/}4]$ and $\chi \equiv 0$ on $[\eta \mathrm{/}2,\eta )$. Define $K:W\to W$ by $K(c(p,s)):=c(H_{\chi (s)}(p),s)$ on the collar image and $K:={\mathrm{id}}_W$ outside. The collar image $V:=c(\mathrm{\partial }W\times [0,\eta ))$ is open, $Z:=c(\mathrm{\partial }W\times [0,\eta \mathrm{/}2])$ is compact with $Z\subseteq V$, and on $V\setminus Z$ one has $\chi (s)=0$, so the collar formula is the identity there and agrees with the outside definition; hence $K$ is smooth. The same computation with $H_{\chi (s)}^{-1}={\mathrm{\Psi }}_{0,\chi (s)}$ in place of $H_{\chi (s)}$ gives a smooth inverse, so $K$ is a diffeomorphism of $W$ that is the identity outside the compact set $Z$ and satisfies $K{\mathrm{\mid }}_{\mathrm{\partial }W}=H_1$, since $\chi (0)=1$.

For every $x\in M$, including its boundary, step 2.1 gives $\frac{d}{dt}{\phi }_t(x)=u_t({\phi }_t(x))$. Uniqueness in [F4] therefore yields $H_t({\phi }_0(x))={\phi }_t(x)$ on the whole attaching region. With step 5.1, $K({\phi }_0(x))={\phi }_1(x)$.

Define $\mathrm{\Phi }:W{\cup }_{{\phi }_0}H\to W{\cup }_{{\phi }_1}H$ by $\mathrm{\Phi }{\mathrm{\mid }}_W:=K$ and $\mathrm{\Phi }{\mathrm{\mid }}_H:={\mathrm{id}}_H$. It is compatible with the two identifications: for $z\in M$ one has $\mathrm{\Phi }(z)=z$ and $\mathrm{\Phi }({\phi }_0(z))=K({\phi }_0(z))={\phi }_1(z)$, and in the target $z$ is identified with ${\phi }_1(z)$. In the seam charts given by the collar data of [F1] and the collar $c$, a source point $c({\phi }_0(z),s)$ with $s\le \eta \mathrm{/}4$ is glued to the handle point $(z,s)$ and is mapped to $c(H_{\chi (s)}({\phi }_0(z)),s)=c({\phi }_1(z),s)$, which is glued to $(z,s)$ in the target; thus $\mathrm{\Phi }$ reads as the identity on a neighbourhood of the seam. Hence $\mathrm{\Phi }$ is smooth with smooth inverse $K^{-1}\bigsqcup {\mathrm{id}}_H$, and it is the identity on $H$ and outside a collar of the region swept by the isotopy.

The attachments in the statement are formed with corners rounded. The map $\mathrm{\Phi }$ preserves the collar data of the corner and therefore carries a compatible rounding of the first presentation to a compatible rounding of the second; by [F5] the rounded attachments are diffeomorphic, and the resulting diffeomorphism is still the identity outside a collar of the swept attaching region in $W$, which is what the swept region corresponds to under the two gluings.

If further handles are attached to the outgoing boundary of the two presentations, then gluing the same handles along attaching data that correspond under $\mathrm{\Phi }$ gives diffeomorphic total manifolds: the map $\mathrm{\Phi }$ on the base together with the identity on the additional handles is compatible with the identifications, exactly as in step 7.1. In particular, attaching data carried into the swept region by the isotopy are transported by $\mathrm{\Phi }$, so the two total manifolds are diffeomorphic.