The surgery gluing has a canonical smooth structure up to diffeomorphism

In the product normal form of $\phi$, a point of $\phi (S^p\times \mathrm{\partial }{D}^q)$ has a chart in which $M$ is an open subset of ${\mathbb{R}}^m$ and the removed piece is the open half-space-product ${\mathbb{R}}^p\times \mathrm{int}{D}^q$; the complement there is locally a closed half-space, so $N$ is a smooth manifold with boundary $\mathrm{\partial }M\bigsqcup \phi (S^p\times S^{q-1})$, and the new boundary component is a closed embedded $(m-1)$-manifold.

Put $B=S^p\times S^{q-1}$ and $P^{\mathrm{\prime }}=D^{p+1}\times S^{q-1}$. The latter has boundary $\mathrm{\partial }{P}^{\mathrm{\prime }}=B$, and $\phi {\mathrm{\mid }}_B$ is a diffeomorphism from $B$ onto the new boundary part of $N$. This uses the supplied embedding on its boundary, rather than identifying that restriction with its derivative framing along the core.

Choose collars $c_N:B\times [0,\epsilon )\to N$ and $c_{P^{\mathrm{\prime }}}:B\times [0,\epsilon )\to P^{\mathrm{\prime }}$, with $c_N(b,0)=\phi (b)$ and $c_{P^{\mathrm{\prime }}}(b,0)=b$. On the quotient $N{\cup }_{\phi {\mathrm{\mid }}_B}{P}^{\mathrm{\prime }}$, define $C(b,s)=[c_N(b,-s)]$ for $s\le 0$ and $C(b,s)=[c_{P^{\mathrm{\prime }}}(b,s)]$ for $s\ge 0$. This is a homeomorphism onto an open seam neighbourhood: each half is a collar homeomorphism, and their relatively open half-images together are saturated in the disjoint union. For each boundary chart $y$ on $B$, use $(y(b),s)$ as a seam chart. Two such charts have transition $(y_2\circ y_1^{-1},s)$; an overlap with a chart away from the seam lies in $s<0$ or $s>0$, where it is a smooth collar-coordinate change. These charts and the original charts away from $B$ generate a smooth atlas. Every seam point is interior, and the remaining boundary is exactly $\mathrm{\partial }M$. A compatible seam smoothing is this signed collar presentation after straightening its collar coordinate.

The quotient is Hausdorff and second countable. Its quotient map from $N\bigsqcup P^{\mathrm{\prime }}$ is closed: saturating a closed set adds only images of its intersections with the compact seam, which are compact and closed in the opposite Hausdorff piece. Distinct quotient points have disjoint finite fibres; finite Hausdorff separation gives disjoint open sets about those fibres, and the complements of the quotient images of their closed complements give disjoint quotient neighbourhoods. The open cover by the two pieces minus their seam and the signed collar has a countable base, by the countable bases of the pieces and of $B\times (-\epsilon ,\epsilon )$. Thus the atlas defines a smooth $m$-manifold. If $M$ is closed, $N$ is compact, so the quotient is compact with empty boundary. This proves (i).

Fix an open seam neighbourhood $U$ in the quotient. On either piece $Q=N$ or $Q=P^{\mathrm{\prime }}$, write $c_0,c_1$ for the old and new collars of its compact boundary part $B$. Extend $c_0$ to the other boundary parts using [F3], and form the smooth boundaryless double $DQ$ using [F5]. Near $B$ in its positive half put $X_i=(c_i{)}_{\ast }{\mathrm{\partial }}_t$. Extend their coordinate components locally across $B$ and combine them by [F9]; this retains the original fields on a smaller positive-side neighbourhood. In the signed $c_0$ coordinate $r$, both $dr(X_i)>0$ along $B$, and hence on a smaller neighbourhood. Choose a smooth $\theta :[0,1]\to [0,1]$ equal to $0$ near $0$ and $1$ near $1$, and flow $X_s=(1-\theta (s))X_0+\theta (s)X_1$ from $b\in B$ for time $t\ge 0$, writing the result as $C_s(b,t)$. Local flow existence, uniqueness and smooth parameter dependence give a jointly smooth family. Its differential in $(b,t)$ at $t=0$ is $(v,a)\mapsto v+a{X}_s(b)$, an isomorphism. Local inverses exist by [F6]; uniqueness and strict increase of $r$ prevent two such trajectories from meeting with different boundary initial points or different flow times. Compactness of $[0,1]\times B$ therefore permits one $\delta >0$ for which all $C_s:B\times [0,\delta ]\to DQ$ are embeddings, constant in $s$ near its endpoints, with $C_0=c_0$ and $C_1=c_1$ on this band. This is the local collar-family construction of the double theorem's proof, rather than an additional assertion of its statement.

Choose an open neighbourhood $W$ of $B$ in $DQ$ disjoint from its other seam parts, with its intersection with the positive half contained in the inverse image of $U$ in $Q$. Shrink $\delta$ so the whole compact swept collar band of step 4.1 lies in $W$; this is possible because $C_s(b,0)=b$ uniformly on the compact set $[0,1]\times B$. Apply [F8] to the compact smooth $m$-manifold with boundary $V=B\times [0,\delta ]$, the boundaryless $m$-manifold $DQ$, and the isotopy $F_s(b,t)=C_s(b,t)$. It gives an ambient isotopy $A_s$ equal to the identity outside $W$ and satisfying $A_s\circ c_0=C_s$ on the band. Every $A_s$ fixes $B$ pointwise, and fixes all other seam parts because they lie outside $W$. A point off the seam cannot cross it during this isotopy: bijectivity and pointwise seam fixing imply $A_s^{-1}(\mathrm{\partial }Q)=\mathrm{\partial }Q$. Thus $A_s$ preserves the positive half. Its time-one restriction $H_Q$ is a boundary-fixing diffeomorphism of $Q$, equal to the identity outside the inverse image of $U$, and $H_Q(c_0(b,t))=c_1(b,t)$ near $B$.

The maps $H_N,H_{P^{\mathrm{\prime }}}$ descend to a bijection of the quotients because they fix the identified boundary points. In the old source and new target seam coordinates it is $(b,s)\mapsto (b,s)$; away from the seam it and its inverse are the smooth maps on the pieces. Consequently it is a diffeomorphism equal to the identity outside $U$. A compatible seam smoothing is straightened into its signed collar presentation as in step 2.1, so the same comparison applies. Since $U$ was arbitrary, this proves (ii).

Finally let ${\phi }_t$ be a smooth isotopy of framed embeddings, constant near the ends, with images in the interior of $M$, and put $F(x,y,t):={\phi }_t(x,y)$ on the compact manifold with boundary $S^p\times D^q$ with values in the boundaryless manifold $N:=\mathrm{int}M$; choose an open neighbourhood $W\subseteq \mathrm{int}M$ of the swept image with $\bar{W}$ compact in $\mathrm{int}M$. Apply [F8] with $V=S^p\times D^q$ and $N=\mathrm{int}M$: there is a smooth $H:\mathrm{int}M\times I\to \mathrm{int}M$ with $H_t\circ {\phi }_0={\phi }_t$ for every $t$, every $H_t$ a diffeomorphism, and $H_t$ the identity outside $W$. Since $H_1$ is the identity outside the compact set $\bar{W}\subseteq \mathrm{int}M$, it extends by the identity across $\mathrm{\partial }M$ to a diffeomorphism $\widehat{H}:M\to M$ agreeing with ${\phi }_1\circ {\phi }_0^{-1}$ on ${\phi }_0(S^p\times D^q)$; hence $\widehat{H}$ carries the complement of ${\phi }_0(S^p\times \mathrm{int}{D}^q)$ onto the complement of ${\phi }_1(S^p\times \mathrm{int}{D}^q)$ and satisfies $\widehat{H}\circ {\phi }_0={\phi }_1$ on the whole product, in particular on the common boundary sphere, so together with the identity on the glued piece $D^{p+1}\times S^{q-1}$ it descends, using collars on the second complement transported by $\widehat{H}$, to the required diffeomorphism $M_{{\phi }_0}\to M_{{\phi }_1}$ supported near the swept region. Other collar choices are compared by (ii); no uniqueness of the resulting diffeomorphism is asserted. This proves (iii), and with it the concluding isotopy-invariance of the construction.