Surgery is reversed by dual surgery

By [F1] the surgered manifold is the union of $N$ with the glued handle $D^{p+1}\times S^{q-1}$ along the boundary $\phi (S^p\times S^{q-1})=S^p\times S^{q-1}$. The dual sphere $S_{\phi }=\{0\}\times S^{q-1}$ lies in that handle, and by [F2] its framing exhibits the product $\{0\}\times S^{q-1}\times D^{p+1}$; the closed glued-in product is a framed product neighbourhood of $S_{\phi }$ in $M_{\phi }$, so the dual surgery of [F2] removes exactly the interior of the glued handle and glues $D^q\times S^p$ along $S^{q-1}\times S^p$.

Removing the interior of the glued handle from $M_{\phi }$ leaves the complement $N$ with boundary $\phi (S^p\times S^{q-1})$, up to a collar; by [F4] the boundary of $D^q\times S^p$ is $S^{q-1}\times S^p$, and the gluing identification is the given framing on that overlap. Hence the surgered manifold of the dual operation is $M_{\text{dual}}=N{\cup }_{\phi {\mathrm{\mid }}_{S^p\times S^{q-1}}}(D^q\times S^p)\mathrm{.}$

The factor swap $(y,x)\mapsto (x,y)$ identifies $D^q\times S^p$ with $S^p\times D^q$ and its boundary with $S^p\times S^{q-1}$. Composing with $\phi$ identifies the quotient in step 2.1 with $N{\cup }_{\phi {\mathrm{\mid }}_B}(S^p\times D^q)=M$, where $B=S^p\times S^{q-1}$. Choose the signed seam collars transported from $M$ for this reconstruction; the map is smooth across the seam and is the identity on $N$. Other collar choices are compared by [F7], with support near the seam. This gives the asserted diffeomorphism and support.

To identify the supporting manifold, use [F6] with $n=m+1$ and $k=p+1$, so $0<k<n$ since $q\ge 1$. Write its elementary band as $H=\{(u,v):-1\le Q\le 1,\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{\mid }v{\mathrm{\mid }}^2\le 2\}$, with $u\in {\mathbb{R}}^{p+1}$ and $v\in {\mathbb{R}}^q$. At $Q=-1$ the coordinates are $(x,v)\mapsto (\sqrt{1+\mathrm{\mid }v{\mathrm{\mid }}^2}x,v)$ for $x\in S^p$, $\mathrm{\mid }v\mathrm{\mid }\le 1$; at $Q=1$ they are $(u,y)\mapsto (u,\sqrt{1+\mathrm{\mid }u{\mathrm{\mid }}^2}y)$ for $\mathrm{\mid }u\mathrm{\mid }\le 1$, $y\in S^{q-1}$. Glue the side to $N\times [-1,1]$ using $\phi {\mathrm{\mid }}_B$ on the sphere coordinates. By [F6] this is $W_{\phi }$ up to diffeomorphism. Now exchange $(u,v)$ with $(v,u)$ and reverse the height on the complementary product. The inequalities defining $H$ are invariant, $Q$ changes to $-Q$, and the old outgoing coordinates become the incoming $S^{q-1}\times D^{p+1}$ coordinates. Their disk derivative along $u=0$ is the product normal framing of [F2]. The same side gluing, read backwards, is therefore the elementary band for that dual framed surgery on $M_{\phi }$. Applying [F6] with $k=q$ identifies it with the dual trace, after absorbing outer collars; compatible roundings are compared by Smooth handle attachment is independent of corner rounding up to diffeomorphism. Thus both traces have the same supporting manifold up to diffeomorphism.