A one-handle between distinct manifold components is a boundary connected sum

Write $h_{-},h_{+}$ for the two feet of the handle, so that the disk identification in the statement is $h_{+}\circ h_{-}^{-1}$. Cut the handle at $M=\{0\}\times D^{n-1}$. The cut pieces have corners along $\mathrm{\partial }M$; they are not yet smooth manifolds with boundary. Use the product seam collars of [F1] and the boundary collars of [F3] throughout. Introduce a corner in $N_i$ along $\mathrm{\partial }{D}_i$, making $D_i$ a distinguished boundary face, and denote the resulting cornered manifold by $N_i^{\mathrm{\angle }}$. Use a corner model that preserves the given disk coordinates at the edge: with $r=\sqrt{s^2+t^2}$, set $(z_1,z_2)=((s^2-t^2)\mathrm{/}r,2st\mathrm{/}r)$ for $r>0$ and $(z_1,z_2)=(0,0)$ at the vertex. This doubles the polar angle and preserves the radius, carrying the quadrant $s,t\ge 0$ homeomorphically onto the half-plane. It is a diffeomorphism off the vertex and restricts to $(s,0)$ on the disk face and $(-t,0)$ on the other face. Its inverse defines the cornered smooth structure; it is not asserted to be smooth at the vertex in the original structure. In particular the original disk coordinate is $z_1=s$, so the handle foot remains smooth up to its edge. A collar of $\mathrm{\partial }{D}_i$ inside $\mathrm{\partial }{N}_i$, followed by the boundary collar, supplies these product coordinates. The face $D_i$ then has a product collar, including its edge. Gluing on the corresponding half-cylinder prolongs that face collar and gives a cornered piece $Z_i$ whose free end is $M$. The coordinate comparison extends smoothly across each attaching seam away from its edge: in the handle-side sector $-\pi \mathrm{/}2\le \theta \le 0$, use a smooth increasing angular map $b$ with $b(\theta )=\theta \mathrm{/}2$ near $0$ and $b(\theta )=\theta$ near $-\pi \mathrm{/}2$, preserving the radius. Such a $b$ is obtained by integrating a positive function, equal to $1\mathrm{/}2$ near $0$ and to $1$ near $-\pi \mathrm{/}2$, whose total integral is $\pi \mathrm{/}2$. It matches the inverse disk-face cornerization across the seam and fixes the outgoing ray. A radial cutoff interpolates its positive angular derivative to that of the identity outside the edge chart. Thus the comparison is a diffeomorphism off the original edge and preserves the given disk parametrization there and at the edge.

Absorb each prolonged face collar before straightening any cut corner. In coordinates $D_i\times [0,\mathrm{\ell })$ the prolonged collar is $D_i\times [-L,\mathrm{\ell })$, where the free end is $t=-L$. Choose a smooth increasing bijection $\alpha :[-L,\mathrm{\ell })\to [0,\mathrm{\ell })$ with $\alpha (-L)=0$, positive derivative, and $\alpha (t)=t$ near $\mathrm{\ell }$, choosing the same $\alpha$ on both halves and $\alpha (t)=a(t+L)$ near $-L$ for a constant $a>0$; integration of a positive smooth scalar function with the required total integral constructs such an $\alpha$. The product map $(x,t)\mapsto (x,\alpha (t))$ is a diffeomorphism of manifolds with corners, including the side face $\mathrm{\partial }{D}_i\times [-L,\mathrm{\ell })$, and extends by the identity at the inner collar edge. Hence $(Z_i,M)\cong (N_i^{\mathrm{\angle }},D_i)$ as cornered pairs, preserving the disk coordinates. This is face-collar absorption, not a diffeomorphism from an unrounded cut piece to smooth $N_i$.

Glue the two cornered pairs of step 2.1 along their distinguished faces using their disk coordinates and signed product collars. The result is precisely $N_1^{\mathrm{\angle }}{\cup }_{h_{+}\circ h_{-}^{-1}}{N}_2^{\mathrm{\angle }}$, the boundary connected sum. At the common edge the two quadrants joined along the distinguished face have coordinates $(s,\tau )$ with $s\ge 0$ and signed normal coordinate $\tau \in \mathbb{R}$. Near that edge the comparison of step 2.1 is $(s,\tau )\mapsto (s,a\tau )$, hence is smooth with smooth inverse, including on the boundary. The same disk coordinates give exactly the specified identification, without any square-root change. Away from that edge the comparison is already a product-collar diffeomorphism. To compare with the original attachment, round the original attaching seams inside the product edge charts before applying their coordinate changes: the rounded profiles avoid the vertices, where the radius-preserving map was singular. On these profiles and their inner sides it is a smooth diffeomorphism. The new smooth side may likewise be pushed inward inside its collar to such a profile and restored by a positive-derivative collar-interval map, so this comparison does not use smoothness at a corner vertex. Thus the glued model is diffeomorphic to a compatible rounded attachment; [F4] compares any other compatible rounding at the original attaching seams. No unrounded cut piece is treated as smooth. Finally the exposed boundary of the handle is $D^1\times S^{n-2}$, joining $\mathrm{\partial }{D}_1$ to $\mathrm{\partial }{D}_2$, while the two attaching disks disappear from $\mathrm{\partial }{N}_1\bigsqcup \mathrm{\partial }{N}_2$. Absorbing this intervening boundary collar gives exactly the boundary description in the statement.

Edge cases deserve the stated conventions. For $n=1$ the disks $D_i$ are single boundary points, the $1$-handle is an interval glued at its two ends, and $\mathrm{\partial }{N}_i$ loses exactly that point; no sphere is glued because $S^{-1}=\mathrm{\varnothing }$, and the conclusion still holds verbatim. For $n=0$ there are no disks and no handles, so the assertion is vacuous. For $n\ge 2$ the attaching regions are genuine disks and the displayed boundary computation applies.