Smooth handle attachment is independent of corner rounding up to diffeomorphism

Attaching a smooth handle with corner rounding: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $X$ be a smooth $n$-manifold with boundary. Attach the handle of def-k-handle-core-cocore-attaching-region-and-belt-sphere by a smooth embedding $h:S^{k-1}\times D^{n-k}\to \mathrm{\partial }X$ that extends to a neighborhood of the disk factor. Form the quotient of $X\bigsqcup (D^k\times D^{n-k})$ identifying $z$ with $h(z)$ in the attaching region. The disk coordinates trivialize the normal bundle of the attaching sphere; this framing is part of the data. Use collars from thm-collar-neighborhood-theorem to give the seam its product smooth charts, then round the compact codimension-two corner. A compatible rounding is a smooth monotone planar profile, transverse to a common diagonal direction, agreeing with the two faces away from a small corner neighborhood. In coordinates along that diagonal it is a graph. This convention fixes the gluing and collar data; changing the attaching embedding is a different question. There is no corner to round when $k=0$ or $k=n$.

The fundamental theorem on flows: Let $X$ be a smooth vector field on $M$. For each $p\in M$, let ${\gamma }_p:I_p\to M$ be the maximal integral curve through $p$, and set $\mathcal{D}:=\{(t,p)\in \mathbb{R}\times M:t\in I_p\},\mathrm{\Phi }(t,p):={\gamma }_p(t)\mathrm{.}$ Then $\mathcal{D}$ is open in $\mathbb{R}\times M$, each fibre ${\mathcal{D}}_p$ is an interval containing $0$, the map $\mathrm{\Phi }:\mathcal{D}\to M$ is smooth, and $\mathrm{\Phi }$ is the unique maximal local flow generated by $X$.

A manifold bump for a compact set inside an open set: Let $M$ be a smooth manifold, let $K\subseteq M$ be compact, and let $W\subseteq M$ be open with $K\subseteq W$. Then there exists a smooth function $\rho :M\to [0,1]$ that equals $1$ on an open neighbourhood of $K$ and satisfies $\mathrm{supp}(\rho )\subseteq W$.