Boundary connected sum with a disk does not change the diffeomorphism type

Smooth collars of a manifold boundary: A smooth collar is a smooth embedding $c:\mathrm{\partial }M\times [0,\epsilon )\to M$ such that $c(p,0)=p$ and whose image is an open neighbourhood of $\mathrm{\partial }M$ in $M$. Locally one may first use a positive smooth width depending on $p$.

Collar neighborhood theorem: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Every smooth manifold with boundary has a smooth collar.

The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold: If $M$ has dimension $n\ge 1$, the restrictions of boundary charts to their faces give $\mathrm{\partial }M$ the structure of a closed embedded smooth boundaryless $(n-1)$-manifold. For $n=0$, $\mathrm{\partial }M=\mathrm{\varnothing }$.

Smooth charts, atlases, and structures with boundary: A boundary chart is a homeomorphism $\phi :U\to V\subseteq {\mathbb{H}}^n$, where $U\subseteq M$ is open and $V$ is relatively open. Two charts are compatible if each transition map is smooth in the local-extension sense. A smooth atlas is a compatible covering atlas; its smooth structure is its maximal compatible atlas.

The smooth inverse function theorem on manifolds: a smooth map with invertible differential has a smooth local inverse. Apply this to the local open extensions of the full-dimensional disk parametrization.

Model straightening. In the half-space ${\mathbb{H}}^n=\{x_n\ge 0\}$ let $\mathrm{\Delta }=D^{n-1}\times \{0\}$ be the flat unit disk. Glue the standard $n$-disk along $\mathrm{\Delta }$ by a diffeomorphism onto $\mathrm{\Delta }$ and round the codimension-two corner of the resulting set. The result is diffeomorphic to ${\mathbb{H}}^n$ by a diffeomorphism equal to the identity outside a compact neighbourhood of $\mathrm{\Delta }$: in suitable coordinates along the rounded corner the glued set is $\{(x^{\mathrm{\prime }},x_n):x_n\ge \gamma (x^{\mathrm{\prime }})\}$ for a compactly supported smooth $\gamma \le 0$ with $\gamma =0$ off a neighbourhood of the disk, and $(x^{\mathrm{\prime }},x_n)\mapsto (x^{\mathrm{\prime }},x_n-\gamma (x^{\mathrm{\prime }})\rho (x_n))$ is the required straightening. Here the added disk is first represented as a sufficiently thin cap, $\rho =1$ near that cap and $\rho =0$ near the inner edge of the chosen collar, and $\mathrm{\parallel }\gamma {\rho }^{\mathrm{\prime }}\mathrm{\parallel }<1$; the normal derivative $1-\gamma {\rho }^{\mathrm{\prime }}$ is positive, so this fibre map is a diffeomorphism and becomes the identity at the inner edge.