In basic surgery theory, it is clear what happens topologically when one "adds a handle", but getting the smooth structure right requires care. For example, in identifying opposite sides of a rectangle with two intervals on the boundary of a disk, the result ``has 270 degree angles''. My favorite treatment of differential topology has been Kosinski's book, which uses explicit diffeomorphisms between subspaces of products of disks to handle smooth structures (pardon the pun). But with this approach one no longer has the interior of the handle as a separate entity at the point-set level.
In recently wanting to apply some elementary surgery theory, I realized that one could use blow-ups as an alternate approach. If $S$ is a framed embedded $k$- sphere in $\partial M$ of codimension $d$ in the boundary, one can blow it up to obtain a manifold with corners $\hat{M}$ with a few nice properties. It is homeomorphic to $M$ but maps back to $M$ with a (closed) stratum $\hat{S} \cong S \times D^d$ mapping to $S$ by projection and otherwise is a diffeomorphism. The stratum $\hat{S}$ can be identified with the ``southern hemisphere'' of the normal bundle of $S$.
The stratum $\hat{S}$ has a collar neighborhood in $M$ which is diffeomorphic to $\hat{S} \times (-1,0]$. The boundary of $D^{k+1}\times D^d$ as a manifold with corners has a component $S^k \times D^d$ which has a collar neighborhood diffeomorphic to $S^k \times D^d \times [0,1) \cong \hat{S} \times [0,1)$. It is thus immediate to place a smooth structure on their identification along $S^k \times D^d \times 0$, and the resulting manifold is the handle attachment along $S$.
There is a similar construction for handle attachment to the interior, where the blow-up is a manifold with boundary and the identifications have tautological smooth structure once collar neighborhoods are chosen.
My question is whether this approach to handle attachment has appeared in the literature.
