In the first section of "A procedure for killing homotopy groups of differentiable manifolds", Milnor gives the surgery construction as follows. Let $W$ be an $n=p+q+1$ dimensional manifold. Given a smooth, orientation preserving embedding:

$$f:S^p\times D^{q+1}\to W$$

we may obtain a new manifold as the disjoint sum

$$(W-f(S^p\times 0))\cup (D^{p+1}\times S^q)$$

modulo some equivalence relation.

My question is: why in this construction is Milnor deleting $S^p\times 0$ from $W$ and not $S^p\times D^{q+1}$? I expected this disjoint sum to be

$$(W-f(S^p\times D^{q+1}))\cup (D^{p+1}\times S^q).$$