Creation of a cancelling handle pair

Choose an embedded closed disc $E\subseteq V$ with $x\in \mathrm{int}E$ and attach an $n$-disc $D^n$ to $W$ along $E$; by [F1], applied to $N=W$ and $D=E$, the boundary connected sum $W\mathrm{♮}{D}^n$ is diffeomorphic to $W$ relative to ${\mathrm{\partial }}_{0}W$, by a diffeomorphism equal to the identity outside a collar of $E$.

By [F2] the standard $n$-disc admits the decomposition $D^n=D^n\cup h^k\cup h^{k+1}$ with the two handles attached in the standard complementary way along a disc of its boundary: the standard $k$-handle is attached along the equatorial embedding and the standard $(k+1)$-handle fills the resulting $S^k\times D^{n-k}$ back to a disc. Hence the attached disc in step 1.1 can be decomposed into the two standard handles supported over $E$ (the upper region lies in the boundary after the lower attachment).

Transporting this decomposition along the absorption diffeomorphism of step 1.1 and adjusting the attaching data by an isotopy inside $V$ using [F3], we obtain attaching embeddings of a $k$-handle $h^k$ and, after it, a $(k+1)$-handle $h^{k+1}$ whose lower attaching region lies in $V$ and whose upper attaching region lies in the boundary region obtained from $V$ after the lower attachment, forming the standard complementary pair on an embedded disc $E\subseteq V$, with $W\cup h^k\cup h^{k+1}$ diffeomorphic to $W$ relative to ${\mathrm{\partial }}_{0}W$.

Consequently every handle presentation of $W$ may be modified by introducing a geometrically cancelling pair of consecutive indices at any prescribed disc of the outgoing boundary, without changing the manifold; the pair is the inverse local modification of the cancellation theorem, and the construction works for every $0\le k\le n-1$ and covers the endpoints through the endpoint conventions of the standard model.