A cancelling zero-one handle pair

Given: Dimension $n\ge 1$, a compact manifold $W$, a $0$-handle $h^0$ attached as a disjoint ball $D^n$ and a $1$-handle $h^1$ attached by an embedding $S^0\times D^{n-1}\to {\mathrm{\partial }}_{+}(W\bigsqcup D^n)$ whose attaching $0$-sphere has one point on the new boundary sphere $S^{n-1}$ of $D^n$ and one point on ${\mathrm{\partial }}_{+}W$.

Geometrically cancelling adjacent handle pair and K handle core cocore attaching region and belt sphere: the belt sphere of a $0$-handle is the new boundary sphere $S^{n-1}$ (disconnected when $n=1$), the attaching sphere of a $1$-handle is a $0$-sphere, and the endpoint convention counts exactly one point of the $0$-sphere lying in the belt sphere, transversality being automatic.

The standard complementary pair fills a ball: for the standard embedding the union of the standard $0$-handle and the standard $1$-handle is an $n$-disc: in its case $k=0$ the standard complementary pair fills the ball, i.e. $(S^0\times D^n)\cup h^1\cong D^n$ for the standard hemisphere embedding.

Handle cancellation, Attaching a smooth handle with corner rounding and The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): assume ${\mathrm{A}\mathrm{C}}_{\omega }$; a geometrically cancelling pair may be deleted, giving a diffeomorphism relative to the incoming boundary; attachments are formed with corners rounded. The assumption is used through the standard-pair model of [F2] and through this deletion.