Dual surgery sphere

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $M$ be a closed smooth $m$-manifold, $0\le p\le m-1$, $q=m-p$, let $\phi$ be a framed embedded surgery sphere in $M$, let $W_{\phi }$ be the trace and let $M_{\phi }$ be the surgered manifold, identified with the outgoing face of the trace (Surgery trace cobordism, The upper boundary of the surgery trace is the surgered manifold). The dual surgery sphere is the belt sphere $S_{\phi }=\{0\}\times S^{q-1}\subseteq D^{p+1}\times S^{q-1}\subseteq M_{\phi },$ of dimension $q-1$, together with the framing of its normal bundle in $M_{\phi }$ induced by the product structure of the handle.

The framing is canonical and not a choice: in the glued handle $D^{p+1}\times S^{q-1}$ the tangent directions of the belt sphere are the $S^{q-1}$-directions, so its normal directions inside $M_{\phi }$ are the $D^{p+1}$-factor directions, and the product trivialization of the disk factor trivializes them. This is the same product data that was used to glue the handle in, read from the other side (K handle core cocore attaching region and belt sphere, Normal and conormal bundles of an embedded submanifold).

The dual operation is the $(q-1)$-surgery on the $m$-manifold $M_{\phi }$ along $S_{\phi }$; after this dual operation its belt sphere identifies with the original underlying sphere in $M$. The construction applies to every datum of the definition, including the endpoint $q=1$, where $S_{\phi }=\{0\}\times S^0$ is a $0$-sphere, a two-point set with a framing of its rank-$m$ normal bundle. The dimension count $(q-1)+1=q$ shows that the dual surgery piece has disk factor $p+1=m-(q-1)$, so the dual operation is again of the form considered in Framed embedded surgery sphere: the sphere dimension is $q-1$ and the disk factor has dimension $p+1$, and the range $0\le q-1\le m-1$ holds because $1\le q\le m$.

In the normal direction the dual sphere carries the framing that the reversal theorem of this page needs; no claim about the diffeomorphism type of the dual surgered manifold is made here, and no orientation of $M$ is used.