p-surgery on a smooth m-manifold

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $M$ be a smooth $m$-manifold, let $0\le p\le m-1$ and $q=m-p$, so that $q\ge 1$, and let $\phi :S^p\times D^q\hookrightarrow M$ be a framed embedded surgery sphere with image in the interior of $M$ (Framed embedded surgery sphere). The $p$-surgery on $M$ along $\phi$, also called a spherical modification of type $(p+1,q)$, is the smooth $m$-manifold $M_{\phi }$ obtained by removing the open tubular piece $\phi (S^p\times \mathrm{int}{D}^q)$ and gluing in $D^{p+1}\times S^{q-1}$ along the common boundary, using the identification induced by $\phi$ on $S^p\times S^{q-1}$: $M_{\phi }=(M\setminus \phi (S^p\times \mathrm{int}{D}^q)){\cup }_{\phi {\mathrm{\mid }}_{S^p\times S^{q-1}}}(D^{p+1}\times S^{q-1})\mathrm{.}$

The two pieces are smooth manifolds with boundary sharing the boundary component $S^p\times S^{q-1}$ under the identification induced by $\phi$; the complement also retains $\mathrm{\partial }M$. They are glued along collars of the shared component and the seam is smoothed by the signed collar charts also used along the boundary of a smooth handle attachment (Attaching a smooth handle with corner rounding, Collar neighborhood theorem). The smooth structure produced this way is independent of the auxiliary collar and smoothing choices up to a diffeomorphism supported near the seam; this is proved by the gluing lemma of this page, and it is the sense in which the construction is well defined (The surgery gluing has a canonical smooth structure up to diffeomorphism ↗).

The core sphere is ${\phi }_0:S^p\to M$, ${\phi }_0(x)=\phi (x,0)$; it lies in the removed piece and is not a submanifold of $M_{\phi }$. The belt sphere is $\{0\}\times S^{q-1}\subseteq D^{p+1}\times S^{q-1}\subseteq M_{\phi }$, a closed embedded $(q-1)$-sphere with the normal data of the disk-factor decomposition (K handle core cocore attaching region and belt sphere). The glued-in disk factor has dimension $p+1$, while the whole piece has dimension $(p+1)+(q-1)=m$. The disk dimension is the index shift recorded by the trace construction of this page.

The boundary identification is the restriction of the supplied product embedding $\phi$, whose derivative along the core induces the normal framing. Changing that embedding or its framing can change the surgery, but distinct normal framings need not give distinct boundary identifications or distinct diffeomorphism types. Nothing in the definition asserts that a framing exists for a given embedded sphere; that condition is the content of the framing lemma of this page.

When $\mathrm{\partial }M\mathrm{\ne }\mathrm{\varnothing }$ the construction takes place in the interior of $M$ and leaves $\mathrm{\partial }M$ unchanged: the removed piece lies in the interior and the glued-in piece meets $\mathrm{\partial }M$ in no point. The case $p=m-1$ (so $q=1$) replaces an open product neighbourhood $S^{m-1}\times (-1,1)$ by the two disks $D^m\times S^0$; the openness of the removed piece and the count of the disk factors are the point of the endpoint formula, which is computed on the B page. No case $p=m$ is included, since then $q=0$ and the boundary being traded would be $S^m\times S^{-1}$; the range $0\le p\le m-1$ excludes it, in accordance with the plan's binding repair.

The smooth structure and boundary conventions are those of Smooth charts, atlases, and structures with boundary, and diffeomorphism means diffeomorphism of smooth manifolds (Diffeomorphisms and local diffeomorphisms of manifolds). Countable Choice is used exactly where the cited collar and handle-attachment suppliers use it, that is, in the existence of the collars along which the two pieces are glued.