Surgery trace cobordism

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $M$ be a closed smooth $m$-manifold, $0\le p\le m-1$, $q=m-p$, and let $\phi :S^p\times D^q\hookrightarrow M$ be a framed embedded surgery sphere (Framed embedded surgery sphere). Regard $\phi$ in the upper face $M\times \{1\}$ of the cylinder $M\times [0,1]$ as the embedding $\phi \times \{1\}$, whose image is contained in the interior of that face. The surgery trace cobordism of $\phi$ is the compact smooth $(m+1)$-manifold $W_{\phi }=(M\times [0,1]){\cup }_{\phi \times \{1\}}(D^{p+1}\times D^q),$ that is, the cylinder $M\times [0,1]$ with the standard $(p+1)$-handle $D^{p+1}\times D^q$ attached along the framed sphere, the corner along $\phi (S^p\times \mathrm{\partial }{D}^q)$ being rounded in the standard way (Attaching a smooth handle with corner rounding). The index shift is part of the construction: a surgery datum of sphere dimension $p$ produces a handle of index $p+1$, whose attaching region is $S^p\times D^q$ and whose outgoing region is $D^{p+1}\times S^{q-1}$ (K handle core cocore attaching region and belt sphere).

The cylinder is the product cobordism with the empty handle presentation, whose incoming face is $M_0=M\times \{0\}$ and whose outgoing face is $M\times \{1\}$; the product presentation and its, possibly empty, collar structure are those of Product cobordisms have critical-point-free presentations, and the collar of the incoming face is the one used whenever the trace is recorded as a cobordism (Smooth collars of a manifold boundary, Smooth cobordism triad for Morse theory).

The trace carries the following fixed pieces of structure, which are not choices made afterwards:

• the incoming face $M\times \{0\}$, identified with $M$ by the product structure;
• the core disk $D^{p+1}\times \{0\}$ of the attached handle;
• the cocore disk $\{0\}\times D^q$ of the attached handle;
• the belt sphere $\{0\}\times S^{q-1}\subseteq D^{p+1}\times S^{q-1}$, a copy of which lies in the outgoing face of $W_{\phi }$.

When $M$ is oriented, an oriented trace additionally requires compatibility of the framing with the orientation: choose the handle orientation so its attaching identification reverses the induced boundary orientations of the handle and the cylinder. The orientation then glues, and $W_{\phi }$ is an oriented bordism from $M$ to its outgoing face, the outgoing face carrying the induced boundary orientation and the incoming face the negative of the orientation of $M$, in the outward-normal-first convention of Induced boundary orientation; this is the oriented bordism relation of Oriented smooth cobordism, and the orientation of $M$ determines the extending orientation when this compatibility holds. For $p\ge 1$ the attaching region is connected, so the handle orientation can always be chosen to match it. For $p=0$ its two components must both match that one handle orientation; an arbitrary pair of interval framings need not do so, and the unoriented trace remains defined in that case. The supplied orientation is the only ambient orientation used (Oriented smooth manifolds and oriented charts).

Nothing is proved here. In particular, the identification of the outgoing face with the surgered manifold $M_{\phi }$ of p-surgery on a smooth m-manifold is the content of the upper-boundary theorem of this page, and the deformation retractions of the trace are not part of the definition.