Framed embedded surgery sphere

Let $M$ be a smooth $m$-manifold and let $0\le p\le m-1$, with $q=m-p$, so that $q\ge 1$. A framed embedded surgery sphere of dimension $p$ in $M$ is a smooth embedding $\phi :S^p\times D^q\hookrightarrow M$ whose image lies in the interior of $M$. Its underlying sphere is the smooth embedding ${\phi }_0:S^p\to M$, ${\phi }_0(x)=\phi (x,0)$ obtained by restricting to the zero of the disk factor (Smooth embeddings, Smooth manifolds and their smooth charts). The differential in the disk directions at $(x,0)$, followed by the quotient $T_{{\phi }_0(x)}M\to T_{{\phi }_0(x)}M\mathrm{/}d{\phi }_0(T_x{S}^p)$, is a linear isomorphism ${\mathbb{R}}^q\to {\nu }_{{\phi }_0,x}$: $d\phi$ is invertible and its sphere directions are precisely the tangent space of the underlying sphere. These isomorphisms vary smoothly and give its normal framing (Normal and conormal bundles of an embedded submanifold). The framing is part of the data and is fixed, not taken up to homotopy: the same underlying sphere with a different trivialization is a different framed embedded surgery sphere.

The existence of such product-embedding data is equivalent to triviality of the normal bundle, as proved in the framing lemma of this page using the tubular neighbourhood theorem. A framing alone does not specify a unique tubular embedding; here the entire embedding $\phi$ is supplied. The disk-factor convention $D^q$ matches the handle vocabulary of K handle core cocore attaching region and belt sphere: the attaching region of the standard handle is a product of a sphere and a disk, and its attaching sphere is the zero of the disk factor.

The range is the one fixed by the plan: $0\le p\le m-1$, equivalently $q\ge 1$. The case $p=m-1$ is included, and then $q=1$: the framing trivializes a normal line bundle, so the normal direction must be orientable. The case $p=m$ is not included, because then the disk factor would be $D^0$ and the construction below would require a surgery on an $S^{-1}$, which is not defined. No orientation of $M$ is assumed, and the definition performs no construction: an existence statement for $\phi$ is not part of it.

The smooth structure and boundary conventions used for $M$ are those of Smooth manifolds and their smooth charts and Smooth charts, atlases, and structures with boundary, and the smooth vector bundle conventions are those of Smooth vector bundles, rank, fibres, and trivial bundles. This definition uses no choice principle.