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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold and let $N \\subset M$ be a compact smooth [[submanifold|VKFW27WXEZ]] without boundary, of codimension $k$, with normal vector bundle $\\nu$.\nChoose an inner product on $\\nu$ and a closed tubular neighbourhood $U$ of $N$.\nLet $e \\colon D(\\nu) \\to U$ be a diffeomorphism from the unit disc bundle that is the identity on the zero section.\nLet $\\operatorname{Th}(\\nu) = D(\\nu) / S(\\nu)$ be the [[Thom space|5MNYT3AFPH]] of $\\nu$ with base point $*$.\nThe *Pontryagin–Thom collapse map* $c \\colon M \\to \\operatorname{Th}(\\nu)$ is the following map:\n$$c(x) = \\begin{cases} [e^{-1}(x)] \u0026 \\text{if } x \\in U, \\\\ * \u0026 \\text{if } x \\notin U. \\end{cases}$$\nSuppose that $\\nu$ comes with a framing, that is, a trivialisation $\\nu \\cong N \\times \\R^k$.\nThe map $\\operatorname{Th}(N \\times \\R^k) \\to D^k / S^{k-1} \\cong S^k$ forgets the point of $N$.\nIts composite with $c$ is a map $M \\to S^k$.\n\n## Remarks\n\nThe map $c$ is continuous because the boundary of $U$ is the image of $S(\\nu)$, which goes to the base point; its homotopy class does not depend on the choices.\nExample: for $N$ a point of $M = S^n$, the normal bundle is $\\R^n$ with its standard framing, and $c \\colon S^n \\to D^n / S^{n-1} \\cong S^n$ has degree $\\pm 1$.\nNon-example of the framed version: the circle $N = \\R P^1$ in $M = \\R P^2$ has the Möbius bundle as normal bundle, which has no framing.\nThe collapse map still exists, with target $\\operatorname{Th}(\\nu) \\cong \\R P^2$, but it does not give a map to $S^1$.\n\nFor $M = S^{n+k}$, Pontryagin's theorem says that the construction gives a bijection between framed cobordism classes of closed framed $n$-dimensional submanifolds of $S^{n+k}$ and the homotopy group $\\pi_{n+k}(S^k)$.\nThe inverse sends a smooth map, made transverse to a point, to the preimage of that point with the pulled back framing.\n\n## Explanations\n\n### L1 human\n\nKeep a thin tube around the submanifold and squash the rest of the space to one point.\nWhen the tube is a product, the tube with its outer wall squashed is a sphere, so the result is a map to a sphere.\nThis dictionary between submanifolds and maps lets homotopy theory count manifolds up to cobordism.\n",
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