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  "body": "\n## Summary\n\nThe Pontryagin–Thom construction starts from a closed submanifold with a tubular neighbourhood.\nIt gives a map from the ambient manifold to the [[Thom space|5MNYT3AFPH]] of the normal bundle.\nIt collapses everything outside the tubular neighbourhood to a point.\nPontryagin used it to identify framed [[cobordism|T8BVGPPSZP]] classes of submanifolds of spheres with [[homotopy groups of spheres|YD6VAXXSBM]].\nThom used it to compute cobordism groups as homotopy groups of Thom spaces, see [[Thom's cobordism theorem|HM23BV561Z]].\n\n## Explanations\n\n### L1 human\n\nPicture a closed curve sitting inside a sphere, with a thin tube around it.\nSquash everything outside the tube to a single point.\nWhat remains is the tube with its outer wall glued to that point, and this only depends on the curve and how the tube is twisted.\nRunning the construction backwards, a map to a sphere gives back a submanifold as the preimage of a point.\nFor a single point in $S^n$ with a small ball around it, the construction gives a map $S^n \\to S^n$ of degree $\\pm 1$.\n",
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