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  "body": "\n## Statement\n\nLet $\\Omega_n^O$ be the group of [[cobordism|T8BVGPPSZP]] classes of closed smooth $n$-manifolds.\nLet $\\Omega_*^O = \\bigoplus_{n \\ge 0} \\Omega_n^O$ be the unoriented [[cobordism ring|C02WQY819R]], with product induced by the Cartesian product of manifolds.\nFor $k \\ge 1$, let $MO(k)$ be the [[Thom space|5MNYT3AFPH]] of the [[universal bundle|1HDH716Z24]] $\\gamma^k$ of rank $k$ over $BO(k)$.\n\n1. For $k \\ge n + 2$, there is an isomorphism $\\Omega_n^O \\cong \\pi_{n+k}(MO(k))$ with the [[homotopy group|P976WYEFMB]] of $MO(k)$.\n2. The ring $\\Omega_*^O$ is a polynomial ring over $\\Z/2\\Z$ with one generator $x_i$ in each degree $i \\ge 2$ that is not of the form $2^j - 1$:\n$$\\Omega_*^O \\cong (\\Z/2\\Z)[x_2, x_4, x_5, x_6, x_8, x_9, \\ldots].$$\n3. Two closed smooth $n$-manifolds are cobordant if and only if all their [[Stiefel–Whitney numbers|JQCK7A210W]] agree.\n\n## Remarks\n\nThe isomorphism in part 1 is the Pontryagin–Thom construction.\nA closed $n$-manifold is embedded in $\\R^{n+k}$, and a tubular neighbourhood of it is mapped to the Thom space of its normal bundle.\nCollapsing the rest of the sphere $S^{n+k}$ gives an element of $\\pi_{n+k}(MO(k))$, and transversality gives the inverse.\nParts 2 and 3 then follow from a computation of the homotopy of $MO(k)$ in the stable range by mod $2$ cohomology and Steenrod squares.\n\nExample: in low degrees, $\\Omega_1^O = 0$, $\\Omega_2^O \\cong \\Z/2\\Z$ generated by $\\R P^2$, $\\Omega_3^O = 0$ and $\\Omega_4^O \\cong (\\Z/2\\Z)^2$, spanned by the classes $x_2^2$ and $x_4$.\nFor even $i$, the generator $x_i$ can be taken to be $\\R P^i$.\n\nThe result was proved by René Thom in 1954.\n\n## Explanations\n\n### L1 human\n\nDeciding whether a manifold is the boundary of another manifold sounds like a hard geometric problem.\nThom's theorem turns it into a computation in homotopy theory, and then solves it completely.\nIn the end, whether a manifold bounds is decided by a finite list of numbers modulo $2$, its Stiefel–Whitney numbers.\n\n### L2 human\n\nThe Pontryagin–Thom construction is the model for many later results that turn geometric questions about manifolds into stable homotopy groups of Thom spectra.\nThe same method computes the oriented cobordism ring through $MSO(k)$, and complex cobordism through $MU(k)$, whose homotopy is described by [[Quillen's theorem|4E518M24MA]].\nPart 3 combines Pontryagin's observation that cobordant manifolds have the same Stiefel–Whitney numbers with Thom's converse.\n",
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