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  "body": "\n## Summary\n\nA Thom spectrum is a [[spectrum|1GHF4EKBXS]] whose spaces are the [[Thom spaces|5MNYT3AFPH]] of a compatible sequence of vector bundles.\nThe basic examples are $MO$, $MSO$ and $MU$, built from the universal bundles over the classifying spaces $BO(n)$, $BSO(n)$ and $BU(n)$.\nBy the Pontryagin–Thom construction, the homotopy groups of these spectra are the [[cobordism|T8BVGPPSZP]] groups of manifolds with the matching structure, as in [[Thom's cobordism theorem|HM23BV561Z]].\nSo Thom spectra turn geometric classification problems into problems of stable homotopy theory.\n\n## Explanations\n\n### L1 human\n\nThe Thom space of a vector bundle is obtained by adding one point at infinity to every fibre and then gluing all these points together.\nIf one adds a trivial line to a bundle, its Thom space becomes the suspension of the old Thom space.\nA Thom spectrum strings together Thom spaces of bigger and bigger bundles with these suspension maps.\nFor example, $MO$ uses the universal $n$-plane bundle over $BO(n)$ for every $n$.\nIts zeroth homotopy group is $\\Z/2$, which counts finite sets of points up to cobordism: a set is a boundary exactly when it has an even number of points.\n",
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