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  "body": "\n## Summary\n\nTopological modular forms is a [[ring spectrum|BAW0T68GWK]], written $\\mathrm{tmf}$, that collects all [[elliptic cohomology|XH68V2BP4K]] theories into one universal object.\nGoerss, Hopkins and Miller built it as the global sections of a sheaf of $E_\\infty$ ring spectra on the moduli stack of elliptic curves.\nIts homotopy groups map to the ring of integral [[modular forms|GWKN5VG78A]], and this map is an isomorphism after tensoring with $\\Q$.\nThe torsion in $\\pi_* \\mathrm{tmf}$ detects many elements of the stable homotopy groups of spheres, and $\\mathrm{tmf}$ receives the Witten genus of string manifolds.\n\n## Explanations\n\n### L1 human\n\nEach elliptic curve gives an elliptic cohomology theory, and modular forms are functions on the space of all elliptic curves.\nTopological modular forms glue all the elliptic cohomology theories together in a homotopy-theoretic way.\nThe result is a single cohomology theory whose coefficients are \"modular forms, plus extra torsion\".\nFor example, $\\pi_3 \\mathrm{tmf} \\cong \\Z/24$, which matches the stable homotopy group $\\pi_3^s \\cong \\Z/24$ of the sphere.\nNo classical modular form explains this group, so $\\mathrm{tmf}$ sees more than modular forms do.\n",
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