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  "body": "\n## Statement\n\nLet $M$ be a closed, oriented, simply connected, [[smooth|F45VQ47GPV]] [[4-manifold|8H58Q6YJCH]].\nLet $Q_M$ be its [[intersection form|WK8GKRKAVD]] on $H_2(M; \\Z) \\cong \\Z^n$.\nIf $Q_M$ is negative definite, then $Q_M$ is isomorphic over $\\Z$ to the diagonal form $-I_n$, that is, $H_2(M; \\Z)$ has a basis $e_1, \\ldots, e_n$ with $Q_M(e_i, e_j) = -\\delta_{ij}$.\n\n## Remarks\n\nSimon Donaldson proved the theorem in 1983 with gauge theory.\nFor a suitable Riemannian metric, the moduli space of anti-self-dual [[instantons|YWBTGC7MFA]] on the $SU(2)$-bundle with $c_2 = 1$ is a $5$-dimensional manifold with singular points.\nIts singular points are the reducible connections, which correspond to the pairs $\\pm e$ with $Q_M(e, e) = -1$, and near each of them it is a cone on $\\C P^2$.\nOne end of the moduli space is a collar on $M$, so $M$ is cobordant to a disjoint union of copies of $\\C P^2$ up to orientation.\nComparing signatures forces many vectors of square $-1$, and this makes the form diagonal.\nDonaldson later removed the hypothesis that $M$ is simply connected.\nReplacing the orientation of $M$ by its opposite gives the positive definite version.\nSmoothness is essential.\nBy [[Freedman's classification|HSD6FKCYRC]], the negative of the $E_8$ form is the intersection form of a closed simply connected topological $4$-manifold.\nThis form is definite but not diagonalizable.\n\n## Explanations\n\n### L1 human\n\nThe intersection form records how surfaces cross inside a four-dimensional space.\nDonaldson's theorem says that in a smooth such space, if all self-crossings have the same sign, the table is as simple as possible: a diagonal of $-1$'s.\nMany tables allowed by algebra and by topology are forbidden by smoothness.\n\n### L2 human\n\nCombined with Freedman's theorem, it shows that most definite unimodular forms, such as $E_8$ and $E_8 \\oplus E_8$, are realized by topological $4$-manifolds that admit no smooth structure.\nIt was the first major application of the Yang–Mills equations to topology and it started [[Donaldson theory|ZQW6M0S6M0]].\nA short proof of the same result was later given with [[Seiberg–Witten invariants|PRDDMSTR4X]].\nIt also leads to the existence of [[exotic $\\R^4$|X44PNC38E2]].\n",
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