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  "body": "\n## Statement\n\nA *unimodular form* is a [[symmetric bilinear form|0AXZNE5XNH]] $Q \\colon \\Z^n \\times \\Z^n \\to \\Z$ whose matrix has determinant $\\pm 1$.\nIt is *even* if $Q(x, x)$ is even for all $x \\in \\Z^n$, and *odd* otherwise.\n\n1. For every unimodular form $Q$ there is a closed, simply connected, topological [[4-manifold|8H58Q6YJCH]] whose [[intersection form|WK8GKRKAVD]] is isomorphic to $Q$.\n2. Two closed, simply connected, topological $4$-manifolds are [[homeomorphic|0QT00PB5G2]] if and only if their intersection forms are isomorphic and their [[Kirby–Siebenmann invariants|E4NJER1RFQ]] in $\\Z/2$ are equal.\n3. If $Q$ is even, there is exactly one such manifold with form $Q$ up to homeomorphism. If $Q$ is odd, there are exactly two, one with each value of the Kirby–Siebenmann invariant.\n\n## Remarks\n\nMichael Freedman proved the theorem in 1982.\nThe main step is his proof that Casson handles are homeomorphic to standard open handles, which makes the topological h-cobordism theorem hold for simply connected $4$-manifolds.\nFor an even form, the Kirby–Siebenmann invariant equals the signature divided by $8$, modulo $2$, so it is determined by $Q$.\nThe manifold with odd form and Kirby–Siebenmann invariant $1$ admits no smooth structure.\nFor example, the form $(1)$ is realized by the complex projective plane $\\C P^2$ and by a fake $\\C P^2$ that is not homeomorphic to it.\nThe [[$E_8$ manifold|BNDRN0G9BR]] realizes the $E_8$ form.\nThe case $Q = 0$ of rank $0$ is the four-dimensional topological Poincaré conjecture: a homotopy $4$-sphere is homeomorphic to $S^4$.\n\n## Explanations\n\n### L1 human\n\nIn dimension four, the way surfaces cross inside a manifold is recorded by a square table of integers, the intersection form.\nFreedman's theorem says that, for manifolds without loops that cannot be contracted, this table and one extra bit decide the manifold up to continuous deformation.\nEvery possible table occurs.\n\n### L2 human\n\nThe theorem reduces the topological classification of closed simply connected $4$-manifolds to the algebra of unimodular forms over $\\Z$.\nIndefinite forms are classified by rank, signature and type, so these manifolds are well understood.\nThe smooth picture is very different: [[Donaldson's theorem|34B7K77P31]] and [[Rokhlin's theorem|12BXVH5W5H]] show that many of Freedman's manifolds have no smooth structure.\nTogether the two results give [[exotic $\\R^4$|X44PNC38E2]].\n",
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