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  "body": "\n## Summary\n\nA $4$-manifold is a [[topological manifold|QQQ9B1DK4P]] of dimension $4$.\nDimension $4$ is special: the methods that classify manifolds of dimension at least $5$ fail, and topological and smooth $4$-manifolds behave very differently.\nFreedman classified simply connected closed topological $4$-manifolds through their [[intersection form|WK8GKRKAVD]], together with the Kirby–Siebenmann invariant when the form is odd.\n[[Donaldson's theorem|34B7K77P31]] and Seiberg–Witten theory show that many of them have no smooth structure, or many smooth structures.\nEven $\\R^4$ has [[exotic smooth structures|X44PNC38E2]], which happens in no other dimension.\n\n## Explanations\n\n### L1 human\n\nA $4$-manifold is a space that looks like $4$-dimensional space near each point.\nExamples are the $4$-sphere, the product $S^2 \\times S^2$ of two spheres, and the complex projective plane.\nDimension $4$ is the hardest one in topology.\nThere is too little room to untangle things as in higher dimensions, and too much room for the tools of dimensions $2$ and $3$.\nIdeas from physics, such as gauge theory, gave the main tools for smooth $4$-manifolds.\n",
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