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  "body": "\n## Summary\n\nAn H-space is a pointed space with a continuous multiplication for which the basepoint is a two-sided unit up to homotopy.\nTopological groups and [[loop spaces|2JFF5F3EGF]] are the main examples.\nThe multiplication gives the cohomology of an H-space extra structure, such as a [[Hopf algebra structure|WJXK0V9NE9]].\nThis strongly restricts which spaces can be H-spaces.\nBy a theorem of Adams, the only spheres that are H-spaces are $S^0$, $S^1$, $S^3$ and $S^7$.\n\n## Explanations\n\n### L1 human\n\nA topological group has a multiplication with an exact identity element.\nAn H-space keeps only part of this: a multiplication with an identity that works up to deformation, and no inverses or associativity are required.\nThe circle $S^1$, seen as the unit complex numbers, is an H-space.\nThe sphere $S^7$, seen as the unit octonions, is an H-space although the octonions are not associative, so $S^7$ is not a group.\n",
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