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  "body": "\n## Definition\n\nAn *H-space* is a [[pointed space|T9DG8DDE75]] $(X, e)$ together with a continuous map $\\mu \\colon X \\times X \\to X$ with these properties.\n\n1. $\\mu(e, e) = e$.\n2. The map $X \\to X$, $x \\mapsto \\mu(e, x)$, is [[homotopic|F2F143KQ2M]] to $\\id_X$ through pointed maps.\n3. The map $X \\to X$, $x \\mapsto \\mu(x, e)$, is homotopic to $\\id_X$ through pointed maps.\n\n## Remarks\n\nSome authors drop the condition that the homotopies fix $e$; for CW complexes the two definitions give the same class of spaces.\nAn H-space is *homotopy associative* if $\\mu \\circ (\\mu \\times \\id) \\simeq \\mu \\circ (\\id \\times \\mu)$, and *homotopy commutative* if $\\mu \\simeq \\mu \\circ \\tau$, where $\\tau$ swaps the factors.\n\nExample: every topological group $G$ with $e$ its identity and $\\mu$ its multiplication is an H-space, since $\\mu(e, x) = \\mu(x, e) = x$ on the nose.\n\nExample: the [[loop space|2JFF5F3EGF]] $\\Omega Y$ with concatenation of loops is an H-space; the constant loop is a unit up to the standard reparametrization homotopy.\n\nNon-example: $S^2$ admits no such $\\mu$.\nIf it did, let $x \\in H^2(S^2; \\Z)$ be a generator; the unit condition forces $\\mu^*(x) = x \\times 1 + 1 \\times x$ in $H^2(S^2 \\times S^2; \\Z)$.\nThen $0 = \\mu^*(x^2) = (x \\times 1 + 1 \\times x)^2 = 2 (x \\times x)$ for the [[cup product|JX47MS1R1H]], which is false since $x \\times x$ generates $H^4(S^2 \\times S^2; \\Z) \\cong \\Z$.\n",
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