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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $(X, e, \\mu)$ be a path-connected [[H-space|EASHTX8EG5]] with the homotopy type of a CW complex.\nAssume that each $H^n(X; k)$ is finite-dimensional.\nBy the Künneth theorem, the cross product gives an isomorphism $H^*(X; k) \\otimes_k H^*(X; k) \\cong H^*(X \\times X; k)$.\nThe *Hopf algebra structure on cohomology* of $X$ consists of:\n\n1. the [[cohomology ring|RDT4785M4D]] $H^*(X; k)$, with the [[cup product|JX47MS1R1H]] as multiplication and $1$ as unit;\n2. the comultiplication $\\Delta \\colon H^*(X; k) \\to H^*(X; k) \\otimes_k H^*(X; k)$, defined as $\\mu^*$ followed by the inverse of the Künneth isomorphism;\n3. the counit $H^*(X; k) \\to k$ given by restriction to the basepoint $e$.\n\nWith these maps, $H^0(X; k) = k$, $\\Delta$ is a counital homomorphism of graded algebras, and $\\Delta(x) = x \\otimes 1 + 1 \\otimes x + \\sum x'_i \\otimes x''_i$ with $0 \u003c \\abs{x'_i}, \\abs{x''_i} \u003c \\abs{x}$ for $\\abs{x} \u003e 0$.\nIf $\\mu$ is homotopy associative, $\\Delta$ is also coassociative and $H^*(X; k)$ is a connected, graded-commutative Hopf algebra.\n\n## Remarks\n\nIn the homotopy associative case the antipode exists automatically, because the bialgebra is connected: $H^0(X; k) = k$.\n\nExample: for $X = S^3$, the unit quaternions, $H^*(S^3; k) = k \\oplus k x$ with $\\abs{x} = 3$.\nBy degrees, $\\Delta(x) = x \\otimes 1 + 1 \\otimes x$, and $\\Delta(x)^2 = 0 = \\Delta(x^2)$, as $x \\otimes x$ and $-x \\otimes x$ cancel by the sign rule.\n\nNon-example: on $H^*(S^2; \\Q) = \\Q[x]/(x^2)$ with $\\abs{x} = 2$, the only candidate $\\Delta(x) = x \\otimes 1 + 1 \\otimes x$ is not an algebra map, since $\\Delta(x)^2 = 2 x \\otimes x \\ne 0 = \\Delta(x^2)$.\nThis reflects the fact that $S^2$ is not an H-space.\n\nHopf's theorem: if $k$ has characteristic $0$ and $H^*(X; k)$ is finite-dimensional, then $H^*(X; k)$ is an exterior algebra on finitely many generators of odd degree.\n",
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