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  "body": "\n## Description\n\nThe sphere spectrum $S$ is the [[spectrum|1GHF4EKBXS]] with $S_n = S^n$, the $n$-sphere, for $n \\ge 0$, and with structure maps the canonical homeomorphisms $\\Sigma S^n \\cong S^{n+1}$.\nIt is the suspension spectrum $\\Sigma^\\infty S^0$ of the two-point space.\nIts homotopy groups are the [[stable homotopy groups of spheres|HGZAYPQBXW]]:\n$$\\pi_k(S) = \\operatorname{colim}_n \\pi_{k+n}(S^n) = \\pi_k^s.$$\n\n## Remarks\n\nOne has $\\pi_k(S) = 0$ for $k \u003c 0$, $\\pi_0(S) = \\Z$, $\\pi_1(S) = \\Z/2$, $\\pi_2(S) = \\Z/2$ and $\\pi_3(S) = \\Z/24$.\nFor $k \u003e 0$ every group $\\pi_k(S)$ is finite, by Serre's finiteness theorem.\nIn the [[stable homotopy category|26GTNMAF7B]], $S$ is the unit for the smash product, and maps $S \\to E$ of degree $k$ compute $\\pi_k(E)$ for every spectrum $E$.\nThe multiplication $S^n \\wedge S^m \\cong S^{n+m}$ makes $S$ a commutative ring spectrum, and $\\pi_*(S)$ is a graded commutative ring.\n\n## Explanations\n\n### L1 human\n\nThe sphere spectrum is the stable version of the two-point space $S^0$, or of the circle, the 2-sphere and all higher spheres at once.\nIt plays the role in stable homotopy theory that the integers play in algebra.\nIts homotopy groups, the stable stems, are among the most studied and least understood objects in topology.\n",
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