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  "body": "\n## Summary\n\nRational homotopy theory studies spaces up to maps that induce isomorphisms on rational homotopy groups, so it ignores all torsion.\nFor simply connected spaces this coarser theory is completely algebraic.\nBy the work of Quillen and Sullivan, rational homotopy types correspond to commutative differential graded algebras over $\\Q$, or to differential graded Lie algebras.\nConcrete invariants such as [[Sullivan minimal models|3QQ2PBGY3N]] make many questions computable that are out of reach integrally.\n\n## Explanations\n\n### L1 human\n\nThe homotopy groups of even a sphere are very hard to compute, mostly because of their finite parts.\nIf one tensors everything with the rationals, the finite parts vanish and what remains is much simpler.\nFor example, $S^3$ rationally has only one homotopy group, $\\Q$ in degree $3$, while $S^2$ rationally has $\\Q$ in degrees $2$ and $3$.\nRational homotopy theory describes such answers with algebra: graded algebras with a differential, in place of spaces.\n",
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