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  "body": "\n## Summary\n\nThe $n$-th homotopy group $\\pi_n(X, x_0)$ of a pointed space consists of the homotopy classes of based maps from the $n$-sphere to $(X, x_0)$.\nFor $n = 1$ it is the [[fundamental group|S1XK3HXHFX]], and for $n \\ge 2$ it is abelian.\nHomotopy groups are invariants of homotopy type, and on CW complexes they detect homotopy equivalences, by the [[Whitehead theorem|6P1E2FJPSD]].\nThey are easy to define and very hard to compute, even for spheres.\n\n## Explanations\n\n### L1 human\n\nThe group $\\pi_n(X)$ records the ways an $n$-dimensional sphere can be placed in a space, where two placements count as the same if one can be deformed into the other.\nA balloon inside a solid ball can always be shrunk to a point, so $\\pi_2$ of a ball is trivial.\nA balloon wrapped once around the surface of a sphere cannot be shrunk on that surface, and $\\pi_2(S^2) \\cong \\Z$ counts how many times it wraps.\n",
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