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  "body": "\n## Summary\n\nA classifying space of a discrete [[group|G5S1M9]] $G$ is a path-connected space $BG$ whose [[fundamental group|S1XK3HXHFX]] is $G$ and whose higher [[homotopy groups|P976WYEFMB]] all vanish.\nIt is the [[Eilenberg–MacLane space|AJHEYJE5XY]] $K(G, 1)$, and among CW complexes it is unique up to [[homotopy equivalence|XZG6MTSWCN]].\nThrough $BG$, group theory and topology meet: the [[homology of the group|CFAG3VRXEA]] is the homology of the space $BG$.\nFor topological groups, the same symbol denotes a space that classifies principal $G$-bundles, see [[principal bundle|8E37E74XVP]].\n\n## Explanations\n\n### L1 human\n\nA classifying space turns a group into a shape.\nThe loops of the shape behave like the elements of the group, and the shape has no other holes of any kind.\nFor the group $\\Z$, the circle works: going around $k$ times is the element $k$.\nFor $\\Z^2$, the torus works, with one kind of loop for each generator.\n",
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