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  "body": "\n## Statement\n\nLet $M$ be an [[orientable|702SNFQG0W]] [[3-manifold|R20TB2KEQQ]] whose second [[homotopy group|P976WYEFMB]] $\\pi_2(M)$ is not trivial.\nThen there is an embedding $S^2 \\to M$ of the $2$-sphere that is not homotopic to a constant map.\n\n## Remarks\n\nPapakyriakopoulos proved the theorem in 1957 with the tower construction also used for [[Dehn's lemma|FH5WSPD79F]].\nFor example, in $M = S^2 \\times S^1$ the sphere $S^2 \\times \\set{1}$ is embedded and generates $\\pi_2(M) \\cong \\Z$.\nA main consequence: if $M$ is orientable and irreducible, meaning that every embedded $2$-sphere bounds a $3$-ball, then $\\pi_2(M) = 0$.\nIf moreover $\\pi_1(M)$ is infinite, then $M$ is aspherical, that is, its universal cover is contractible.\nOrientability cannot be dropped: in the non-orientable case one must also allow embedded two-sided projective planes, as in Epstein's version of the theorem.\n\n## Explanations\n\n### L1 human\n\nThe second homotopy group records spheres in a space that cannot be shrunk to a point.\nThe sphere theorem says that, in an orientable $3$-manifold, if there is any such sphere, then there is one that does not cross itself.\nSo a nontrivial $\\pi_2$ is always witnessed by an embedded sphere.\n\n### L2 human\n\nThe theorem explains why irreducible orientable $3$-manifolds have trivial $\\pi_2$.\nWith infinite fundamental group, they are then Eilenberg–MacLane spaces, determined up to homotopy by $\\pi_1$.\nThis is the starting point for the study of $3$-manifolds through their fundamental groups.\n",
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