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  "body": "\n## Example\n\nIdentify the $3$-sphere $S^3$ with the group $SU(2)$ of unit quaternions.\nThe double cover $SU(2) \\to SO(3)$ maps onto the rotation group of a regular icosahedron, a group of order $60$ isomorphic to $A_5$.\nIts preimage $I^*$ in $SU(2)$ is the binary icosahedral group, of order $120$.\nThe Poincaré homology sphere is the quotient $P = S^3 / I^*$ of $S^3$ by the action of $I^*$ by left multiplication.\n\nThis action is free, so $P$ is a closed orientable [[3-manifold|R20TB2KEQQ]] with [[fundamental group|S1XK3HXHFX]] $I^*$.\nThe group $I^*$ is perfect, so $H_1(P; \\Z)$, the abelianization of $I^*$, is $0$.\nBy [[Poincaré duality|9VY90HJVZX]], $H_2(P; \\Z) \\cong H^1(P; \\Z) = \\Hom(H_1(P; \\Z), \\Z) = 0$, and $H_3(P; \\Z) = \\Z$.\nSo $P$ has the homology of $S^3$ and is a [[homology sphere|HK8NS02NNE]].\nIt is not homeomorphic to $S^3$, because its fundamental group is not trivial.\n\n## Remarks\n\nThe same manifold is obtained from a solid regular dodecahedron by gluing each face to the opposite face after a turn of $1/10$ of a full turn.\nIt is also the link of the singularity of $x^2 + y^3 + z^5 = 0$ in $\\C^3$, often written $\\Sigma(2, 3, 5)$.\nPoincaré found it while testing whether homology characterises the $3$-sphere, and this example led him to the [[Poincaré conjecture|3R064KFF68]], which uses the fundamental group instead.\n\n## Explanations\n\n### L1 human\n\nThe Poincaré homology sphere looks like the $3$-sphere to homology: it has no holes of dimension $1$ or $2$.\nStill, it contains loops that cannot be shrunk to a point.\nIt shows that counting holes with homology is not enough to recognise the sphere.\n",
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