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  "body": "\n## Description\n\nThe *$E_8$ manifold* is the closed, simply connected, topological [[4-manifold|8H58Q6YJCH]] whose [[intersection form|WK8GKRKAVD]] is the form of the $E_8$ lattice.\nThis form is positive definite, even and unimodular, of rank $8$ and [[signature|07EZA80NEH]] $8$.\nIts existence and its uniqueness up to [[homeomorphism|0QT00PB5G2]] follow from [[Freedman's classification|HSD6FKCYRC]], since the form is even.\nOne construction plumbs together eight disk bundles over the $2$-sphere with Euler number $2$, following the pattern of the $E_8$ Dynkin diagram.\nThe boundary of this plumbing is the [[Poincaré homology sphere|NHERCK7C43]].\nFreedman showed that this homology sphere bounds a contractible topological $4$-manifold, and gluing it to the plumbing along the common boundary gives the $E_8$ manifold.\n\n## Remarks\n\nThe $E_8$ manifold admits no [[smooth structure|R2TQ82PCNB]].\nIndeed its intersection form is even, so a smooth structure would give a closed smooth spin $4$-manifold of signature $8$, against [[Rokhlin's theorem|12BXVH5W5H]], which demands signature divisible by $16$.\nIt also follows from [[Donaldson's theorem|34B7K77P31]], since the $E_8$ form is definite but not diagonalizable over $\\Z$.\nIts Kirby–Siebenmann invariant is $1$, equal to the signature divided by $8$ modulo $2$.\nAndrew Casson showed that it admits no triangulation at all.\n\n## Explanations\n\n### L1 human\n\nThe $E_8$ manifold is a four-dimensional space that can be built by gluing topological pieces but that has no smooth version.\nIts fingerprint is the $E_8$ pattern, a famous $8 \\times 8$ table of numbers that records how surfaces inside it cross.\nIt was one of the first proofs that topological and smooth manifolds really differ in dimension four.\n",
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