The $E_8$ manifold is the unique compact, simply connected topological 4-manifold with intersection form the E8 lattice.
The $E_{8}$ manifold has no smooth structure, nor even is triangulable as a simplicial complex.
(Stub for the moment)
Robion Kirby and Paul Melvin, The E8â€“manifold, singular fibers and handlebody decompositions, Geometry & Topology Monographs Volume 2: Proceedings of the Kirbyfest, pp. 233 - 258.
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