E8 manifold

Construction

The manifold can be constructed by first plumbing together disc bundles of Euler number 2 over the sphere, according to the Dynkin diagram for ${\displaystyle E_8}$. This results in ${\displaystyle P_{E_8}}$, a 4-manifold whose boundary is homeomorphic to the Poincaré homology sphere. Freedman's theorem on fake 4-balls then says we can cap off this homology sphere with a fake 4-ball to obtain the ${\displaystyle E_8}$ manifold.