E8 manifold explained

In low-dimensional topology, a branch of mathematics, the E₈ manifold is the unique compact, simply connected topological 4-manifold with intersection form the E₈ lattice.

History

The

E _{8} manifold was discovered by Michael Freedman in 1982. Rokhlin's theorem shows that it has no smooth structure (as does Donaldson's theorem), and in fact, combined with the work of Andrew Casson on the Casson invariant, this shows that the E _{8} manifold is not even triangulable as a simplicial complex.

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

E _{8} . This results in 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 E _{8} manifold.

References

• Freedman . Michael Hartley . Michael Freedman. The topology of four-dimensional manifolds . 679066 . 1982 . . 0022-040X . 17 . 3 . 357–453.
• Book: Scorpan, Alexandru . The Wild World of 4-manifolds. . 2005. 0-8218-3749-4.