E₈ manifold

In mathematics, the E₈ manifold is the unique compact, simply connected topological 4-manifold with intersection form the E₈ lattice.

History

The ${\displaystyle E_{8}}$ 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 ${\displaystyle E_{8}}$ 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 ${\displaystyle E_{8}}$. This results in ${\displaystyle P_{E_{8}}}$, a 4-manifold with boundary equal 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.

See also

• E₈ (mathematics)
• Glossary of topology
• List of geometric topology topics

References

• Freedman, Michael Hartley (1982). "The topology of four-dimensional manifolds". Journal of Differential Geometry. 17 (3): 357–453. ISSN 0022-040X. MR 0679066.
• Scorpan, Alexandru (2005). The Wild World of 4-manifolds. American Mathematical Society. ISBN 0-8218-3749-4.