E8 manifold: Difference between revisions
Added {{short description|}} |
Duckmather (talk | contribs) added context template |
||
Line 1: | Line 1: | ||
{{DISPLAYTITLE:''E''<sub>8</sub> manifold}} |
{{DISPLAYTITLE:''E''<sub>8</sub> manifold}} |
||
{{short description|A topological manifold that does not admit a smooth structure.}} |
{{short description|A topological manifold that does not admit a smooth structure.}}{{Context|date=July 2021}} |
||
In [[mathematics]], the '''''E''<sub>8</sub> manifold''' is the unique [[Compact space|compact]], [[simply connected]] topological [[4-manifold]] with [[Intersection form (4-manifold)|intersection form]] the [[E8 lattice|''E''<sub>8</sub> lattice]]. |
In [[mathematics]], the '''''E''<sub>8</sub> manifold''' is the unique [[Compact space|compact]], [[simply connected]] topological [[4-manifold]] with [[Intersection form (4-manifold)|intersection form]] the [[E8 lattice|''E''<sub>8</sub> lattice]]. |
||
Revision as of 22:18, 24 July 2021
This article provides insufficient context for those unfamiliar with the subject.(July 2021) |
In mathematics, the E8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the E8 lattice.
History
The 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 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 . This results in , 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 manifold.
See also
- E8 (mathematics) – 248-dimensional exceptional simple Lie group
- 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.