E8 manifold: Difference between revisions

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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 (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.

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+{{Short description|Topological manifold in mathematics}}
 {{DISPLAYTITLE:''E''<sub>8</sub> manifold}}
+In [[low-dimensional topology]], a branch of [[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]].
-{{short description|A topological manifold that does not admit a smooth structure.}}
-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]].
 ==History==
@@ Line 9: / Line 9: @@
 ==Construction==
-The manifold can be constructed by first plumbing together disc bundles of [[Euler number (topology)|Euler number]] 2 over the [[sphere]], according to the [[Dynkin diagram]] for <math>E_8</math>. This results in <math>P_{E_8}</math>, a 4-manifold with boundary equal to the [[Poincaré homology sphere]].  Freedman's theorem on [[fake 4-ball]]s then says we can cap off this homology sphere with a fake 4-ball to obtain the <math>E_8</math> manifold.
+The manifold can be constructed by first plumbing together disc bundles of [[Euler number (topology)|Euler number]] 2 over the [[sphere]], according to the [[Dynkin diagram]] for <math>E_8</math>. This results in <math>P_{E_8}</math>, a 4-manifold whose boundary is homeomorphic to the [[Poincaré homology sphere]].  Freedman's theorem on [[fake 4-ball]]s then says we can cap off this homology sphere with a fake 4-ball to obtain the <math>E_8</math> manifold.
 ==See also==