E8 manifold
Encyclopedia
In mathematics
, the E8 manifold is the unique compact
, simply connected topological 4-manifold
with intersection form
the E8 lattice
.
The E8 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 E8 manifold is not even triangulable
as a simplicial complex
.
The manifold can be constructed by first plumbing together disc bundles of Euler number 2 over the sphere
, according to the Dynkin diagram for E8. This results in PE8, a 4-manifold with boundary equal to the Poincare 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 E8 manifold.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the E8 manifold is the unique compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, simply connected topological 4-manifold
4-manifold
In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...
with intersection form
Intersection form (4-manifold)
In mathematics, the intersection form of an oriented compact 4-manifold is a special symmetric bilinear form on the 2nd cohomology group of the 4-manifold...
the E8 lattice
E8 lattice
In mathematics, the E8 lattice is a special lattice in R8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8...
.
The E8 manifold was discovered by Michael Freedman
Michael Freedman
Michael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...
in 1982. Rokhlin's theorem
Rokhlin's theorem
In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, compact 4-manifold M has a spin structure , then the signature of its intersection form, a quadratic form on the second cohomology group H2, is divisible by 16...
shows that it has no smooth structure
Smooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold....
(as does Donaldson's theorem
Donaldson's theorem
In mathematics, Donaldson's theorem states that a definite intersection form of a simply connected smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive definite, it can be diagonalized to the identity matrix...
), and in fact, combined with the work of Andrew Casson
Andrew Casson
Andrew John Casson FRS is a mathematician, an expert on geometric topology.Casson is the Philip Schuyler Beebe Professor of Mathematics at Yale University in the United States where he served as department chair between 2004 and 2007. His Ph.D. advisor at the University of Liverpool was C. T. C...
on the Casson invariant
Casson invariant
In 3-dimensional topology, a part of the mathematical field of geometric topology, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson....
, this shows that the E8 manifold is not even triangulable
Triangulation (topology)
In mathematics, topology generalizes the notion of triangulation in a natural way as follows:A triangulation of a topological space X is a simplicial complex K, homeomorphic to X, together with a homeomorphism h:K\to X....
as a simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
.
The manifold can be constructed by first plumbing together disc bundles of Euler number 2 over the sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
, according to the Dynkin diagram for E8. This results in PE8, a 4-manifold with boundary equal to the Poincare homology sphere. Freedman's theorem on fake 4-ball
Fake 4-ball
In mathematics, a fake 4-ball is a compact contractible topological 4-manifold. Michael Freedman proved that every three-dimensional homology sphere bounds a fake 4-ball. His construction involves the use of Casson handles and so does not work in the smooth category.-References:*Alexandru...
s then says we can cap off this homology sphere with a fake 4-ball to obtain the E8 manifold.