In
low-dimensional topology, a branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the ''E''
8 manifold is the unique
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
,
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
topological
4-manifold with
intersection form the
''E''8 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
In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
.
Construction
The manifold can be constructed by first plumbing together disc bundles of
Euler number
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
2 over the
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, according to the
Dynkin diagram for
. This results in
, 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
manifold.
See also
*
*
*
References
*
*
{{DEFAULTSORT:E8 Manifold
4-manifolds
Geometric topology
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...