---
tag: BNDRN0G9BR
kind: object
name: E8 manifold
msc: [57K40, 57R55]
instance_of: [8H58Q6YJCH]
sources:
  - url: https://en.wikipedia.org/wiki/E8_manifold
same_as:
  wikidata: Q2788118
  wikipedia: E8 manifold
---

## Description

The *$E_8$ manifold* is the closed, simply connected, topological [[4-manifold|8H58Q6YJCH]] whose [[intersection form|WK8GKRKAVD]] is the form of the $E_8$ lattice.
This form is positive definite, even and unimodular, of rank $8$ and [[signature|07EZA80NEH]] $8$.
Its existence and its uniqueness up to [[homeomorphism|0QT00PB5G2]] follow from [[Freedman's classification|HSD6FKCYRC]], since the form is even.
One construction plumbs together eight disk bundles over the $2$-sphere with Euler number $2$, following the pattern of the $E_8$ Dynkin diagram.
The boundary of this plumbing is the [[Poincaré homology sphere|NHERCK7C43]].
Freedman showed that this homology sphere bounds a contractible topological $4$-manifold, and gluing it to the plumbing along the common boundary gives the $E_8$ manifold.

## Remarks

The $E_8$ manifold admits no [[smooth structure|R2TQ82PCNB]].
Indeed its intersection form is even, so a smooth structure would give a closed smooth spin $4$-manifold of signature $8$, against [[Rokhlin's theorem|12BXVH5W5H]], which demands signature divisible by $16$.
It also follows from [[Donaldson's theorem|34B7K77P31]], since the $E_8$ form is definite but not diagonalizable over $\Z$.
Its Kirby–Siebenmann invariant is $1$, equal to the signature divided by $8$ modulo $2$.
Andrew Casson showed that it admits no triangulation at all.

## Explanations

### L1 human

The $E_8$ manifold is a four-dimensional space that can be built by gluing topological pieces but that has no smooth version.
Its fingerprint is the $E_8$ pattern, a famous $8 \times 8$ table of numbers that records how surfaces inside it cross.
It was one of the first proofs that topological and smooth manifolds really differ in dimension four.