nLab E₈ manifold

Idea

The $E_8$ manifold is the unique compact, simply connected topological 4-manifold with intersection form the E₈ lattice.

The $E_{8}$ manifold has no smooth structure, nor even is triangulable as a simplicial complex.

Related entries

• h-cobordism theorem

Articles on geometry and topology of 4-manifolds:

• Basic concepts:

  • Donaldson theory and related articles
  • Seiberg-Witten theory and related articles
  • intersection form
  • Kirby-Siebenmann invariant
  • signature

• Important examples:

  • Barlow surface$\mathbb{C}P^2\# 8\overline{\mathbb{C}P^2}$
  • Dolgachev surface$\mathbb{C}P^2\# 9\overline{\mathbb{C}P^2}$
  • E₈ manifold$M_{E_8}$
  • exotic R^4
  • fake second complex projective space$\mathbb{C}P^2$

• Central results:

  • Donaldson theorem (Fields Medal 1986)
  • Freedman classification (Fields Medal 1986)
  • Furuta theorem (10/8 theorem)
  • Hirzebruch signature theorem
  • Hitchin-Thorpe inequality
  • Quinn theorem
  • Rokhlin theorem
  • van der Blij lemma
  • Wall theorems
  • Wu formula

• Open problems:

  • 3/2 conjecture
  • 11/8 conjecture
  • simple type conjecture
  • Poincaré conjecture

References

• Robion Kirby, Paul Melvin: The $E_8$–manifold, singular fibers and handlebody decompositions, Geometry & Topology Monographs 2 (1999) 233-258 [arXiv:math/9911253, GTMon2:13]

See also:

• Wikipeida, E₈ manifold