nLab exotic R^4

Idea

There exists exotic smooth structure on the Euclidean space $\mathbb{R}^4$ of dimension 4.

(For all other dimensions there is a unique smooth structure, the standard one, see at smooth structure).

Related entries

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 $\mathbb{R}^n$
  • 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

• Robert Gompf, Three exotic $\mathbb{R}^4$ and other anomalies, J. Differential Geom. 18 2 (1983), pp. 317-328 [doi:10.4310/jdg/1214437666.full]

• Robert Gompf, An infinite set of exotic $\mathbb{R}^4$‘s, J. Differential Geom. 21 2 (1985) [doi:10.4310/jdg/1214439566 pdf]

See also:

• Wikipedia, Exotic R4