nLab Barlow surface

Idea

Barlow surfaces provide countably infinitely many non-diffeomorphic smooth structures on the complex surface $\mathbb{C}P^2\# 8\overline{\mathbb{C}P^2}$, which is the eight-fold blow-up of the second complex projective space. (One blow-up more yields the Dolgachev surfaces $\mathbb{C}P^2\# 9\overline{\mathbb{C}P^2}$.)

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 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

• Rebecca Barlow, Some new surfaces with $p_g=0$, Duke Mathematical Journal 51 4 (1984), pp. 889–904 [doi:10.1215/S0012-7094-84-05139-1 ISSN 0012-7094 MR 0771386]

• Dieter Kotschick, On manifolds homeomorphic to $\mathbb{C}P^2\#8\overline{\mathbb{C}P^2}$, Invent. Math. 95 3, pp. 591-600 (1989) [doi:10.1007/BF01393892]

See also:

• Wikipedia, Barlow surface