nLab fake second complex projective space

Idea

The fake second complex projective space is a topological 4-manifold with very similar properties as the second complex projective space

Definition

Freedman's classification shows, that up to homeomorphism there is a unique fake second complex projective space $*\mathbb{C}P^2$ and orientation-reversed fake second complex projective space $*\overline{\mathbb{C}P^2}\coloneqq\overline{*\mathbb{C}P^2}$, which are also simply connected closed topological 4-manifolds with respective intersection form $Q_{\mathbb{C}P^2}=[+1]$ and $Q_{\overline{\mathbb{C}P^2}}=[-1]$, but have a switched Kirby-Siebenmann invariant:

$$ks(*\mathbb{C}P^2)=1;$$

$$ks(*\overline{\mathbb{C}P^2})=1.$$

Hence $*\mathbb{C}P^2$ and $*\overline{\mathbb{C}P^2}$ are not smoothable (as well as their products with $\mathbb{R}$ and $S^1$).

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

  • 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

(Scorpan 05, p. 241-242)

References

• Alexandru Scorpan, The Wild World of 4-Manifolds, American Mathematical Society (2005) [ISBN 978-1470468613]