nLab Quinn theorem

Idea

Quinn’s theorem shows the importance of the purely topological property of compactness for non-smoothable topological 4-manifolds.

According to Moise's theorem, non-smoothable topological 1-manifolds?, 2-manifolds and 3-manifolds are impossible. However, many non-smoothable topological 4-manifolds like the E8 manifold or the fake second complex projective space exist as a consequence of multiple results like Rokhlin's theorem, Freedman's classification or Donaldson's theorem. All of them include compactness.

Statement

Proposition

(Quinn’s theorem) Every open (non-compact without boundary) topological 4-manifold is smoothable.

(Freedman & Quinn 90, p. 116 Scorpan 05, p. 222)

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

References

• Michael Freedman, Frank Quinn, Topology of 4-Manifolds (1990) [doi:10.1515/9781400861064]

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