nLab 3/2 conjecture

Idea

The 3/2 conjecture arises from the (still not fully answered) question: Which symmetric unimodular bilinear forms are the intersection forms of oriented closed smooth 4-manifolds?

According to Donaldson's theorem, a definite form is the intersection form of an oriented closed smooth 4-manifolds if and only if it’s either $\oplus n[-1]$ or $\oplus n[+1]$, with both realized by the even simply connected examples $\#n\mathbb{C}P^2$ or $\#n\overline{\mathbb{C}P^2}$. According to Serre’s classification theorem, an indefinite odd form is isomorphic to $\oplus m[-1]\oplus n[+1]$, which are all realized by the even simply connected examples $\#m\mathbb{C}P^2\#m\overline{\mathbb{C}P^2}$. (According to Freedman's theorem, all these simply connected examples are even unique up to homeomorphism.)

Left over is, which indefinite even forms are realized. It comes down to the forms:

$$\pm 2 m E_8 \oplus n H$$

according to Serre’s classification theorem with Rokhlin's theorem requiring an even number of definite $\pm E_8$ and indefiniteness requiring $n\geq 1$. Simon Donaldson furthermore showed, that if $m\neq 0$, then $n\geq 3$.

Statement

The open 3/2 conjecture assumes:

Every smooth 4-manifold $M$ which is:

1. oriented

2. simply-connected (cf. Scorpan 2005, ftn. 1 on p. 111)

3. closed

4. irreducible (not splitting in a non-trivial connected sum)

5. with indefinite even intersection form

satisfies:

$$\chi(M) \geq \frac{3}{2}{|\sigma(M)|} \mathrlap{\,.}$$

Alternatively, for the above structure of the intersection form, the 3/2 conjecture is equivalent to:

$$n\geq 4{|m|}-1.$$

This follows from:

$$\chi(M) = b_2(M)+2 = 16{|m|}+2n+2;$$

$${|\sigma(M)|} =16{|m|}.$$

(Scorpan 05, p. 247)

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

• Alexandru Scorpan, The Wild World of 4-Manifolds, American Mathematical Society (2005) [ISBN 978-1470468613, §3: pdf]