Contents

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:

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 irreducible (not splitting in a non-trivial connected sum) oriented closed smooth 4-manifold $M$ with indefinite even intersection form fulfills:

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

This follows from:

(Scorpan 05, p. 247)

Related entries

References

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