Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $X$ be any simply connected smooth 4-manifold with a fixed Euler characteristic $e$, signature $\sigma$ and boundary $Y$. Assume that the determinant of the intersection form $Q_{X}$ is equal to a fixed integer $k \neq 0, \pm 1$. Is it true that the number of possible intersection forms for such $X$ is finite? Any reference would be appreciated.

share|cite|improve this question

2 Answers 2

up vote 4 down vote accepted

The classification of integral quadratic forms is discussed in Chapter 15 of Conway-Sloane. In particular, the discussion there implies that there are only finitely many integral quadratic forms of a given determinant and dimension. Section 11 in the chapter discusses methods for computing the number of such forms.

share|cite|improve this answer
@Agol Thanks a lot for the reference! –  user25085 Jul 20 '12 at 22:33

As the question is posed, the answer is not. If you take connected sum with $\Bbb{CP}^2$ you preserve the determinant, but change the intersection form.

share|cite|improve this answer
Thanks Daniele. I should have added that the Euler characteristic and the signature of $X$ are fixed as well. –  user25085 Jul 12 '12 at 21:24

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.