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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.

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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.

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@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.

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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

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