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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2 Answers
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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.
answered Jul 20, 2012 at 22:06
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$\begingroup$ @Agol Thanks a lot for the reference! $\endgroup$ Commented Jul 20, 2012 at 22:33
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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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$\begingroup$ Thanks Daniele. I should have added that the Euler characteristic and the signature of $X$ are fixed as well. $\endgroup$ Commented Jul 12, 2012 at 21:24