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. 



Intersection forms of 4-manifolds with boundary
The classification of integral quadratic forms is discussed in [Chapter 15 of Conway-Sloane](http://books.google.com/books?id=upYwZ6cQumoC&lpg=PR5&ots=_L-R4Qi9DS&dq=conway%2520sloane&lr&pg=PA352#v=onepage&q&f=false). 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. 


