For a simply connected four-dimension manifold, we know the Freedmen's work. 
My question is: For every integer N, Is the number of simply connected 4-manifolds which the second betti number is smaller than N finite or infinite?


Or equivalently: Can we classify the bilinear forms of integer coefficients? Or what's the recently progress of this classification problem?



The number of simply connected 4-dimension manifold
This was essentially answered by Agol in the comments. For any fixed second Betti number $N$, there are finitely many homeomorphism types, because there are finitely many isomorphism types of unimodular lattices of rank $N$.


As far as classification is concerned, the number of isomorphism types of definite unimodular lattices grows slowly until you reach rank about 24, then there is a phase transition to superexponential growth. In other words, the general classification problem is hopeless, even though for any given rank the lattices can be classified in finite time by an automaton.


**Edit:** The [Smith-Minkowski-Siegel mass formula](http://en.wikipedia.org/wiki/Smith-Minkowski-Siegel_mass_formula) is essentially:


$$\sum\_{\Lambda} \frac{1}{|\operatorname{Aut} \Lambda|} = \pi^{-\frac{n^2+n}{4}}\prod\_{j=1}^n \Gamma\left(\frac{j}{2}\right) \cdot (\text{minor corrections})$$


where the sum is over unimodular lattices $\Lambda$ of rank $n$. Since all lattices have automorphism group at least 2, the number of summands (hence the number of homeomorphism types of simply connected 4-manifolds) is at least twice the product on the right. If you can put an upper bound on the orders of automorphism groups, you can also get an upper bound. A rough estimate of the asymptotics is given by taking logarithm of the right side, to get roughly $n^2(\frac{\log n}{4} - \frac{5}{6})$. When $n$ is greater than $e^{10/3} \sim 28$, you get rapid growth, eventually faster than $e^{n^2}$.


