  In this article, we study the rational cohomology rings of Voisin's punctual
Hilbert schemes $X^{[n]}$ associated to a symplectic compact fourfold $X$. We
prove that these rings can be universally constructed from $H^*(X,\mathbb{Q})$
and $c_1(X)$, and that Ruan's crepant resolution conjecture holds if $c_1(X)$
is a torsion class. Next, we prove that for any almost-complex compact fourfold
$X$, the complex cobordism class of $X^{[n]}$ depends only on the cobordism
class of $X$.
