Intersection form of a 4-manifold with boundary

Question

For a closed oriented 4-manifold $X$, the bilinear intersection form $H_2(X)\times H_2(X)\to \Bbb Z$, $(a,b)\mapsto \langle PD(a)\cup PD(b), [X]\rangle$ is unimodular, which can be shown by Poincare duality. Suppose $X$ is a compact oriented 4-manifold with nonempty boundary, and with second betti number $\geq 1$. Can the intersection form of $X$ be trivial (so the intersection form is represented by the zero matrix)?

Answer 1

Yes, for example $S^2\times D^2$ is such a 4-manifold.