Every closed connected oriented $4$-manifold has a signature, defined via a cohomological intersection form. In Turaev's book Quantum Invariants of Knots and 3-Manifolds the definition of a certain invariant of $3$-manifolds in chapter II uses the signature of a $4$-manifold which is not closed (actually it arises from attaching $2$-handles to a $4$-ball), which is defined using a homological intersection form.
Q1) How is this homological intersection form defined?
Q2) Why is the signature of $S^2 \times D^2$ zero? (Actually I don't know if this is true, but this seems to be used in a calculation in Turaev's book.)