The definition of the positive and negative parts of the second Betti number which I know is via the diagonalized intersection form, and possible for $4$-manifolds $M$.
$b^\pm_2:= \dim H^2_\pm( M;\mathbb{R})$.
Is it as well possible, or even more common, to define it as the dimension of the selfdual and anti-selfdual harmonic forms $\mathcal{H}^{2,\pm}(M)$?
Why are these definitions equivalent?
Let's assume we're dealing with a smooth, oriented, compact $4$-manifold $X$. Each cohomology class in $H^2(X)$ has a unique harmonic representative, and Poincare duality is given by the perfect pairing $i(\alpha, \beta) \to \int_X \alpha \wedge * \beta$; identifying $H^4(X)$ with $\mathbb{R}$ gives the intersection form. This means that, in a suitable basis where the intersection form is diagonal, self-dual elements have $i(\alpha, \alpha) > 0$ and anti-self-dual elements have $i(\alpha, \alpha) < 0$.