Some context:

In the theory of compact, oriented Riemannian Einstein 4-manifolds, there is a a fundamental topological constraint that is implied by the Einstein equations. To wit, if $\chi$ and $\tau$ denote the Euler characteristic and signature of a fixed 4-manifold $M$, then $$2\chi\pm3\tau=\textstyle\frac{1}{8\pi^2}\displaystyle\int_M\,\big(\displaystyle\frac{R^2}{12}-\mathring{|Rc|}^2+|W^{\pm}|^2 \big)\,dV$$ for $any$ Riemannian metric on $M$. As the traceless Ricci tensor $\mathring{Rc}$ vanishes identically for an Einstein metric, these 2 (well, 1 if $\tau=0$) topological invariants are non-negative if an Einstein metric exists on $M$. This is known as the Hitchin-Thorpe inequality.

One can calculate that in fact $2\chi \pm 3\tau=p_1(\Lambda^2_{\pm})$ by using the connection which is induced on $\Lambda^2_{\pm}$ by the Levi-Civita connection, so the presence of an Einstein metric on $M$ means that the first Pontryagin number of the bundles $\Lambda^2_{\pm}$ is non-negative.

My question is plainly this: Sticking with a 4-manifold $M$, what does it mean in a tangible, geometric sense, to say that a vector bundle over $M$ has non-negative/positive first Pontryagin number? Or perhaps more generally, what does the quantity $p_1$ in fact quantify? Does it say something about generic sections of the bundle (I vaguely recall that the Steifel-Whitney classes quantify something like this), or something else palpable?

I'd be perfectly happy if one were to restrict discussion to the bundles $\Lambda^2_{\pm}$ above or maybe general rank-3 bundles if either simplifies the discussion.