Appearances not withstanding this is a talk about rational curves because they're the cause of the failure. Similarly, since homotopy groups are constant on the fibres of topological fibrations, a counterexample has to be in positive or mixed characteristic, and the specific one which I'll discuss is bi-disc quotients over Spec Z. The example also has considerable logical implications for studying boundedness of rational curves on surfaces of general type, i.e. it cannot be implied by any theorem in $ACF_0$. Conversely, and more substantially, this boundedness can be proved uniformly in sufficiently large primes $p$ in $ACF_0$ provided the surface enjoys $c_1^2> c_2$.