  We compute the integral cohomology of the minimal non-trivial nilpotent orbit
in a complex simple (or quasi-simple) Lie algebra. We find by a uniform
approach that the middle cohomology group is isomorphic to the fundamental
group of the sub-root system generated by the long simple roots. The modulo
$\ell$ reduction of the Springer correspondent representation involves the sign
representation exactly when $\ell$ divides the order of this cohomology group.
The primes dividing the torsion of the rest of the cohomology are bad primes.
