  The low energy dynamics of the anti-ferromagnetic Heisenberg spin $S$ chain
in the semiclassical limit $S\to\infty$ is known to map onto the O(3) nonlinear
$\sigma$ model with a $\theta$ term in 1+1 dimension. Guided by the underlying
dual symmetry of the spin chain, as well as the recently established
topological significance of "dangling edge spins," we report an {\em exact}
mapping onto the O(3) model that avoids the conventional large $S$
approximation altogether. Our new methodology demonstrates all the super
universal features of the $\theta$ angle concept that previously arose in the
theory of the quantum Hall effect. It explains why Haldane's original ideas
remarkably yield the correct answer in spite of the fundamental complications
that generally exist in the idea of semiclassical expansions.
