  We uncover new topology-related features of the $\pi$-flux saddle-point
solution of the $D$=2+1 Heisenberg antiferromagnet. We note that symmetries of
the spinons sustain a built-in competition between antiferromagnetic (AF) and
valence-bond-solid (VBS) orders, the two tendencies central to recent
developments on quantum criticality. An effective theory containing an analogue
of the Wess-Zumino-Witten term is derived, which generates quantum phases
related to AF monopoles with VBS cores, and reproduces Haldane's hedgehog Berry
phases. The theory readily generalizes to $\pi$-flux states for all $D$.
