In certain topological effects the accumulation of a quantum phase shift is
accompanied by a local observable effect. We show that such effects manife
st a complementarity between nonlocal and local attributes of the topology,
which is reminiscent but different from the usual wave-particle: complemen
tarity. This complementarity is not a consequence of noncommutativity, rath
er it is due to the noncanonical nature of the observables, We suggest that
a local/nonlocal complementarity is a general Feature of topological effec
ts that an "dual" to the Aharonov-Bohm effect.