Complementarity between local and nonlocal topological effects

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.