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In this article we consider m-acyclic maps with respect to a field F and prove the existence of chain approximation for such maps. Furthermore we provide a unified approach to the Borsuk-Ulam theorem and the Bourgin-Yang generalization. Finally we prove the existence of A-systems for certain m-acyclic maps and define a fixed point index

Borsuk-Ulam theorem, fixed point index and chain approximations for maps with multiplicity