Parametrized Borsuk-Ulam theorems for multivalued maps

By combining parametrized Borsuk-Ulam theorems proved by Dold with methods using the Vietoris mapping theorem we show that Dold’s results can be extended to multivalued maps. Such methods were invented by Eilenberg and Montgomery who applied them to multivalued fixed-point theorems, and they were used by Jaworowski to prove a multivalued version of the Borsuk-Ulam theorem. Subsequently they were extended and refined in various ways by Górniewicz and others. We also indicate how our results can be proved in the relative case, for pairs of spaces rather than for single spaces only. This allows us to obtain positive results for bundles over manifolds with boundary; for instance, over a closed interval.