A parametrized version of the Borsuk Ulam theorem

The main result of this note is a parametrized version of the BorsukUlam theorem. Loosely speaking, it states that for a map X × S → S (with k < n), considered as a family of maps from S to S, the set of solutions to the Borsuk-Ulam problem (i.e. opposite points in S with the same value) depends continuously on X. We actually formuate this for correspondences. The “continuity” is measured by the existence of a non-trivial fundamental class in Cech homology. An interesting construction used in the proof context is a canonical 1/2-squaring construction in Cech homology with Z/2-coefficients. The main theorem implies a special case of a conjecture of Simon. This conjecture is be relevant in connection with new existence results for equilibria in repeated 2-player games with incomplete information. 1 A parametrized Borsuk-Ulam theorem