A parametrized version of the Borsuk-Ulam theorem

We show that for a 'continuous' family of Borsuk-Ulam situations, parametrized by points of a compact manifold W, its solution set also depends 'continuously' on the parameter space W. By such a family we understand a compact set Z subset of W x S-m x R-m, the solution set consists of points (w, x, v) is an element of Z such that also (w, -x, v) is an element of Z. Here, 'continuity' means that the solution set supports a homology class that maps onto the fundamental class of W. We also show how to construct such a family starting from a 'continuous' family Y subset of partial derivative W x R-m when W is a compact top-dimensional subset in Rm+1. This solves a problem related to a conjecture that is relevant for the construction of equilibrium strategies in repeated two-player games with incomplete information. A new method (of independent interest) used in this context is a canonical symmetric squaring construction in Cech homology with Z/2-coefficients.