A parametrized version of the Borsuk–Ulam theorem
Abstract
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 ⊂ W × S m ×ℝ m , the solution set consists of points ( w , x , v )∈ Z such that also ( w ,− x , v )∈ 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 ⊂∂ W ×ℝ m when W is a compact top-dimensional subset in ℝ m +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 Čech homology with ℤ/2-coefficients.