A parametrized version of the Borsuk Ulam theorem

TL;DR

This paper develops a parametrized version of the Borsuk-Ulam theorem, showing the continuous dependence of solution sets on parameters, with applications to game theory and a new homology construction.

Contribution

It introduces a parametrized Borsuk-Ulam theorem and a novel symmetric squaring method in Cech homology, addressing a problem in game theory.

Findings

Solution sets depend continuously on parameters

Constructs continuous families from boundary data

Introduces a new symmetric squaring in Cech homology

Abstract

The main result of this note is a parametrized version of the Borsuk-Ulam theorem. We show that for a continuous family of Borsuk-Ulam situations, parameterized by points of a compact manifold W, its solution set also depends continuously on the parameter space W. Continuity here means that the solution set supports a homology class which maps onto the fundamental class of W. When W is a subset of Euclidean space, we also show how to construct such a continuous family starting from a family depending in the same way continuously on the points of the boundary of W. This solves a problem related to a conjecture which 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 coefficients in Z/2Z.