A parametrized version of the Borsuk Ulam theorem
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.
Keywords
Cite
@article{arxiv.0709.1774,
title = {A parametrized version of the Borsuk Ulam theorem},
author = {Thomas Schick and Robert Simon and Stanislav Spiez and Henryk Torunczyk},
journal= {arXiv preprint arXiv:0709.1774},
year = {2012}
}
Comments
15 pages, v2: typos corrected, v3: missing bibliography added, v4: completely rewritten version with (hopefully) much clearer exposition. Section on relation to game theory added. v5+v6: small typographical and stylistic correction, v7: reference to related work added. to appear in Bulletin of the London Math. Society