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A parametrized version of the Borsuk-Ulam theorem

Schick, Thomas, Simon, Robert Samuel, Spiez, Stanislav and Torunczyk, Henryk (2011) A parametrized version of the Borsuk-Ulam theorem. Bulletin of the London Mathematical Society, 43 (6). pp. 1035-1047. ISSN 0024-6093

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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×Sm×ℝm, 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×ℝm 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 Čech homology with ℤ/2-coefficients.

Item Type: Article
Official URL: http://blms.oxfordjournals.org/
Additional Information: © 2011 London Mathematical Society
Library of Congress subject classification: Q Science > QA Mathematics
Sets: Departments > Mathematics
Rights: http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx
Date Deposited: 23 Jan 2012 15:39
URL: http://eprints.lse.ac.uk/41652/

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