Schick, Thomas and Simon, Robert Samuel and 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

## 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 |
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Official URL: | http://blms.oxfordjournals.org/ |

Additional Information: | © 2011 London Mathematical Society |

Subjects: | Q Science > QA Mathematics |

Sets: | Departments > Mathematics |

Date Deposited: | 23 Jan 2012 15:39 |

Last Modified: | 02 Jun 2014 08:55 |

URI: | http://eprints.lse.ac.uk/id/eprint/41652 |

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