A parametrized version of the Borsuk Ulam theorem
| Autor: | Schick, Thomas, Simon, Robert, Spiez, Stanislav, Torunczyk, Henryk |
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| Rok vydání: | 2007 |
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| Zdroj: | Bull. Lond. Math. Soc. 43 (2011), no. 6, 1035-1047 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1112/blms/bdr037 |
| Popis: | 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. Comment: 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 |
| Databáze: | arXiv |
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