A parametrized version of the Borsuk Ulam theorem

Details A parametrized version of the Borsuk Ulam theorem A parametrized version of the Borsuk Ulam theorem

2007-09-12

arxiv.org/abs/0709.1774v7

Mathematics

Algebraic Topology

15 pages, v2: typos corrected, v3: missing bibliography added, v4: completely rewritten version with (hopefully) much clearer

Scientific paper

10.1112/blms/bdr037

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.

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