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Preprint Number 186

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186. Ikumitsu Nagasaki, Tomohiro Kawakami, Yasuhiro Hara and Fumihiro Ushitaki
A generalized Borsuk-Ulam theorem in a real closed field
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Submission date: 20 May 2009.

Abstract:

Let G be the cyclic group of order k and N=(R, + , • , < , ... ) an o-minimal expansion of a real closed field R. Let X be a definably connected definable set with a free definable G-action. Assume that there exists a positive integer n such that H_q(X; Z/kZ) q for 1 \leq q \leq n. If Y is a definable set with a free definable G-action such that H_{n+1}(Y/G, Z/kZ)=0, then there is no definable G-map from X to Y. We also prove the topological version of this definable version.

Mathematics Subject Classification: 57S10, 57S17, 55M20, 55M35, 03C64

Keywords and phrases: The Borsuk-Ulam theorem, o-minimal, real closed fields, finite groups, definable $C_k$-maps, continuous $C_k$-maps.

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