Real Algebraic and Analytic Geometry

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287. Ikumitsu Nagasaki, Tomohiro Kawakami, Yasuhiro Hara and Fumihiro Ushitaki:
Smith homology and Borsuk-Ulam type theorems.

e-mail: , , ,

Submission: 2009, June 11.

Abstract:
Let k be a positive integer greater than 1 and C_k the cyclic group of order k. Let X be an arcwise connected free C_k-space and Y a Hausdorff free C_k-space. If there exists a positive integer n such that H_q(X; \bZ/k\bZ)=0 for 1 \le q \le n and H_{n+1}(Y/C_k; \bZ/k\bZ)=0, then there is no continuous C_k-map from X to Y.
We also prove a definable version of this topological version in an o-minimal expansion of a real closed field R. .

Mathematics Subject Classification (2000): 57S10, 57S17, 55M20, 55M35, 03C64.

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

Full text, 8p.: dvi 48k, pdf 347k.


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