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Borsuk–Ulam theorem for filtered spaces

  • Carlos Biasi , Alice Kimie Miwa Libardi EMAIL logo , Denise de Mattos and Sergio Tsuyoshi Ura
From the journal Forum Mathematicum

Abstract

Let X and Y be pathwise connected and paracompact Hausdorff spaces equipped with free involutions T:XX and S:YY, respectively. Suppose that there exists a sequence

(Xi,Ti)hi(Xi+1,Ti+1)for 1ik,

where, for each i, Xi is a pathwise connected and paracompact Hausdorff space equipped with a free involution Ti, such that Xk+1=X, and hi:XiXi+1 is an equivariant map, for all 1ik. To achieve Borsuk–Ulam-type theorems, in several results that appear in the literature, the involved spaces X in the statements are assumed to be cohomological n-acyclic spaces. In this paper, by considering a more wide class of topological spaces X (which are not necessarily cohomological n-acyclic spaces), we prove that there is no equivariant map f:(X,T)(Y,S) and we present some interesting examples to illustrate our results.

MSC 2010: 55M20; 55M35

Communicated by Frederick R. Cohen


Funding statement: The first three authors named are partially supported by Projeto Temático: Topologia Algébrica, Geométrica e Diferencial, FAPESP, process number: 2016/24707-4. The fourth author is supported by FAPESP, process number: 2018/17240-8.

Acknowledgements

The authors would like to thank Edivaldo Lopes dos Santos for stimulating discussions which improved the paper.

References

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Received: 2019-10-24
Revised: 2020-10-14
Published Online: 2021-01-09
Published in Print: 2021-03-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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