Abstract
In this work we study a generalization of the Borsuk–Ulam Theorem. Namely, we replace the sphere \({\mathbb {S}}^n\) by a product of two closed surfaces \(M^2 \times N^2\) equipped with the diagonal involution \(T \times S\) where T and S are free involutions on \(M^2\) and \(N^2\), respectively, and the indexes \(i(M^2, T)=i(N^2, S)=2\). Then we compute the index of the pair \((M^2 \times N^2,T \times S)\) and we obtain a Borsuk-Ulam Theorem for \(M^2 \times N^2\).
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Acknowledgements
We would like to thank the referee for his careful reading and for his suggestions and comments which includes proposing the problem that appears at the end of the introduction. The presentation of this work has been substantially improved after the revision.
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The first author was partially supported by the FAPESP Projeto Temático “Topologia Algébrica, Geométrica e Diferencial” 2016/24707-4 (Brazil). The second author was supported by CAPES–Ciência sem Fronteiras, Processo: 88881.068125/2014-01.
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Gonçalves, D.L., dos Santos, A.P. Diagonal Involutions and the Borsuk–Ulam Property for Product of Surfaces. Bull Braz Math Soc, New Series 50, 771–786 (2019). https://doi.org/10.1007/s00574-018-0098-4
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DOI: https://doi.org/10.1007/s00574-018-0098-4