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Diagonal Involutions and the Borsuk–Ulam Property for Product of Surfaces

  • Daciberg Lima Gonçalves
  • Anderson Paião dos SantosEmail author
Article
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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 \(M^2 \times N^2\).

Keywords

Borsuk–Ulam Theorem Surfaces Involution Equivariant map Covering space Index 

Mathematics Subject Classification

Primary 55M20 secondary 55M35 

Notes

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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Copyright information

© Sociedade Brasileira de Matemática 2018

Authors and Affiliations

  • Daciberg Lima Gonçalves
    • 1
  • Anderson Paião dos Santos
    • 2
    Email author
  1. 1.Departamento de MatemáticaIME-Universidade de São PauloSão PauloBrazil
  2. 2.Departamento de MatemáticaUniversidade Tecnológica Federal do Paraná, UTFPR-CP, Avenida Alberto Carazzai, 1640Cornélio ProcópioBrazil

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