Borsuk-Ulam Theorems and Their Parametrized Versions for \(\mathbb {F}P^m\times \mathbb {S}^3\)

  • Somorjit Konthoujam Singh
  • Hemant Kumar Singh
  • Tej Bahadur Singh


Let \(G=\mathbb {Z}_p,\) \(p>2\) a prime, act freely on a finitistic space X with mod p cohomology ring isomorphic to that of \(\mathbb {F}P^m\times \mathbb {S}^3\), where \(m+1\not \equiv 0\) mod p and \(\mathbb {F}=\mathbb {C}\) or \(\mathbb {H}\). We wish to discuss the nonexistence of G-equivariant maps \(\mathbb {S}^{2q-1}\rightarrow X\) and \( X\rightarrow \mathbb {S}^{2q-1}\), where \(\mathbb {S}^{2q-1}\) is equipped with a free G-action. These results are analogues of the celebrated Borsuk-Ulam theorem. To establish these results first we find the cohomology algebra of orbit spaces of free G-actions on X. For a continuous map \(f\!:\! X\rightarrow \mathbb {R}^n\), a lower bound of the cohomological dimension of the partial coincidence set of f is determined. Furthermore, we approximate the size of the zero set of a fibre preserving G-equivariant map between a fibre bundle with fibre X and a vector bundle. An estimate of the size of the G-coincidence set of a fibre preserving map is also obtained. These results are parametrized versions of the Borsuk-Ulam theorem.


Free action Finitistic space Leray-Serre spectral sequence Parametrized Borsuk-Ulam theorem Characteristic polynomial Partial coincidence set 

Mathematics Subject Classification

Primary 57S99 Secondary 55T10 55M20 



We are thankful to the referee for his valuable suggestions, which have brought significant improvement in the original paper.


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Copyright information

© Sociedade Brasileira de Matemática 2017

Authors and Affiliations

  • Somorjit Konthoujam Singh
    • 1
  • Hemant Kumar Singh
    • 1
  • Tej Bahadur Singh
    • 1
  1. 1.Department of MathematicsUniversity of DelhiDelhiIndia

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