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
Article

Abstract

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.

Keywords

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 

Notes

Acknowledgements

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

References

  1. Bourgin, D.J.: On some separation and mapping theorems. Comment. Math. Helv. Soc. 29, 199–214 (1955)MathSciNetCrossRefMATHGoogle Scholar
  2. Cohen, F., Lusk, E.L.: Configuration-like spaces and the Borsuk-Ulam theorem. Proc. Am. Math. Soc. 56, 313–317 (1976)MathSciNetCrossRefMATHGoogle Scholar
  3. Conner, P.E., Floyd, E.E.: Fixed point free involutions and equivariant maps. Bull. Am. Math. Soc. 66, 416–441 (1960)MathSciNetCrossRefMATHGoogle Scholar
  4. Davis, J. F., Kirk, P.: Lecture notes in algebraic topology, Graduate studies in Mathematics, Vol. 35, American Mathematical Society (2001)Google Scholar
  5. Dold, A.: Parametrized Borsuk-Ulam theorems. Comment. Math. Helv. 63, 275–285 (1988)MathSciNetCrossRefMATHGoogle Scholar
  6. Dotzel, R.M., et al.: The cohomology rings of the orbit spaces of free transformation groups on the product of two spheres. Proc. Am. Math. Soc. 129, 921–930 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. Fadell, E., Husseini, S.: Relative cohomological index theories. Adv. Math. 64, 1–31 (1987)MathSciNetCrossRefMATHGoogle Scholar
  8. Fadell, E., Husseini, S.: An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems. Ergodic Theory Dynam. Syst. 8, 73–85 (1988)MathSciNetCrossRefMATHGoogle Scholar
  9. Jaworowski, J.: Bundles with periodic maps and mod \(p\) Chern polynomial. Proc. Am. Math. Soc. 132, 1223–1228 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. Koikara, B.S., Mukerjee, H.K.: A Borsuk-Ulam theorem type for a product of spheres. Topol. Appl. 63, 39–52 (1995)CrossRefMATHGoogle Scholar
  11. de Mattos, D., et al.: Borsuk-Ulam Theorems and their Parametrized versions for spaces of type (a, b). Algebraic Geomet. Topol. 13, 2827–2843 (2013)Google Scholar
  12. de Mattos, D., de Santos, E.L.: A parametrized Borsuk-Ulam Theorem for a product spheres with free \(\mathbb{Z}_p\)- action and free \(\mathbb{S}^1\)- action. Algebraic Geomet. Topol. 7, 1791–1804 (2007)CrossRefMATHGoogle Scholar
  13. de Mattosa, D., et al.: Zero sets of equivariant maps from products of spheres to Euclidean spaces. Topol. Appl. 202, 720 (2016)MathSciNetGoogle Scholar
  14. McCleary, J.: A user’s guide to spectral sequences, IInd edn, Cambridge University Press, Cambridge (2001)Google Scholar
  15. Swan, R.G.: A new method in fixed point theory. Comment. Math. Helv. 34, 1–16 (1960)MathSciNetCrossRefMATHGoogle Scholar
  16. Singh, H.K., Singh, T.B.: On the cohomology of orbit space of free \(\mathbb{Z}_p\)- actions on lens space. Proc. Indian Acad. Sci. (Math. Sci.) 117, 287–292 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. Singh, M.: Parametrized Borsuk-Ulam problem for projective space bundle. Fund. Math. 211, 135–147 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. Singh, M.: Orbit spaces of free involutions on the product of two projective spaces. Results Math. 57, 53–67 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. Singh, M.: Cohomology algebra of orbit spaces of free involutions on lens spaces. J. Math. Soc. Japan 65, 1055–1078 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. Singh, S.K., et al.: A Borsuk-Ulam type theorem for the product of a projective space and \(3\)- Sphere. Topol. Appl. 225, 112–129 (2017)Google Scholar
  21. Quillen, D.G.: The spectrum of an equivariant cohomology ring I. Ann. Math. 94, 549–572 (1971)MathSciNetCrossRefMATHGoogle Scholar
  22. Volovikov, AYu.: A theorem of Bourgin-Yang type for \(\mathbb{Z}_p^{n}\)- action. Sb. math. 183, 115–144 (1992)MATHGoogle Scholar
  23. Volovikov, AYu.: On the index of \(G\)- spaces. Sb. Math. 191, 1259–1277 (2000)MathSciNetCrossRefMATHGoogle Scholar
  24. Volovikov, AYu.: On the Cohen-Lusk Theorem. J. Math. Sci. 159, 790–793 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. Yang, C.T.: On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobo and Dyson-I. Ann. Math. 60, 262–282 (1954)MathSciNetCrossRefMATHGoogle Scholar

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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