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Computations in \(C_{pq}\)-Bredon cohomology

  • Samik BasuEmail author
  • Surojit Ghosh
Article
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Abstract

In this paper, we compute the \(RO(C_{pq})\)-graded cohomology of \(C_{pq}\)-orbits. We deduce that in all the cases the Bredon cohomology groups are a function of the fixed point dimensions of the underlying virtual representations. Further, when thought of as a Mackey functor, the same independence result holds in almost all cases. This generalizes earlier computations of Stong and Lewis for the group \(C_p\). The computations of cohomology of orbits are used to prove a freeness theorem. The analogous result for the group \(C_p\) was proved by Lewis. We demonstrate that certain complex projective spaces and complex Grassmannians satisfy the freeness theorem.

Keywords

Bredon cohomology Mackey functor Grassmann manifolds Equivariant cohomology 

Mathematics Subject Classification

Primary 55N91 55P91 Secondary 57S17 14M15 

Notes

References

  1. 1.
    Bredon, G.E.: Equivariant Cohomology Theories, Lecture Notes in Mathematics, vol. 34. Springer, Berlin (1967)CrossRefGoogle Scholar
  2. 2.
    Caruso, J.L.: Operations in equivariant \({\bf Z}{/}p\)-cohomology. Math. Proc. Camb. Philos. Soc. 126, 521–541 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dress, A.W.M.: Contributions to the theory of induced representations. In: Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Math, vol. 342, pp. 183–240. Springer, Berlin (1973)Google Scholar
  4. 4.
    Ferland, K.K.: On the RO(G)-graded equivariant ordinary cohomology of generalized G-cell complexes for G = Z/p, ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.). Syracuse University (1999)Google Scholar
  5. 5.
    Ferland, K.K., Lewis, L.G., Jr.: The \(R{\rm O}(G)\)-graded equivariant ordinary homology of \(G\)-cell complexes with even-dimensional cells for \(G={\mathbb{Z}}/p\), Memoirs of the American Mathematical Society, vol. 167, pp. viii+129 (2004)Google Scholar
  6. 6.
    Greenlees, J.P.C., May, J.P.: Equivariant Stable Homotopy Theory, in Handbook of Algebraic Topology, pp. 277–323. North-Holland, Amsterdam (1995)zbMATHGoogle Scholar
  7. 7.
    Hill, M.A., Hopkins, M.J., Ravenel, D.C.: On the nonexistence of elements of Kervaire invariant one. Ann. Math. (2) 184, 1–262 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lewis Jr., L.G.: The \(R{\rm O}(G)\)-graded equivariant ordinary cohomology of complex projective spaces with linear \({\bf Z}/p\) actions, in algebraic topology and transformation groups (Göttingen, vol. 1361 of Lecture Notes in Math. 1988, pp. 53–122. Springer, Berlin (1987)Google Scholar
  9. 9.
    Lewis Jr., L.G.: The equivariant Hurewicz map. Trans. Am. Math. Soc. 329, 433–472 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lewis, L.G., Jr.: The category of Mackey functors for a compact Lie group, in Group representations: cohomology, group actions and topology. In: Seattle, WA: vol. 63 of Proc. Sympos. Pure Math., Amer. Math. Soc. Providence, RI, vol. 1998, pp. 301–354 (1996)Google Scholar
  11. 11.
    Lewis, L.G. Jr., May, J.P., Steinberger, M., McClure, J.E.: Equivariant stable homotopy theory. With contributions by J. E. McClure. Lecture Notes in Mathematics, vol. 1213, x+538 pp, Springer-Verlag, Berlin (1986)Google Scholar
  12. 12.
    Lewis Jr., L.G.: The theory of green functors. Mimeographed notes (1981)Google Scholar
  13. 13.
    May, J.P.: Equivariant homotopy and cohomology theory, vol. 91 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With contributions by M. Cole, G. Comezaña, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. WanerGoogle Scholar
  14. 14.
    Wasserman, A.G.: Equivariant differential topology. Topology 8, 127–150 (1969)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Stat-Math UnitIndian Statistical InstituteKolkataIndia
  2. 2.Department of MathematicsUniversity of HaifaHaifaIsrael

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