SpringerLink
Log in

Algebraic realization for simple groups

  • Research Article
  • Published:
European Journal of Mathematics Aims and scope Submit manuscript

Abstract

Let G be a finite simple group with elementary abelian Sylow 2 subgroup. We show that every closed smooth G manifold has a strongly algebraic model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akbulut, S., King, H.: The topology of real algebraic sets with isolated singularities. Ann. Math. (2) 113(3), 425–446 (1981)

    Article  MathSciNet  Google Scholar 

  2. Akbulut, S., King, H.: On approximating submanifolds by algebraic sets and a solution to the Nash conjecture. Invent. Math. 107(1), 87–89 (1992)

    Article  MathSciNet  Google Scholar 

  3. Atiyah, M.F.: \(K\)-Theory. W.A. Benjamin, New York (1967)

    MATH  Google Scholar 

  4. Benedetti, R., Tognoli, A.: On real algebraic vector bundles. Bull. Sci. Math. 104(1), 89–112 (1980)

    MathSciNet  MATH  Google Scholar 

  5. Bochnak, J., Coste, M., Roy, M.-F.: Géométrie Algébrique Réelle. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 12. Springer, Berlin (1987)

  6. Borel, A.: Topics in the Homology Theory of Fibre Bundles. Lecture Notes in Mathematics, vol. 36. Springer, Berlin (1967)

  7. Bredon, G.: Introduction to Compact Transformation Groups. Pure and Applied Mathematics, vol. 46. Academic Press, New York (1972)

  8. Cho, J.-H., Kim, S.S., Masuda, M., Suh, D.Y.: Classification of equivariant complex vector bundles over a circle. J. Math. Kyoto Univ. 41(3), 517–534 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Cho, J.-H., Kim, S.S., Masuda, M., Suh, D.Y.: Classification of equivariant real vector bundles over a circle. J. Math. Kyoto Univ. 42(2), 223–242 (2001)

    MathSciNet  MATH  Google Scholar 

  10. Cho, J.-H., Kim, M.K., Suh, D.Y.: On extensions of representations for compact Lie groups. J. Pure Appl. Algebra 178(3), 245–254 (2003)

    Article  MathSciNet  Google Scholar 

  11. Conner, T., Leemans, D.: An atlas of subgroup lattices of finite almost simple groups. Ars Math. Contemp. 8(2), 259–266 (2015)

    Article  MathSciNet  Google Scholar 

  12. Dovermann, K.H.: Strongly algebraic realization of dihedral group actions. Pacific J. Math. 305(2), 563–576 (2020)

    Article  MathSciNet  Google Scholar 

  13. Dovermann, K.H., Flores, D.J., Giambalvo, V.: Algebraic realization of actions of some finite groups. Manuscripta Math. 165(1–2), 239–254 (2021)

    Article  MathSciNet  Google Scholar 

  14. Dovermann, K.H., Giambalvo, V.: Algebraic realization for projective special linear actions. Abh. Math. Semi. Univ. Hambg. 91(1), 15–28 (2021)

    Article  MathSciNet  Google Scholar 

  15. Dovermann, K.H., Hanson, J., Little, R.D.: Examples of algebraically realized maps. Geom. Dedicata 186, 1–25 (2017)

    Article  MathSciNet  Google Scholar 

  16. Dovermann, K.H., Masuda, M.: Algebraic realization of manifolds with group actions. Adv. Math. 113(2), 304–338 (1995)

    Article  MathSciNet  Google Scholar 

  17. Dovermann, K.H., Masuda, M.: Uniqueness questions in real algebraic transformation groups. Topology Appl. 119(2), 147–166 (2002)

    Article  MathSciNet  Google Scholar 

  18. Dovermann, K.H., Masuda, M., Suh, D.Y.: Algebraic realization of equivariant vector bundles. J. Reine Angew. Math. 448, 31–64 (1994)

    MathSciNet  MATH  Google Scholar 

  19. Dovermann, K.H., Wasserman, A.G.: Algebraic realization for cyclic group actions with one isotropy type. Transform. Groups 25(2), 483–515 (2020)

    Article  MathSciNet  Google Scholar 

  20. Dovermann, K.H., Wasserman, A.G.: Algebraic realization for cyclic group actions. Transform. Groups (to appear)

  21. Gorenstein, D., Lyons, R., Solomon, R.: The Classification of the Finite Simple Groups. No 3. Part I. Chapter A. Mathematical Surveys and Monographs, vol. 40.3. American Mathematical Society, Providence (1998)

  22. Herstein, I.N.: Topics in Algebra, 2nd edn. Wiley, New York (1975)

    MATH  Google Scholar 

  23. Hirzebruch, F.: Topological Methods in Algebraic Geometry. 3rd edn. Grundlehren der Mathematischen Wissenschaften, vol. 131. Springer, New York (1966)

  24. Ivanov, N.V.: Approximation of smooth manifolds by real algebraic sets. Russian Math. Surveys 37(1), 1–59 (1982)

    Article  MathSciNet  Google Scholar 

  25. Kleidman, P.B.: The maximal subgroups of the Chevalley groups \(G_2(q)\) with \(q\) odd, the Ree groups \(^2G_2(q)\), and their automorphism groups. J. Algebra 117(1), 30–71 (1988)

    Article  MathSciNet  Google Scholar 

  26. Levchuk, V.M., Nuzhin, Ya.N.: The structure of Ree groups. Algebra i Logika 24(1), 26–41 (1985) (in Russian)

  27. Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Annals of Mathematics Studies, vol. 76. Princeton University Press, Princeton (1974)

  28. Nash, J.: Real algebraic manifolds. Ann. Math. (Series 2) 56, 405–421 (1952)

    MathSciNet  MATH  Google Scholar 

  29. Ree, R.: A family of simple groups associated with the simple Lie algebra of type \((G_{2})\). Bull. Amer. Math. Soc. 66, 508–510 (1960)

    Article  MathSciNet  Google Scholar 

  30. Ree, R.: A family of simple groups associated with the simple Lie algebra of type \((G_{2})\). Amer. J. Math. 83(3), 432–462 (1961)

    Article  MathSciNet  Google Scholar 

  31. Schwarz, G.: Algebraic quotients of compact group actions. J. Algebra 244(2), 365–378 (2001)

    Article  MathSciNet  Google Scholar 

  32. Segal, G.: Equivariant Kd-theory. Inst. Hautes Études Sci. 34, 129–151 (1968)

    Article  Google Scholar 

  33. Stong, R.E.: Unoriented Bordism and Actions of Finite Groups. Memoirs of the American Mathematical Society, vol. 103. American Mathematical Society, Providence (1970)

  34. Suh, D.Y.: Quotients of real algebraic \(G\) varieties and algebraic realization problems. Osaka J. Math. 33(2), 399–410 (1996)

    MathSciNet  MATH  Google Scholar 

  35. Tognoli, A.: Su una congettura di Nash. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 27, 167–185 (1973)

  36. Walter, J.H.: The characterization of finite groups with abelian Sylow \(2\)-subgroups. Ann. Math. (Series 2) 89, 405–514 (1969)

    MathSciNet  MATH  Google Scholar 

  37. Wasserman, A.G.: Equivariant differential topology. Topology 8, 127–150 (1969)

    Article  MathSciNet  Google Scholar 

  38. Wasserman, A.G.: Simplifying group actions. Topology Appl. 75(1), 13–31 (1997)

    Article  MathSciNet  Google Scholar 

  39. Wasserman, A.G.: Extending algebraic actions. Rev. Mat. Complut. 12(2), 463–474 (1999)

    Article  MathSciNet  Google Scholar 

  40. Whitney, H.: Elementary structures of real algebraic varieties. Ann. Math. (Series 2) 66, 545–556 (1957)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank Ron Solomon and Dimitri Leemans for their generous help with the group theory. Critical questions of Mikiya Masuda encouraged us to clean up some arguments.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Hawaii, Honolulu, Hawaii, 96822, USA

    Karl Heinz Dovermann & Vincent Giambalvo

  2. University of Connecticut, Storrs, Connecticut, 06269, USA

    Vincent Giambalvo

Authors
  1. Karl Heinz Dovermann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vincent Giambalvo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karl Heinz Dovermann.

Ethics declarations

Funding:

N/A

Conflict of interest

N/A

DAS

The manuscript has no associated data or materials.

Code Availability

N/A

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dovermann, K.H., Giambalvo, V. Algebraic realization for simple groups. European Journal of Mathematics 8, 475–513 (2022). https://doi.org/10.1007/s40879-022-00532-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40879-022-00532-w

Keywords

Mathematics Subject Classification

Search

Navigation