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ALGEBRAIC REALIZATION FOR CYCLIC GROUP ACTIONS

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Abstract

Suppose G is a finite cyclic group and M a closed smooth G-manifold. We will show that there is a nonsingular real algebraic G-variety X that is equivariantly diffeomorphic to M so that all G-vector bundles over X are strongly algebraic.

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References

  1. S. Akbulut, H. King, The topology of real algebraic sets with isolated singularities, Ann. of Math. (2) 113 (1981), no. 3, 425–446.

  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  MATH  Google Scholar 

  3. R. Benedetti, A. Tognoli, On real algebraic vector bundles, Bull. Sci. Math. (2) 104 (1980), no. 1, 89–112.

  4. J. Bochnak, M. Coste, M.-F. Roy. Géométrie Algébrique Réelle, Ergebn. Math. Grenzg. (3), Bd. 12, Springer Verlag, Berlin, 1987.

  5. A. Borel, Topics in the Homology Theory of Fibre Bundles, Lect. Notes Math., Vol. 36, Springer Verlag, Berlin, 1967.

  6. G. Bredon, Introduction to Compact Transformation Groups, Pure Appl. Math., Vol 46, Academic Press, New York, 1972.

  7. P. E. Conner, Differentiable Periodic Maps, 2nd Edition, Lect. Notes Math., Vol. 738, Springer Verlag, Berlin, 1979.

  8. P. E. Conner, E. E. Floyd, Maps of odd period, Ann. of Math. (2) 84, 1966, 132–156.

  9. Costenoble, S.R.: Unoriented bordism for odd-order groups. Topology Appl. 28, 277–287 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. K. H. Dovermann, Equivariant algebraic realization of smooth manifolds and vector bundles, in: Real Algebraic Geometry and Topology (East Lansing, MI, 1993), Contemp. Math. 182, Amer. Math. Soc., Providence, RI, 1995, pp. 29–46.

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Dovermann, K.H., Flores, D., Giambalvo, V.: Algebraic realization of actions of some finite groups. Manuscr. Math. 165, 239–254 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  13. K. H. Dovermann, V. Giambalvo, Algebraic realization for simple groups, European J. Math., https://doi.org/https://doi.org/10.1007/s40879-022-00532-w.

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Dovermann, K.H., Hanson, J.S.: Tensor products of symmetric functions over2. Cent. Eur. J. Math. 3(2), 1–9 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. K. H. Dovermann, M. Masuda, Algebraic realization of closed smooth manifolds with cyclic actions, Mathematica Göttingensis, Heft 29 (1993), available at http://www.math.hawaii.edu/_heiner/researchpapers.html.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. 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 

  21. 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  MATH  Google Scholar 

  22. J. S. Hanson, Bordism and Algebraic Realization, Thesis, University of Hawaii at Manoa, 1998.

  23. F. Hirzebruch, Topological Methods in Algebraic Geometry, Grundl. math. Wissenschaften, Bd. 131, Springer Verlag, Berlin, 1966.

  24. Kosniowski, C.: Actions of Finite Abelian Groups, Research Notes in Mathematics, vol. 18. Pitman, London (1978)

    MATH  Google Scholar 

  25. C. N. Lee, A. G. Wasserman, Equivariant characteristic numbers, in: Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971), Part I, Lect. Notes Math., Vol. 298, Springer, Berlin, 1972, pp. 191–216.

  26. Milnor, J.: On the Stiefel–Whitney numbers of complex manifolds and of spin-manifolds. Topology. 3, 223–230 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  27. J. W. Milnor, J. D. Stasheff, Characteristic Classes, Annals of Mathematics Studies, Vol. 76, Princeton University Press, Princeton, NJ, 1974.

  28. J. Nash, Real algebraic manifolds, Annals of Math. (2) 56 (1952), 405–421.

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Serre, J.-P.: Cohomologie modulo 2 des complexes d’Eilenberg–MacLane. Comm. Math. Helv. 27, 198–232 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  31. R. E. Stong, Unoriented Bordism and Actions of Finite Groups, Memoirs of the Amer. Math. Soc., Vol. 103, 1970.

  32. R. E. Stong, All in the family, preprint.

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

    MathSciNet  MATH  Google Scholar 

  34. Tognoli, A.: Su una Congettura di Nash. Ann. Scuola Norm. Sup. di Pisa. 27, 167–185 (1973)

    MathSciNet  MATH  Google Scholar 

  35. Wasserman, A.G.: Equivariant differential topology. Topology. 8, 127–150 (1967)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  37. H. Whitney, Algebraic structure of real algebraic varieties, Ann. of Math. (2) 66 (1957), 545–556.

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Correspondence to KARL HEINZ DOVERMANN.

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DOVERMANN, K.H., WASSERMAN, A.G. ALGEBRAIC REALIZATION FOR CYCLIC GROUP ACTIONS. Transformation Groups 28, 1561–1593 (2023). https://doi.org/10.1007/s00031-022-09728-6

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