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ALMOST FIXED POINTS OF FINITE GROUP ACTIONS ON MANIFOLDS WITHOUT ODD COHOMOLOGY

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Abstract

If X is a smooth manifold and \( \mathcal{G} \) is a subgroup of Diff(X) we say that (X, \( \mathcal{G} \)) has the almost fixed point property if there exists a number C such that for any finite subgroup G ≤ \( \mathcal{G} \) there is some x ∈ X whose stabilizer Gx ≤ \( \mathcal{G} \) satisfies [G : Gx] ≤ C. We say that X has no odd cohomology if its integral cohomology is torsion free and supported in even degrees.

We prove that if X is compact and possibly with boundary and has no odd cohomology then (X, Diff(X)) has the almost fixed point property. Combining this with a result of Petrie and Randall we conclude that if Z is a non-necessarily compact smooth real affine variety, and Z has no odd cohomology, then (Z, Aut(Z)) has the almost fixed point property, where Aut(Z) is the group of algebraic automorphisms of Z lifting the identity on Spec ℝ.

The main ingredients in the proof are: (1) the Jordan property for diffeomorphism groups of compact manifolds with nonzero Euler characteristic, and (2) the study of λ-stability, a condition on actions of finite abelian groups on manifolds that we introduce in this paper.

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References

  1. A. Bak, M. Morimoto, The dimension of spheres with smooth one fixed point actions, Forum Math. 17 (2005), no. 2, 199–216.

    Article  MathSciNet  Google Scholar 

  2. A. Bialynicki-Birula, J. B. Carrell, W. M. McGovern, Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action, Encyclopaedia of Mathematical Sciences, Vol. 131, Subseries Invariant Theory and Algebraic Transformation Groups, Vol. II, Springer-Verlag, Berlin, 2002.

  3. E. M. Bloomberg, Manifolds with no periodic homeomorphism, Trans. Amer. Math. Soc. 202 (1975), 67–78.

    Article  MathSciNet  Google Scholar 

  4. A. Borel, Seminar on Transformation Groups, Ann. of Math. Studies, Vol. 46, Princeton University Press, New Jersey, 1960.

  5. G. E. Bredon, Introduction to Compact Transformation Groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York, 1972.

  6. N. P. Buchdahl, S. Kwasik, R. Schultz, One fixed point actions on low-dimensional spheres, Invent. Math. 102 (1990), no. 3, 633–662.

    Article  MathSciNet  Google Scholar 

  7. A. Čap, J. Slovák, Parabolic Geometries. I. Background and General Theory, Mathematical Surveys and Monographs, Vol. 154, American Mathematical Society, Providence, RI, 2009.

  8. P. E. Conner, F. Raymond, P. J. Weinberger, Manifolds with no periodic maps, in: Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971), Part II, Lecture Notes in Math., Vol. 299, Springer, Berlin, 1972, pp. 81–108.

  9. C. W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, AMS Chelsea Publishing, Providence, RI, 2006.

    Book  Google Scholar 

  10. K. H. Dovermann, M. Masuda, T. Petrie, Fixed point free algebraic actions on varieties diffeomorphic to ℝn, in: Topological Methods in Algebraic Transformation Groups (New Brunswick, NJ, 1988), Progr. Math., Vol. 80, Birkhäuser Boston, Boston, MA, 1989, pp. 49–80.

  11. T. tom Dieck, Transformation Groups, de Gruyter Studies in Mathematics, Vol. 8, Walter de Gruyter, Berlin, 1987.

  12. H. M. Farkas, I. Kra, Riemann Surfaces, 2nd ed., Graduate Texts in Mathematics, Vol 71, Springer-Verlag, New York, 1992.

  13. O. Haution, On rational fixed points of finite group actions on the affine space, Trans. Amer. Math. Soc. 369 (2017), no. 11, 8277–8290.

    Article  MathSciNet  Google Scholar 

  14. R. Haynes, S. Kwasik, J. Mast, R. Schultz, Periodic maps on ℝ7 without fixed points, Math. Proc. Cambridge Philos. Soc. 132 (2002), no. 1, 131–136.

    Article  MathSciNet  Google Scholar 

  15. M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, Vol. 33, Springer, New York, 1976.

  16. C. Jordan, Mémoire sur les équations différentielles linéaires à intégrale algébrique, J. Reine Angew. Math. 84 (1878) 89–215.

    MATH  Google Scholar 

  17. H. Kraft, Algebraic automorphisms of affine space, in: Topological Methods in Algebraic Transformation Groups (New Brunswick, NJ, 1988), Progr. Math., Vol. 80, Birkhäuser Boston, Boston, MA, 1989, pp. 81–105.

  18. H. Kraft, Challenging problems on affine n-space, Séminaire Bourbaki, Vol. 1994/95, Astérisque 237 (1996), Exp. No. 802, 5, 295–317.

  19. H. Kraft, G. Schwarz, Finite automorphisms of affine N-space, in: Automorphisms of Affine Spaces (Curaçao, 1994), Kluwer, Dordrecht, 1995, pp. 55–66.

  20. S. D. Liao, A theorem on periodic transformations of homology spheres, Ann. of Math. 56 (1952) 68–83.

    Article  MathSciNet  Google Scholar 

  21. L. N. Mann, J. C. Su, Actions of elementary p-groups on manifolds, Trans. Amer. Math. Soc. 106 (1963), 115–126.

    MathSciNet  MATH  Google Scholar 

  22. H. Minkowski, Zur Theorie der positiven quadratischen Formen, J. für die reine und angew. Math. 101 (1887), 196–202. (See also Collected Works, I, Chelsea Publ., 1967, pp. 212–218.)

  23. I. Mundet i Riera, Finite groups actions on manifolds without odd cohomology, preprint arXiv:1310.6565v4 (2014).

  24. I. Mundet i Riera, Finite group actions on 4-manifolds with nonzero Euler characteristic, Math. Z. 282 (2016), 25–42.

    Article  MathSciNet  Google Scholar 

  25. I. Mundet i Riera, Finite group actions on homology spheres and manifolds with nonzero Euler characteristic, J. Topology 12 (2019), 743–757.

    Article  MathSciNet  Google Scholar 

  26. I. Mundet i Riera, Finite subgroups of Ham and Symp, Math. Annalen 370 (2018), 331–380.

    Article  MathSciNet  Google Scholar 

  27. I. Mundet i Riera, A. Turull, Boosting an analogue of Jordan’s theorem for finite groups, Adv. Math. 272 (2015), 820–836.

    Article  MathSciNet  Google Scholar 

  28. T. Petrie, J. D. Randall, Finite-order algebraic automorphisms of affine varieties, Comment. Math. Helv. 61 (1986), no. 2, 203–221.

    Article  MathSciNet  Google Scholar 

  29. V. L. Popov, On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, in: Peter Russells Festschrift, Proceedings of the Conference on Affine Algebraic Geometry Held in Professor Russells Honour, 1–5 June 2009, Centre de Recherches Mathématiques CRM Proc. and Lect. Notes, Vol. 54, McGill University, Montreal, 2011, pp. 289–311.

  30. V. Puppe, Do manifolds have little symmetry?, J. Fixed Point Theory Appl. 2 (2007), no. 1, 85–96.

    Article  MathSciNet  Google Scholar 

  31. J.–P. Serre, Bounds for the orders of the finite subgroups of G(k), in: Group Representation Theory, EPFL Press, Lausanne, 2007, pp. 405–450.

  32. J.–P. Serre, How to use finite fields for problems concerning infinite fields, in: Arithmetic, Geometry, Cryptography and Coding Theory, Contemp. Math., Vol. 487, Amer. Math. Soc., Providence, RI, 2009, pp. 183–193.

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  1. Facultat de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007, Barcelona, Spain

    IGNASI MUNDET I RIERA

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This work has been partially supported by the (Spanish) MEC Projects MTM2012-38122-C03-02 and MTM2015-65361-P.

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MUNDET I RIERA, I. ALMOST FIXED POINTS OF FINITE GROUP ACTIONS ON MANIFOLDS WITHOUT ODD COHOMOLOGY. Transformation Groups 25, 1269–1288 (2020). https://doi.org/10.1007/s00031-019-09534-7

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