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An Obstruction to Fundamental Groups of Positively Ricci Curved Manifolds

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Abstract

In this note we propose a conjecture concerning fundamental groups of Riemannian n-manifolds with positive Ricci curvature. We prove a partial result under an extra condition on a lower bound of sectional curvature. Our main tool is the theory of Hausdorff convergence. We also extend Fukaya and Yamaguchi's resolution of a conjecture of Gromov to limit spaces which may have singular points.

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References

  1. Burago, Y., Gromov, M. and Perelman, G.: A. D. Alexandrov’s spaces with curvatures bounded from below, Uspekhi Mat. Nauk 47 (1992), 3–51.

    Google Scholar 

  2. Buser, P. and Karcher, H.: Gromov’s almost flat manifolds, Astérisque 81 (1981), 1–148.

    Google Scholar 

  3. Cheeger, J. and Colding, T. H.: Lower bounds on Ricci curvature and almost rigidity of warped products, Ann. of Math. 144 (1996), 189–237.

    Google Scholar 

  4. Davermann, R. J.: Decompositions of Manifolds, Academic Press, New York, 1986.

    Google Scholar 

  5. Fukaya, K.: Theory of convergence for Riemannian orbifolds, Japan. J. Math. 12 (1986), 121–160.

    Google Scholar 

  6. Fukaya, K.: Hausdorff convergence of Riemannian manifolds and its applications, in Ochiai, T. (ed.), Recent Topics in Differential and Analytic Geometry, Adv. Stud. Pure Math., Vol. 18-I, Kinokuniya, Tokyo, 1990, pp. 143–238.

  7. Fukaya, K. and Yamaguchi, T.: The fundamental group of almost nonnegatively curved manifolds, Ann. of Math. 136 (1992), 253–333.

    Google Scholar 

  8. Fukaya, K. and Yamaguchi, T.: Isometry group of singular spaces, Math. Z. 216 (1994), 31–44.

    Google Scholar 

  9. Gromov, M.: Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), 179–195.

    Google Scholar 

  10. Gromov, M.: Almost flat manifolds, J. Differential Geom. 13 (1978), 231–241.

    Google Scholar 

  11. Gromov, M.: Groups of polynomial growth and expanding maps, Publ. Math. IHES 53 (1981), 53–73.

    Google Scholar 

  12. Grove, K. and Petersen V, P.: Manifolds near the boundary of existence, J. Differential Geom. 33 (1991), 379–394.

    Google Scholar 

  13. Grove, K. and Petersen V, P.: A pinching theorem for homotopy spheres, J. Amer. Math. Soc. 3 (1990), 671–677.

    Google Scholar 

  14. Gromov, M., Lafontaine, J. and Pansu, P.: Structure métrique pour les variétés riemanniennes, Cedic/Fernand Nathan, Paris, 1981.

    Google Scholar 

  15. Grove, K., Petersen, P. and Wu, J.-Y.: Geometric finiteness theorems via controlled topology, Invent. Math. 99 (1990), 205–211.

    Google Scholar 

  16. Gao, L. Z. and Yau, S.-T.: The existence of negatively Ricci curved metrics of three manifolds, Invent. Math. 85 (1986), 637–652.

    Google Scholar 

  17. Hamilton, R.: Three manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), 255–306.

    Google Scholar 

  18. Hu, S. T.: Theory of Retract, Wayne State University Press, Detroit, 1965.

    Google Scholar 

  19. Lohkamp, J.: Metrics of negative Ricci curvature, Ann. of Math. 140 (1994), 655–683.

    Google Scholar 

  20. Milnor, J.: A note on curvature and fundamental group, J. Differential Geom. 2 (1968), 1–7.

    Google Scholar 

  21. O’Neill, B.: Fundamental equations for a submersion, Michigan J. Math. 13 (1966), 459–469.

  22. Perelman, G.: Alexandrov’s spaces with curvatures bounded from below, II, preprint, 1991.

  23. Perelman, G.: The beginnings of Morse theory on Alexandrov spaces, St. Petersburg Math. J., to appear.

  24. Petersen V, P.: Gromov–Hausdorff convergence of metric spaces, in Yau, S.-T. and Green, R. (eds.), Differential Geometry, Proc. Symp. Pure Math., Vol. 54, Part 3, AMS, Providence, RI, 1993, pp. 489–504.

    Google Scholar 

  25. Stolz, S.: A conjecture concerning positive Ricci curvature and the Witten genus, Math. Ann. 304 (1996), 785–800.

    Google Scholar 

  26. Wu, J.-Y.: Hausdorff convergence and sphere theorems, in Yau, S.-T. and Green, R. (eds.), Differential Geometry, Proc. Symp. Pure Math., Vol. 54, Part 3, AMS, Providence, RI, 1993, pp. 685–692.

    Google Scholar 

  27. Wu, J.-Y.: On the structure of almost nonnegatively curved manifolds, J. Differential Geom. 35 (1992), 385–397.

    Google Scholar 

  28. Wu, J.-Y.: Convergence of Riemannian 3-manifolds under a Ricci curvature bound, Amer. J. Math. 116 (1994), 1019–1029.

    Google Scholar 

  29. Wolf, J. A.: Spaces of Constant Curvature, McGraw-Hill, New York, 1966.

    Google Scholar 

  30. Yamaguchi, T.: Collapsing and pinching under a lower curvature bound, Ann. of Math. 133 (1991), 317–357.

    Google Scholar 

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  1. Department of Mathematics, National Chung Cheng University, Chia-Yi, 621, Taiwan, R.O.C.

    Jyh-Yang Wu

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  1. Jyh-Yang Wu
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Wu, JY. An Obstruction to Fundamental Groups of Positively Ricci Curved Manifolds. Annals of Global Analysis and Geometry 16, 371–382 (1998). https://doi.org/10.1023/A:1006511918006

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