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Category bounds for nonnegative Ricci curvature manifolds with infinite fundamental group


Author: John Oprea
Journal: Proc. Amer. Math. Soc. 130 (2002), 833-839
MSC (1991): Primary 53P99; Secondary 55P99
DOI: https://doi.org/10.1090/S0002-9939-01-06121-4
Published electronically: June 21, 2001
MathSciNet review: 1866039
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Abstract:

This brief note presents refinements of the bounds on the first Betti number and the polynomial growth degree of the fundamental group for manifolds with nonnegative Ricci curvature and infinite fundamental group. These refinements are then sharpened when applied to symplectic manifolds.


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  • [Ad] J. F. Adams, Infinite Loop Spaces, Annals of Math. Studies 90, Princeton Univ. Press 1978. MR 80d:55001
  • [Au] M. Audin, Exemples de variétés presque complexes, Enseign. Math. 37 (1991) 175-190. MR 92g:53024
  • [Cai] M. Cai, A splitting theorem for manifolds of almost nonnegative Ricci curvature, Ann. Global Anal. Geom. 11 no. 4 (1993) 373-385. MR 94j:53043
  • [CG] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of non-negative Ricci curvature, J. Diff. Geom. 6 (1971) 119-128. MR 46:2597
  • [DoC] M. Do Carmo, Riemannian Geometry, Birkhäuser Boston, 1992. MR 92i:53001
  • [Fe] Y. Felix, La Dichotomie Elliptique-Hyperbolique en Homotopie Rationnelle, Asterisque 176, Soc. Math. France, 1989. MR 91c:55016
  • [GHL] S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Universitext, Springer-Verlag, 1990. MR 91j:53001
  • [Go] D. Gottlieb, Splitting off tori and the evaluation subgroup of the fundamental group, Israel J. Math. 66 (1989) 216-222. MR 90j:55019
  • [Gro] M. Gromov, Metric Structures for Riemannian and non-Riemannian Spaces, Birkhäuser Boston, 1999. MR 2000d:53065
  • [Ja] I. James, On category in the sense of Lusternik-Schnirelmann, Topology 17 (1978), 331-348. MR 80i:55001
  • [LM] H. B. Lawson and M. L. Michelsohn, Spin Geometry, Princeton Math. Series 38, Princeton Univ. Press, 1989. MR 91g:53001
  • [LO] G. Lupton, and J. Oprea, Cohomologically symplectic spaces: toral actions and the Gottlieb group, Trans. Amer. Math. Soc. 347 no. 1 (1995), 261-288. MR 95f:57056
  • [LS] L. Lusternik and L. Schnirelmann, Methodes Topologiques Dans Les Problemes Variationnels, Actualites Scientifiques et Industrielles 188, Hermann, 1934.
  • [Mil] J. Milnor, A note on curvature and fundamental group, J. Diff. Geom. 2 (1968) 1-7. MR 38:636
  • [MS] D. McDuff and D. Salamon, Introduction to Symplectic Topology, 2nd ed., Oxford University Press, 1998. MR 2000g:53098
  • [Op] J. Oprea, A homotopical Conner-Raymond theorem and a question of Gottlieb, Can. Math. Bull. 33 (1990), 219-229. MR 91e:55019
  • [OP] J. Oprea and J. Pak, Principal bundles over tori and maps which induce the identity on homotopy, Top. Appl. 52 (1993), 11-22. MR 95e:55024
  • [OW] J. Oprea and J. Walsh, Quotient maps, group actions and Lusternik-Schnirelmann category, preprint 2000, to appear in Top. Appl.
  • [Pet] P. Petersen, Riemannian Geometry, Grad. Texts in Math., 171, Springer-Verlag, 1998. MR 98m:53001
  • [RO] Y. Rudyak and J. Oprea, On the Lusternik-Schnirelmann category of symplectic manifolds and the Arnold conjecture, Math. Zeit. 230 no. 4 (1999) 673-678. MR 2000b:53115
  • [TO] A. Tralle and J. Oprea, Symplectic Manifolds with no Kähler Structure, Lecture Notes in Math., 1661, Springer-Verlag, 1997. MR 98k:53038
  • [Wh] G. Whitehead, Elements of Homotopy Theory, Grad. Texts in Math., 61, Springer-Verlag, 1978. MR 80b:55001
  • [Wol] J. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Diff. Geom. 2 (1968), 421-426. MR 40:1939
  • [Yau] S.-T. Yau, Calabi's conjecture and some new results in algebraic geometry, Proc. Natl. Acad. Sci. 74 (1977), 1798-1799. MR 56:9467

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Additional Information

John Oprea
Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115
Email: oprea@math.csuohio.edu

DOI: https://doi.org/10.1090/S0002-9939-01-06121-4
Received by editor(s): June 9, 2000
Received by editor(s) in revised form: August 24, 2000
Published electronically: June 21, 2001
Additional Notes: I wish to thank Mladen Bestvina for helpful emails pointing out several relevant results in [Gro]
Communicated by: Wolfgang Ziller
Article copyright: © Copyright 2001 American Mathematical Society

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