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Obstructions to nonnegative curvature and rational homotopy theory


Authors: Igor Belegradek and Vitali Kapovitch
Journal: J. Amer. Math. Soc. 16 (2003), 259-284
MSC (2000): Primary 53C20, 55P62
DOI: https://doi.org/10.1090/S0894-0347-02-00418-6
Published electronically: December 3, 2002
MathSciNet review: 1949160
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Abstract: We establish a link between rational homotopy theory and the problem which vector bundles admit a complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if $C$ lies in the class and $T$ is a torus of positive dimension, then ``most'' vector bundles over $C\times T$ admit no complete nonnegatively curved metrics.


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  • [Baz96] Ya. V. Baza{\u{\i}}\kern.15emkin, On a family of $13$-dimensional closed Riemannian manifolds of positive curvature, Sibirsk. Mat. Zh. 37 (1996), no. 6, 1219-1237, ii. MR 98c:53045
  • [BK] I. Belegradek and V. Kapovitch, Obstructions to nonnegative curvature and rational homotopy theory, preprint, 2000, available electronically at the xxx-achive: http://arxiv.org/abs/math.DG/0007007.
  • [BK01a] I. Belegradek and V. Kapovitch, Finiteness theorems for nonnegatively curved vector bundles, Duke Math. J. 108 (2001), no. 1, 109-134. MR 2002c:53050
  • [BK01b] I. Belegradek and V. Kapovitch, Topological obstructions to nonnegative curvature, Math. Ann. 320 (2001), no. 1, 167-190. MR 2002d:53044
  • [CG72] J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. 96 (1972), 413-443. MR 46:8121
  • [Che73] J. Cheeger, Some examples of manifolds of nonnegative curvature, J. Differential Geom. 8 (1973), 623-628. MR 49:6085
  • [Dav94] J. Davis, Manifold aspects of the Novikov conjecture, Princeton Univ. Press, 1994, Surveys on surgery theory, Vol. 1, 195-224, Ann. of Math. Stud., 145. MR 2002a:57037
  • [Esc82] J.-H. Eschenburg, New examples of manifolds with strictly positive curvature, Invent. Math. 66 (1982), no. 3, 469-480. MR 83i:53061
  • [Esc92a] J.-H. Eschenburg, Cohomology of biquotients, Manuscripta Math. 75 (1992), no. 2, 151-166. MR 93e:57070
  • [Esc92b] J.-H. Eschenburg, Inhomogeneous spaces of positive curvature, Differential Geom. Appl. 2 (1992), no. 2, 123-132. MR 94j:53044
  • [Fél89] Y. Félix, La dichotomie elliptique-hyperbolique en homotopie rationnelle, Astérisque (1989), no. 176, 1-187. MR 91c:55016
  • [FHT01] Y. Félix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, Springer-Verlag, 2001. MR 2002d:55014
  • [GH83] K. Grove and S. Halperin, Contributions of rational homotopy theory to global problems in geometry, Inst. Hautes Études Sci. Publ. Math. 56 (1982) (1983), 171-177. MR 84b:58030
  • [GHV76] W. Greub, S. Halperin, and R. Vanstone, Connections, curvature, and cohomology, Academic Press, 1976, Volume III: Cohomology of principal bundles and homogeneous spaces, Pure and Applied Mathematics, Vol. 47-III. MR 53:4110
  • [Gri94] P.-P. Grivel, Algèbres de Lie de dérivations de certaines algèbres pures, J. Pure Appl. Algebra 91 (1994), no. 1-3, 121-135. MR 95a:55024
  • [GW00] L. Guijarro and G. Walschap, The metric projection onto the soul, Trans. Amer. Math. Soc. 352 (2000), 55-69. MR 2000c:53034
  • [GZ00] K. Grove and W. Ziller, Curvature and symmetry of Milnor spheres, Ann. of Math. (2) 152 (2000), no. 1, 331-367. MR 2001i:53047
  • [Hae61] A. Haefliger, Plongements différentiables de variétés dans variétés, Comment. Math. Helv. 36 (1961), 47-82. MR 26:3069
  • [Hal77] S. Halperin, Finiteness in the minimal models of Sullivan, Trans. Amer. Math. Soc. 230 (1977), 173-199. MR 57:1493
  • [Mar90] M. Markl, Towards one conjecture on collapsing of the Serre spectral sequence, Proceedings of the Winter School on Geometry and Physics (Srní, 1989); Rend. Circ. Mat. Palermo (2) Suppl., no. 22, 1990, pp. 151-159. MR 91c:55033
  • [Mei83] W. Meier, Some topological properties of Kähler manifolds and homogeneous spaces, Math. Z. 183 (1983), no. 4, 473-481. MR 84j:55005
  • [MZ87] J. McCleary and W. Ziller, On the free loop space of homogeneous spaces, Amer. J. Math. 109 (1987), 765-781. MR 88k:58023
  • [Oni94] A. L. Onishchik, Topology of transitive transformation groups, Johann Ambrosius Barth Verlag GmbH, 1994. MR 95e:57058
  • [ÖW94] M. Özaydin and G. Walschap, Vector bundles with no soul, Proc. Amer. Math. Soc. 120 (1994), no. 2, 565-567. MR 94d:53057
  • [Rig78] A. Rigas, Geodesic spheres as generators of the homotopy groups of ${\rm {O}}$, ${\rm {BO}}$, J. Differential Geom. 13 (1978), no. 4, 527-545 (1979). MR 81e:57043
  • [Sin93] W. Singhof, On the topology of double coset manifolds, Math. Ann. 297 (1993), no. 1, 133-146. MR 94k:57054
  • [ST87] H. Shiga and M. Tezuka, Rational fibrations, homogeneous spaces with positive Euler characteristics and Jacobians, Ann. Inst. Fourier (Grenoble) 37 (1987), no. 1, 81-106. MR 89g:55019
  • [Sto92] S. Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. 136 (1992), no. 3, 511-540. MR 93i:57033
  • [Sul77] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 269-331 (1978). MR 58:31119
  • [TO97] A. Tralle and J. Oprea, Symplectic manifolds with no Kähler structure, Springer-Verlag, 1997. MR 9k:53038
  • [Wal72] N. R. Wallach, Compact homogeneous riemannian manifolds with strictly positive curvature, Ann. of Math. 96 (1972), 277-295. MR 46:6243
  • [Wal99] C. T. C. Wall, Surgery on compact manifolds, second ed., American Mathematical Society, 1999, edited and with a foreword by A. A. Ranicki. MR 2002a:57089
  • [Wil00] B. Wilking, On fundamental groups of manifolds of nonnegative curvature, Differential Geom. Appl. 13 (2000), no. 2, 129-165. MR 2001g:53076
  • [Wil02] B. Wilking, Manifolds with positive sectional curvature almost everywhere, Invent. Math. 148 (2002), no. 1, 117-141. MR 2003a:53049
  • [Yam] T. Yamaguchi, A rational condition on fiber in fibrations, preprint, 2002.
  • [Yan95] D. Yang, On complete metrics of nonnegative curvature on $2$-plane bundles, Pacific J. Math. 171 (1995), no. 2, 569-583. MR 96k:53034

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Additional Information

Igor Belegradek
Affiliation: Department of Mathematics, 253-37, California Institute of Technology, Pasadena, California 91125
Email: ibeleg@its.caltech.edu

Vitali Kapovitch
Affiliation: Department of Mathematics, University of California Santa Barbara, Santa Barbara, California 93106
Email: vitali@math.ucsb.edu

DOI: https://doi.org/10.1090/S0894-0347-02-00418-6
Keywords: Nonnegative curvature, soul, derivation, Halperin's conjecture
Received by editor(s): October 28, 2001
Published electronically: December 3, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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