Obstructions to nonnegative curvature and rational homotopy theory

Belegradek, Igor¹ ; Kapovitch, Vitali²

¹ Department of Mathematics, 253-37, California Institute of Technology, Pasadena, California 91125
² Department of Mathematics, University of California Santa Barbara, Santa Barbara, California 93106

Journal of the American Mathematical Society, Tome 16 (2003) no. 2, pp. 259-284

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-02-00418-6

Bibliographie

[1] Bazaä­Kin, Ya. V. On a family of 13-dimensional closed Riemannian manifolds of positive curvature Sibirsk. Mat. Zh. 1996

[2] Belegradek, Igor, Kapovitch, Vitali Finiteness theorems for nonnegatively curved vector bundles Duke Math. J. 2001 109 134

[3] Belegradek, Igor, Kapovitch, Vitali Topological obstructions to nonnegative curvature Math. Ann. 2001 167 190

[4] Cheeger, Jeff, Gromoll, Detlef On the structure of complete manifolds of nonnegative curvature Ann. of Math. (2) 1972 413 443

[5] Cheeger, Jeff Some examples of manifolds of nonnegative curvature J. Differential Geometry 1973 623 628

[6] Davis, James F. Manifold aspects of the Novikov conjecture 2000 195 224

[7] Eschenburg, J.-H. New examples of manifolds with strictly positive curvature Invent. Math. 1982 469 480

[8] Eschenburg, J.-H. Cohomology of biquotients Manuscripta Math. 1992 151 166

[9] Eschenburg, J.-H. Inhomogeneous spaces of positive curvature Differential Geom. Appl. 1992 123 132

[10] Fã©Lix, Yves La dichotomie elliptique-hyperbolique en homotopie rationnelle Astérisque 1989 187

[11] Fã©Lix, Yves, Halperin, Stephen, Thomas, Jean-Claude Rational homotopy theory 2001

[12] Grove, Karsten, Halperin, Stephen Contributions of rational homotopy theory to global problems in geometry Inst. Hautes Études Sci. Publ. Math. 1982

[13] Greub, Werner, Halperin, Stephen, Vanstone, Ray Connections, curvature, and cohomology 1976

[14] Grivel, Pierre-Paul Algèbres de Lie de dérivations de certaines algèbres pures J. Pure Appl. Algebra 1994 121 135

[15] Guijarro, Luis, Walschap, Gerard The metric projection onto the soul Trans. Amer. Math. Soc. 2000 55 69

[16] Grove, Karsten, Ziller, Wolfgang Curvature and symmetry of Milnor spheres Ann. of Math. (2) 2000 331 367

[17] Haefliger, Andr㩠Plongements différentiables de variétés dans variétés Comment. Math. Helv. 1961 47 82

[18] Halperin, Stephen Finiteness in the minimal models of Sullivan Trans. Amer. Math. Soc. 1977 173 199

[19] Markl, Martin Towards one conjecture on collapsing of the Serre spectral sequence Rend. Circ. Mat. Palermo (2) Suppl. 1990 151 159

[20] Meier, Willi Some topological properties of Kähler manifolds and homogeneous spaces Math. Z. 1983 473 481

[21] Mccleary, John, Ziller, Wolfgang On the free loop space of homogeneous spaces Amer. J. Math. 1987 765 781

[22] Onishchik, Arkadi L. Topology of transitive transformation groups 1994

[23] ÖZaydin, Murad, Walschap, Gerard Vector bundles with no soul Proc. Amer. Math. Soc. 1994 565 567

[24] Rigas, A. Geodesic spheres as generators of the homotopy groups of 𝑂, 𝐵𝑂 J. Differential Geometry 1978

[25] Singhof, W. On the topology of double coset manifolds Math. Ann. 1993 133 146

[26] Shiga, H., Tezuka, M. Rational fibrations, homogeneous spaces with positive Euler characteristics and Jacobians Ann. Inst. Fourier (Grenoble) 1987 81 106

[27] Stolz, Stephan Simply connected manifolds of positive scalar curvature Ann. of Math. (2) 1992 511 540

[28] Sullivan, Dennis Infinitesimal computations in topology Inst. Hautes Études Sci. Publ. Math. 1977

[29] Birkhoff, Garrett, Ward, Morgan A characterization of Boolean algebras Ann. of Math. (2) 1939 609 610

[30] Wallach, Nolan R. Compact homogeneous Riemannian manifolds with strictly positive curvature Ann. of Math. (2) 1972 277 295

[31] Gontcharoff, M. Sur quelques séries d’interpolation généralisant celles de Newton et de Stirling Uchenye Zapiski Moskov. Gos. Univ. Matematika 1939 17 48

[32] Wilking, Burkhard On fundamental groups of manifolds of nonnegative curvature Differential Geom. Appl. 2000 129 165

[33] Wilking, Burkhard Manifolds with positive sectional curvature almost everywhere Invent. Math. 2002 117 141

[34] Yang, Dagang On complete metrics of nonnegative curvature on 2-plane bundles Pacific J. Math. 1995 569 583