Obstructions to nonnegative curvature and rational homotopy theory

Creators: Belegradek, Igor; Kapovitch, Vitali

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 x T admit no complete nonnegatively curved metrics.

Additional Information

© 2002 American Mathematical Society. Received by the editors October 28, 2001. Article electronically published on December 3, 2002. The first author is grateful to McMaster University and California Institute of Technology for support and excellent working conditions. It is our pleasure to thank Gregory Lupton and Samuel Smith for insightful discussions on rational homotopy theory, Stefan Papadima for Lemma 5.1, Ian Hambleton for Lemma A.1, Toshihiro Yamaguchi for Example 9.7, Alexander Givental for incisive comments on deformation theory, and Burkhard Wilking and Wolfgang Ziller for countless discussions and insights related to this work. We are grateful to the referee for helpful advice on the exposition. As always, the authors are solely responsible for possible mistakes. The present paper grew out of our earlier preprint [BK] written in the summer of 2000. In [BK] we proved much weaker results, for example, Theorem 1.3 is stated there as an open question. Most of the results of the present paper were obtained in May and early June of 2001 and reported by the first author during the Oberwolfach geometry meeting on June 12, 2001. On June 29, 2001, we received a preprint by Jianzhong Pan where he independently proves Theorem 1.3 in response to our question in [BK].

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