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Fundamental Groups of Manifolds with Positive Sectional Curvature and Torus Symmetry

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Abstract

In 1965, Chern posed a question concerning the extent to which fundamental groups of manifolds admitting positive sectional curvature look like spherical space form groups. The original question was answered in the negative by Shankar in 1998, but there are a number of positive results in the presence of symmetry. These classifications fall into categories according to the strength of their conclusions. We give an overview of these results in the case of torus symmetry and prove new results in each of these categories.

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Acknowledgements

This work began as part of the author’s Ph.D. thesis, and he is pleased to thank his advisor, Wolfgang Ziller, for multiple discussions and comments on earlier drafts. The author is also grateful to Daryl Cooper and Darren Long for directing him to the work of Jim Davis and Shmuel Weinberger, and to Jim Davis for a discussion of this work. The author is grateful for support from NSF Grants DMS-1045292 and DMS-1404670/1622541. The author would also like to thank the referee for helpful comments.

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  1. Department of Mathematics, University of Oklahoma, Norman, Oklahoma, 73019, USA

    Lee Kennard

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Kennard, L. Fundamental Groups of Manifolds with Positive Sectional Curvature and Torus Symmetry. J Geom Anal 27, 2894–2925 (2017). https://doi.org/10.1007/s12220-017-9787-2

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