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Transactions of the American Mathematical Society

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Sphere bundles with $ 1/4$-pinched fiberwise metrics


Authors: Thomas Farrell, Zhou Gang, Dan Knopf and Pedro Ontaneda
Journal: Trans. Amer. Math. Soc. 369 (2017), 6613-6630
MSC (2010): Primary 57R22; Secondary 53C44
DOI: https://doi.org/10.1090/tran/6993
Published electronically: May 16, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that all smooth sphere bundles that admit fiberwise $ 1/4$-pinched metrics are induced bundles of vector bundles, so their structure groups reduce from $ \mathrm {DIFF}(\mathbb{S}^n)$ to $ \mathrm {O}(n+1)$. This result implies the existence of many smooth $ \mathbb{S}^n$-bundles over $ \mathbb{S}^k$ that do not support strictly $ 1/4$-pinched positively curved Riemannian metrics on their fibers.


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Additional Information

Thomas Farrell
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing, People’s Republic of China
Email: farrell@math.tsinghua.edu.cn

Zhou Gang
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Address at time of publication: Department of Mathematical Sciences, Binghamton University, Binghamton, New York 13902
Email: gzhou@caltech.edu

Dan Knopf
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Email: danknopf@math.utexas.edu

Pedro Ontaneda
Affiliation: Department of Mathematical Sciences, Binghamton University, Binghamton, New York 13902
Email: pedro@math.binghamton.edu

DOI: https://doi.org/10.1090/tran/6993
Received by editor(s): April 21, 2016
Received by editor(s) in revised form: June 8, 2016
Published electronically: May 16, 2017
Additional Notes: The first and fourth authors thank NSF for support in DMS-1206622
The second author thanks NSF for support in DMS-1308985 and DMS-1443225
The third author thanks NSF for support in DMS-1205270
Article copyright: © Copyright 2017 American Mathematical Society

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