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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On closed $\delta$-pinched manifolds with discrete abelian group actions
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by Yusheng Wang
Proc. Amer. Math. Soc. 137 (2009), 265-272
DOI: https://doi.org/10.1090/S0002-9939-08-09454-9
Published electronically: July 28, 2008

Abstract:

Let $M^{n}$ be a closed odd $n$-manifold with sectional curvature $\delta <\sec _{M}\le 1$, and let $M$ admit an effective isometric $\mathbb {Z}^{k}_{p}$-action with $p$ prime. The main results in the paper are: (1) if $\delta >0$ and $n\ge 5$, then there exists a constant $p(n,\delta )$, depending only on $n$ and $\delta$, such that $p\ge p(n,\delta )$ implies that (i) $k\le \frac {n+1}{2}$, (ii) the universal covering space of $M$ is homeomorphic to $S^{n}$ if $k>\frac {3}{8}n+1$, (iii) the fundamental group $\pi _{1}(M)$ is cyclic if $k>\frac {n+1}{4}+1$; (2) if $\delta =0$ and $n=3$, then $k\le 4$ for $p=2$ and $k\le 2$ for $p\ge 3$, and $\pi _{1}(M)$ is cyclic if $p\ge 5$ and $k=2$.
References
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Bibliographic Information
  • Yusheng Wang
  • Affiliation: School of Mathematical Sciences (& Lab. Math. Com. Sys.), Beijing Normal University, Beijing 100875, People’s Republic of China
  • Email: wyusheng@163.com, wwyusheng@gmail.com
  • Received by editor(s): December 20, 2006
  • Received by editor(s) in revised form: December 12, 2007
  • Published electronically: July 28, 2008
  • Additional Notes: The author was supported in part by NSFC Grant #10671018.
  • Communicated by: Alexander N. Dranishnikov
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 265-272
  • MSC (2000): Primary 53C20
  • DOI: https://doi.org/10.1090/S0002-9939-08-09454-9
  • MathSciNet review: 2439449