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On closed $ \delta $-pinched manifolds with discrete abelian group actions

Author: Yusheng Wang
Journal: Proc. Amer. Math. Soc. 137 (2009), 265-272
MSC (2000): Primary 53C20
Published electronically: July 28, 2008
MathSciNet review: 2439449
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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$.

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Yusheng Wang
Affiliation: School of Mathematical Sciences (& Lab. Math. Com. Sys.), Beijing Normal University, Beijing 100875, People’s Republic of China

Keywords: $\delta $-pinched manifold, group action, fundamental group
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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