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

HTML articles powered by AMS MathViewer

- 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
- PDF | Request permission

## 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

- Glen E. Bredon,
*Introduction to compact transformation groups*, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR**0413144** - Jeff Cheeger and David G. Ebin,
*Comparison theorems in Riemannian geometry*, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR**0458335** - Fuquan Fang and Xiaochun Rong,
*Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank*, Math. Ann.**332**(2005), no. 1, 81–101. MR**2139252**, DOI 10.1007/s00208-004-0618-y - Fuquan Fang and Xiaochun Rong,
*Positively curved manifolds with maximal discrete symmetry rank*, Amer. J. Math.**126**(2004), no. 2, 227–245. MR**2045502** - P. Frank; X. Rong; Y. Wang,
*Fundamental groups of positively curved manifolds with symmetry*, preprint. - Michael Hartley Freedman,
*The topology of four-dimensional manifolds*, J. Differential Geometry**17**(1982), no. 3, 357–453. MR**679066** - Michael Gromov,
*Curvature, diameter and Betti numbers*, Comment. Math. Helv.**56**(1981), no. 2, 179–195. MR**630949**, DOI 10.1007/BF02566208 - Karsten Grove and Catherine Searle,
*Positively curved manifolds with maximal symmetry-rank*, J. Pure Appl. Algebra**91**(1994), no. 1-3, 137–142. MR**1255926**, DOI 10.1016/0022-4049(94)90138-4 - Richard S. Hamilton,
*Three-manifolds with positive Ricci curvature*, J. Differential Geometry**17**(1982), no. 2, 255–306. MR**664497** - Shoshichi Kobayashi,
*Transformation groups in differential geometry*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70, Springer-Verlag, New York-Heidelberg, 1972. MR**0355886** - John Milnor,
*Groups which act on $S^n$ without fixed points*, Amer. J. Math.**79**(1957), 623–630. MR**90056**, DOI 10.2307/2372566 - Peter Orlik,
*Seifert manifolds*, Lecture Notes in Mathematics, Vol. 291, Springer-Verlag, Berlin-New York, 1972. MR**0426001** - Xiaochun Rong,
*Positively curved manifolds with almost maximal symmetry rank*, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II (Haifa, 2000), 2002, pp. 157–182. MR**1950889**, DOI 10.1023/A:1021242512463 - Xiaochun Rong,
*On the fundamental groups of manifolds of positive sectional curvature*, Ann. of Math. (2)**143**(1996), no. 2, 397–411. MR**1381991**, DOI 10.2307/2118648 - Xiaochun Rong,
*On fundamental groups of positively curved manifolds with local torus actions*, Asian J. Math.**9**(2005), no. 4, 545–559. MR**2216245**, DOI 10.4310/AJM.2005.v9.n4.a6 - Xiaochun Rong and Xiaole Su,
*The Hopf conjecture for manifolds with abelian group actions*, Commun. Contemp. Math.**7**(2005), no. 1, 121–136. MR**2129791**, DOI 10.1142/S0219199705001660 - Stephen Smale,
*Generalized Poincaré’s conjecture in dimensions greater than four*, Ann. of Math. (2)**74**(1961), 391–406. MR**137124**, DOI 10.2307/1970239 - Burkhard Wilking,
*Torus actions on manifolds of positive sectional curvature*, Acta Math.**191**(2003), no. 2, 259–297. MR**2051400**, DOI 10.1007/BF02392966

## 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