RESEARCH ARTICLE

A generalized π2-diffeomorphism finiteness theorem

  • Xiaochun RONG 1 ,
  • Xuchao YAO , 2
Expand
  • 1. Department of Mathematics, Rutgers University, New Brunswick, NJ 08903 USA
  • 2. School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Received date: 28 Feb 2020

Accepted date: 17 Apr 2020

Published date: 15 Apr 2020

Copyright

2020 Higher Education Press

Abstract

The π2-diffeomorphism finiteness result of F. Fang-X. Rong and A. Petrunin-W. Tuschmann (independently) asserts that the diffeomorphic types of compact n-manifolds M with vanishing first and second homotopy groups can be bounded above in terms of n; and upper bounds on the absolute value of sectional curvature and diameter of M: In this paper, we will generalize this π2-diffeomorphism finiteness by removing the condition that π1(M) = 0 and asserting the diffeomorphism finiteness on the Riemannian universal cover of M:

Cite this article

Xiaochun RONG , Xuchao YAO . A generalized π2-diffeomorphism finiteness theorem[J]. Frontiers of Mathematics in China, 2020 , 15(2) : 399 -418 . DOI: 10.1007/s11464-020-0834-0

1
Anderson M. Convergence and rigidity of manifolds under Ricci curvature bounds. Invent Math, 1990, 102: 429–445

DOI

2
Anderson M, Cheeger J. Ca-compactness for manifolds with Ricci curvature and injectivity radius bounded below. J Differential Geom, 1992, 35: 265–281

DOI

3
Cheeger J. Finiteness theorems for Riemannian manifolds. Amer J Math, 1970, 92: 61–75

DOI

4
Cheeger J, Colding T.Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann of Math, 1996, 144: 189–237

DOI

5
Cheeger J, Colding T.On the structure of space with Ricci curvature bounded below I. J Differential Geom, 1997, 46: 406–480

DOI

6
Cheeger J, Fukaya K, Gromov M. Nilpotent structures and invariant metrics on collapsed manifolds. J Amer Math Soc, 1992, 5: 327–372

DOI

7
Cheeger J,Gromoll D. The splitting theorem for manifolds of nonnegative Ricci curvature. J Differential Geom, 1971, 6: 119–128

DOI

8
Cheeger J, Gromov M. Collapsing Riemannian manifolds while keeping their curvature bounded I. J Differential Geom, 1986, 23: 309–346

DOI

9
Cheeger J, Gromov M. Collapsing Riemannian manifolds while keeping their curvature bounded II. J Differential Geom, 1990, 32: 269–298

DOI

10
Cheeger J, Jiang W, Naber A. Rectifiability of singular sets in noncollapsed spaces with Ricci curvature bounded below. arXiv: 1805.07988

11
ColdingT, Naber A. Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications. Ann of Math, 2012, 176: 1172–1229

DOI

12
Eschenburg J-H.New examples of manifolds with strictly positive curvature. Invent Math, 1982, 66: 469–480

DOI

13
Fang F, Rong X. Positive pinching, volume and second Betti number. Geom Funct Anal, 1999, 9: 641–674

DOI

14
Fang F, Rong X. The twisted second Betti number and convergence of collapsing Riemannian manifolds. Invent Math, 2002, 150: 61–109

DOI

15
Fukaya K. Collapsing Riemannian manifolds to ones of lower dimensions. J Differential Geom, 1987, 25: 139–156

DOI

16
Fukaya K. A boundary of the set of Riemannian manifolds with bounded curvature and diameters. J Differential Geom, 1988, 28: 1–21

DOI

17
Fukaya K. Collapsing Riemannian manifolds to ones with lower dimension II. J Math Soc Japan, 1989, 41: 333–356

DOI

18
Fukaya K, Yamaguchi T. The fundamental groups of almost nonnegatively curved manifolds. Ann of Math, 1992, 136: 253–333

DOI

19
Greene R E, Wu H. Lipschitz convergence of Riemannian manifolds. Pacific J Math, 1988, 131: 119–141

DOI

20
Gromov M. Almost flat manifolds. J Differential Geom, 1978, 13: 231–242

DOI

21
Gromov M, Lafontaine J,Pansu P.Structures métriques pour les variétés riemanniennes.Paris: Cedic/Fernand Nathan, 1981

22
Grove K, Karcher H. How to conjugate C1-close group actions. Math Z, 1973, 132: 11–20

DOI

23
Grove K, Petersen P, Wu J. Geometric finiteness theorems via controlled topology. Invent Math, 1990, 99: 205–213

DOI

24
Grove K, Petersen P, Wu J. Erratum to Geometric finiteness theorems via controlled topology. Invent Math, 1991, 104: 221–222

DOI

25
Huang H. Fibrations and stability of compact group actions on manifolds with local bounded Ricci covering geometry. Front Math China, 2020, 15: 69–89

DOI

26
Kapovitch V, Wilking B. Structure of fundamental groups of manifolds with Ricci curvature bounded below. arXiv: 1105.5955

27
Kirby R C, Siebenmann L C. Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. Ann of Math Stud, Vol 88. Princeton: Princeton Univ Press, 1977

DOI

28
Palais R S. Equivalence of nearby differentiable actions of a compact group. Bull Amer Math Soc, 1961, 67: 362–364

DOI

29
Perelman G. Alexandrov's spaces with curvatures bounded from below II. Preprint

30
Peters S. Cheeger's finiteness theorem for diffeomorphism classes of Riemannian manifolds. J Reine Angew Math, 1984, 349: 77–82

DOI

31
Petrunin A, Tuschmann W. Diffeomorphism finiteness, positive pinching, and second homotopy. Geom Funct Anal, 1999, 9: 736–774

DOI

32
Rong X.Convergence and collapsing theorems in Riemannian geometry. In: Ji L, Li P, Schoen R, Simon L, eds. Handbook of Geometric Analysis, Vol II. Adv Lect Math (ALM), Vol 13. Beijing/Boston: Higher Education Press/Int Press, 2010, 193{299

33
Switzer R. Algebraic Topology-Homotopy and Homology. Grundlehren Math Wiss, Vol 212. Berlin: Springer-Verlag, 1975

DOI

Outlines

/