Topological regularity theorems for Alexandrov spaces

Jyh-Yang WU

doi:10.2969/jmsj/04940741

Jyh-Yang WU

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JOURNAL FREE ACCESS

1997 Volume 49 Issue 4 Pages 741-757

DOI https://doi.org/10.2969/jmsj/04940741

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Published: 1997 Received: January 20, 1995 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : BGP) Yu. Burago, M. Gromov and G. Perelman, A.D. Alexandrov's spaces with curvatures bounded from below, Usp. Mat. Nauk, 47 (1992), 3-51.
CG) J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. Math., 96 (1974), 413-443.
D) R.J. Daverman, Decompositions of manifolds, New York, Academic Press, 1986.
EK) R. Edwards and R. Kirby. Deformations of spaces of embeddings, Ann. Math., 93 (1971), 63-88.
FY) K. Fukaya and T. Yamaguchi, Isometry groups of singular spaces, Math. Zeit., 216 (1994), 31-44.
G1) M. Gromov, Groups of polynomial growth and expanding maps, IHES Publ. Math.,53 (1981), 183-215.
G2) M. Gromov, Structure métrique pour les variétés riemanniennes, Cedic/Fernand Nathan, Paris, 1981.
G3) M. Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv., 56 (1981), 179-195.
GP1) K. Grove and P. Petersen V, Manifolds near the boundary of existence, J. Differ. Geom., 33 (1991), 379-394.
GP2) K. Grove and P. Petersen V, A radius sphere theorem, Invent. Math., 112 (1993), 577-583.
GP3) K. Grove and P. Petersen V, On the excess of metric spaces and manifold, preprint. [GPW) K. Grove, P. Petersen V and J.-Y. Wu, Geometric finiteness theorems via controlled topology, Invent. Math., 99 (1990), 205-213.
Pe) G. Perelman, Alexandrov's spaces with curvatures bounded from below, II, preprint. [Pt) C. Plaut, Spaces of Wald-Berestovskii curvature bounded below, J. Geom. Analysis, 6 (1996), 113-134.
Q1) F. Quinn, Ends of maps, I, Ann. Math., 110 (1979), 275-331.
Q2) F. Quinn, Ends of maps, III, J. Differ. Geom., 17 (1982), 503-521.[Q3) F. Quinn, Resolutions of homology manifolds, and the topological characterizations of manifolds, Invent. Math., 72 (1983), 267-284.
Q4) F. Quinn, An obstruction to the resolution of homology manifolds, Michigan Math. J., 34 (1987), 285-291.
S) L.C. Siebenmann, Deformation of homeomorphisms on stratified sets, Comment. math. Helv., 47 (1972), 123-163.
Sh) Z. Shen, A regularity theorem for Alexandrov spaces, Math. Nachr., to appear..
SW) Z. Shen and G. Wei, On Riemannian manifolds with almost nonnegative curvature, Indiana Univ. Math. J., 40 (1991), 551-563.
W1) J.-Y. Wu, Hausdorff convergence and sphere theorems, Proc. Symp. Pure Math., ed. by R. Greene and S.-T. Yau, 54 part 3 (1993), 685-692.
W2) J.-Y. Wu, On the structure of almost nonnegatively curved manifolds, J. Differ. Geom., 35 (1992), 385-397.
W3) J.-Y. Wu, Convergence of Riemannian 3-manifolds under Ricci curvature bound, Amer. J. Math., Vol. 116 (1994), 1019-1029.
Wh) G.W. Whitehead, Elements of homotopy theory, GTM 61.
Y) T. Yamaguchi, Collaping and pinching under a lower curvature bound, Ann. Math., 133 (1991), 317-357.

Right : [BGP] Yu. Burago, M. Gromov and G. Perelman, A.D. Alexandrov's spaces with curvatures bounded from below, Usp. Mat. Nauk, 47 (1992), 3-51.
[CG] J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. Math., 96 (1974), 413-443.
[D] R. J. Daverman, Decompositions of manifolds, New York, Academic Press, 1986.
[EK] R. Edwards and R. Kirby. Deformations of spaces of embeddings, Ann. Math., 93 (1971), 63-88.
[FY] K. Fukaya and T. Yamaguchi, Isometry groups of singular spaces, Math. Zeit., 216 (1994), 31-44.
[G1] M. Gromov, Groups of polynomial growth and expanding maps, IHES Publ. Math.,53 (1981), 183-215.
[G2] M. Gromov, Structure métrique pour les variétés riemanniennes, Cedic/Fernand Nathan, Paris, 1981.
[G3] M. Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv., 56 (1981), 179-195.
[GP1] K. Grove and P. Petersen V, Manifolds near the boundary of existence, J. Differ. Geom., 33 (1991), 379-394.
[GP2] K. Grove and P. Petersen V, A radius sphere theorem, Invent. Math., 112 (1993), 577-583.
[GP3] K. Grove and P. Petersen V, On the excess of metric spaces and manifold, preprint.
[GPW] K. Grove, P. Petersen V and J. -Y. Wu, Geometric finiteness theorems via controlled topology, Invent. Math., 99 (1990), 205-213.
[Pe] G. Perelman, Alexandrov's spaces with curvatures bounded from below, II, preprint.
[Pt] C. Plaut, Spaces of Wald-Berestovskii curvature bounded below, J. Geom. Analysis, 6 (1996), 113-134.
[Q1] F. Quinn, Ends of maps, I, Ann. Math., 110 (1979), 275-331.
[Q2] F. Quinn, Ends of maps, III, J. Differ. Geom., 17 (1982), 503-521.
[Q3] F. Quinn, Resolutions of homology manifolds, and the topological characterizations of manifolds, Invent. Math., 72 (1983), 267-284.
[Q4] F. Quinn, An obstruction to the resolution of homology manifolds, Michigan Math. J., 34 (1987), 285-291.
[S] L. C. Siebenmann, Deformation of homeomorphisms on stratified sets, Comment. math. Helv., 47 (1972), 123-163.
[Sh] Z. Shen, A regularity theorem for Alexandrov spaces, Math. Nachr., to appear..
[SW] Z. Shen and G. Wei, On Riemannian manifolds with almost nonnegative curvature, Indiana Univ. Math. J., 40 (1991), 551-563.
[W1] J. -Y. Wu, Hausdorff convergence and sphere theorems, Proc. Symp. Pure Math., ed. by R. Greene and S. -T. Yau, 54 part 3 (1993), 685-692.
[W2] J. -Y. Wu, On the structure of almost nonnegatively curved manifolds, J. Differ. Geom., 35 (1992), 385-397.
[W3] J. -Y. Wu, Convergence of Riemannian 3-manifolds under Ricci curvature bound, Amer. J. Math., Vol. 116 (1994), 1019-1029.
[Wh] G. W. Whitehead, Elements of homotopy theory, GTM 61.
[Y] T. Yamaguchi, Collaping and pinching under a lower curvature bound, Ann. Math., 133 (1991), 317-357.

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