The Journal of Geometric Analysis

, Volume 6, Issue 3, pp 475–494 | Cite as

CAT(0) 4-manifolds possessing a single tame point are Euclidean

  • Paul Thurston


The concepts of geometric and topological tame point are introduced for a space of nonpositive curvature. These concepts are applied to the characterization problem forCAT(0) 4-manifolds. It is shown that everyCAT(0)M 4 having a single (geometric or topological) tame point is homeomorphic toR 4. Davis and Januszkiewicz have recently constructedCAT(0)n-manifolds,M n withn ≥ 5 such that the set of tame points form a dense open subset ofM n , butM n R n .

Math Subject Classification

51H20 51K10 57N13 57N10 

Key Words and Phrases

Curvature bounded from above convex metric tame point cell-like map homology manifold 1-LCC embedding 


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  1. [1]
    Aleksandrov, A. D. Über eine verallgemeinerung der riemannschen geometrie.Schrifttenreihe der Institut fur Mathematik 1, 33–84 (1957).Google Scholar
  2. [2]
    Aleksandrov, A. D. Berestovsky, V. N., and Nikolaev, I. G. Generalized riemannian spaces.Russian Math. Surveys 41(3), 1–54 (1986).CrossRefMATHGoogle Scholar
  3. [3]
    Alexander, Stephanie, and Bishop, Richard L. The Hadamard-Cartan theorem in locally convex metric spaces.L’Enseignement Mathématique 36, 309–320 (1990).MathSciNetMATHGoogle Scholar
  4. [4]
    Ancel, F. D., and Guilbault, C. R., Interiors of compact contractiblen-manifolds are hyperbolic (n ≥ 5), preprint (1993).Google Scholar
  5. [5]
    Andrews, J. J., and Curtis, M. L.n-space modulo an arc.Ann. of Math. 75, 1–7 (1962).CrossRefMathSciNetGoogle Scholar
  6. [6]
    Armentrout, Cellular decompositions of 3-manifolds that yield 3-manifolds.Mem. Am. Math. Soc. 107 (1971).Google Scholar
  7. [7]
    Ballman, W., Haefliger, A., de la Harpe, P., Salem, E., Strebel, R., and Troyanov, M. Sur les Groupes hyperboliques d’apres Mikhael Gromov.Progress in Math. 83, Birkhauser, Stuttgart, 1990.Google Scholar
  8. [8]
    Berestovsky, V. N. Introduction of a Riemannian structure in certain metric spaces.Siberian J. Math. 16, 499, 507 (1975).CrossRefGoogle Scholar
  9. [9]
    Berestovsky, V. N. Borsuk’s problem on the metrization of a polyhedron.Soviet Math. Dokl. 27, 56–59 (1983).Google Scholar
  10. [10]
    Bing, R. H. An alternative proof that 3-manifolds can be triangulated.Ann. of Math. 69, 37–65 (1959).CrossRefMathSciNetGoogle Scholar
  11. [11]
    Bing, R. H. A convex metric with unique segments.Proc. Am. Math. Soc. 4, 167–174 (1953).CrossRefMathSciNetMATHGoogle Scholar
  12. [12]
    Blumenthal, L. M.Theory and Applications of Distance Geometry. Oxford University Press, Oxford and New York, 1953.MATHGoogle Scholar
  13. [13]
    Bowers, Philip L. On convex metric spaces VII: The four point properties and weak nonpositive curvature, preprint (1993).Google Scholar
  14. [14]
    Brown, Morton. The monotone union of open cells is an open cell.Proc. Am. Math. Soc. 12, 812–814 (1961).CrossRefMATHGoogle Scholar
  15. [15]
    Burago, Y., Gromov, M., and Perlman, G. Aleksandrov’s spaces with curvature bounded from below, I.Russian Math. Surveys 47(2), 1–58 (1992).CrossRefMathSciNetMATHGoogle Scholar
  16. [16]
    Busemann, Herbert. Metric methods in Finsler spaces and in the foundations of geometry.Ann. Math. Stud. No. 8, Princeton University Press, Princeton, 1942.Google Scholar
  17. [17]
    Busemann, Herbert. On spaces in which two points determine a geodesic.Trans. Am. Math. Soc. 54, 171–184 (1943).CrossRefMathSciNetMATHGoogle Scholar
  18. [18]
    Busemann, Herbert.The Geometry of Geodesics. Academic Press, New York, 1955.MATHGoogle Scholar
  19. [19]
    do Carmo, M.Riemannian Geometry. Birkhäuser, Boston and Berlin, 1992.Google Scholar
  20. [20]
    Cheeger, J., and Ebin, D.Comparison Theorems in Riemannian Geometry. North-Holland, Amsterdam, 1975.MATHGoogle Scholar
  21. [21]
    Daverman, R. J. Singular regular neighborhoods and local flatness in codimension one.Proc. Am. Math. Soc. 57, 357–362 (1976).CrossRefMathSciNetMATHGoogle Scholar
  22. [22]
    Daverman, R. J.Decompositions of Manifolds. Academic Press, New York, 1986.MATHGoogle Scholar
  23. [23]
    Daverman, R. J. Decompositions of manifolds into codimension one submanifolds.Compositio Math. 55, 185–207 (1985).MathSciNetMATHGoogle Scholar
  24. [24]
    Daverman, R. J., and Preston, D. K. Shrinking certain sliced decompositions of En+1.Proc. Am. Math. Soc. 79, 477–483 (1980).CrossRefMathSciNetMATHGoogle Scholar
  25. [25]
    Daverman, R. J., and Tinsley, F. Acyclic maps whose mapping cylinders embed in 5-manifolds.Houston J. Math. 16, 255–270 (1990).MathSciNetMATHGoogle Scholar
  26. [26]
    Daverman, R. J., and Walsh, J. J., Acylic decompostions of manifolds.Pacific J. Math. 109, 291–303 (1983).MathSciNetMATHGoogle Scholar
  27. [27]
    Davis, Michael W., and Januszkiewicz, Tadeusz. Hyperbolization of polyhedra.J. Diff. Geo. 34, 347–388 (1991).MathSciNetMATHGoogle Scholar
  28. [28]
    Freedman, M. H. The topology of 4-dimensional manifolds.J. Diff. Geo. 17, 357–454 (1982).MATHGoogle Scholar
  29. [29]
    Freedman, M. H., and Quinn, Frank.The Topology of 4-Manifolds. Princeton University Press, Princeton, 1990.Google Scholar
  30. [30]
    Gromov, M. Hyperbolic manifolds, groups and actions. InRiemann Surfaces and Related Topics (I. Kra and B. Maskit, eds.),Ann. of Math. Studies, No. 97, pp. 183–213, Princeton University Press, Princeton, 1981.Google Scholar
  31. [31]
    Grove, K., Petersen, P., and Wu, H. Geometric finiteness theorems via controlled topology.Invent. Math. 99, 205–213 (1990); Correction.Invent. Math 104, 221–222 (1991).CrossRefMathSciNetMATHGoogle Scholar
  32. [32]
    Hadamard, Jacques. Les surfaces à courbures opposées et leur lignes géodésiques.J. Math. Pure. Appl. 4, 27–73 (1898).Google Scholar
  33. [33]
    Hopf, H., and Rinow, W. Über den Begriff der vollständigen differentialgeometrischen Flächen.Comm. Math. Helv. 3, 209–225 (1931).CrossRefMathSciNetMATHGoogle Scholar
  34. [34]
    Hurewicz, Witold, and Wallman, Henry.Dimension Theory. Princeton University Press, Princeton, 1941.Google Scholar
  35. [35]
    Kirby, R. C. On the set of non-locally flat points of a submanifold of codimension one.Ann. of Math. 88, 281–290 (1968).CrossRefMathSciNetGoogle Scholar
  36. [36]
    Lacher, R. C. Cellularity criteria for maps.Mich. Math. J. 17, 385–396 (1970).CrossRefMathSciNetMATHGoogle Scholar
  37. [37]
    Lacher, C., and Wright, A. Mapping cylinders and 4-manifolds. InTopology of Manifolds (J. C. Cantrell and C. H. Edwards, eds.).Proceedings of the University of Georgia Topology of Manifolds Institute, pp. 424–427, Markham, Chicago, 1970.Google Scholar
  38. [38]
    McMillan, D. R., Jr. A criterion for cellularity in a manifold.Ann. of Math. 79, 327–337 (1964).CrossRefMathSciNetGoogle Scholar
  39. [39]
    McMillan, D. R., Jr. Compact, acyclic subsets of 3-manifolds.Mich. Math. J. 16, 129–136 (1969).CrossRefMathSciNetMATHGoogle Scholar
  40. [40]
    McMillan, D. R., Jr. Acyclicity in 3-manifolds.Bull. Am. Math. Soc. 76, 942–964 (1971).CrossRefMathSciNetGoogle Scholar
  41. [41]
    Mitchell, W. J. R. Defining the boundary of a homology manifold.Proc. Am. Math. Soc. 110, 509–513 (1990).CrossRefMATHGoogle Scholar
  42. [42]
    Moise, E. E. Affine structures in 3-manifolds, V. The triangulation theorem and Hauptervermutung.Ann. of Math. 56, 96–114 (1952).CrossRefMathSciNetGoogle Scholar
  43. [43]
    Moore, R. L. Concerning upper semicontinous collections of continua.Trans. Am. Math. Soc. 27, 416–428 (1925).CrossRefMATHGoogle Scholar
  44. [44]
    Munkres, James R.Elements of Algebraic Topology. Addison-Wesley, New York, 1984.MATHGoogle Scholar
  45. [45]
    Nikolaev, I. G. Smoothness of the metric spaces with bilaterally bounded curvature in the sense of A. D. Aleksandrov.Siberian J. Math. 24, 247–263 (1983).CrossRefMATHGoogle Scholar
  46. [46]
    Plaut, Conrad. Almost Riemannian spaces.J. Diff. Geo. 34, 515–537 (1991).MathSciNetMATHGoogle Scholar
  47. [47]
    Plaut, Conrad. A metric characterization of manifolds with boundary.Compositio Math. 81, 337–354 (1992).MathSciNetMATHGoogle Scholar
  48. [48]
    Raymond, Frank. Separation and union theorems for generalized manifolds with boundary.Mich. Math. J. 7, 7–21 (1960).CrossRefMathSciNetMATHGoogle Scholar
  49. [49]
    Rolfsen, Dale. Strongly convex metrics in cells.Bull. Am. Math. Soc. 78, 171–175 (1968).CrossRefMathSciNetGoogle Scholar
  50. [50]
    Rinow, W.Die Innere Geometrie der Metrischen Räume. Springer-Verlag, Berlin-Heidelberg-New York, 1961.MATHGoogle Scholar
  51. [51]
    Roberts, J. H., and Steenrod, N. E. Monotone transformations of 2-dimensional manifolds.Ann. of Math. 39, 851–862 (1938).CrossRefMathSciNetGoogle Scholar
  52. [52]
    Row, H. Compact subsets of three-manifolds definable by cubes-with-handles. Ph.D. dissertation, University of Wisconsin, 1969.Google Scholar
  53. [53]
    Rushing, T. B.Topological Embeddings. Academic Press, New York, 1973.MATHGoogle Scholar
  54. [54]
    Spanier, Edwin H.Algebraic Topology. Springer-Verlag, Berlin-Heidelberg-New York, 1966.MATHGoogle Scholar
  55. [55]
    Thurston, P. The topology of 4-dimensional G-spaces and a study of 4-manifolds of non-positive curvature. Ph.D. thesis, Knoxville, Tennessee, 1993.Google Scholar
  56. [56]
    Tits, Jacques. Etude de certains espaces métriques.Bull. Soc. Math. Belgique 5, 44–52 (1952).MathSciNetGoogle Scholar
  57. [57]
    Wilder, Ray. Topology of manifolds.AMS Colloq. Publ. 32, Am. Math. Soc., Providence, RI, 1963.Google Scholar
  58. [58]
    White, Paul. Some characterizations of generalized manifolds with boundaries.Can. J. Math. 4, 329–342 (1952).MATHGoogle Scholar
  59. [59]
    Wright, A. Monotone mappings of compact 3-manifolds. Ph.D. dissertation, University of Wisconsin, 1969.Google Scholar

Copyright information

© Mathematica Josephina, Inc. 1996

Authors and Affiliations

  • Paul Thurston
    • 1
  1. 1.Department of MathematicsCornell UniversityIthaca

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