Skip to main content
SpringerLink
Log in

Poincaré conjecture and related statements

  • Published:
Russian Mathematics Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. H. Poincaré, Selected Works. Fifth Supplement to “Analysis Situs.” New Methods of Celestial Mechanics. Topology. Number Theory (Nauka, Moscow, 1972), Vol. II, pp. 674–734 [Russian translation].

    Google Scholar 

  2. J. Hempel, 3-Manifolds (Princeton Univ. Press, 1976), Ann. of Math. Studies 86.

  3. K. Johanson, Homotopy Equivalences of 3-Manifolds with Boundaries (Springer-Verlag, Berlin and New-York, 1978) Lect. Notes in Math. 761.

    Google Scholar 

  4. W. J. Jaco and P. B. Shalen, “Seifert Fibered Spaces in 3-Manifolds,” Mem. Amer. Math. Soc. 21(220), 1–192 (1979).

    MathSciNet  Google Scholar 

  5. W. P. Thurston, “Three-Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry,” Bull. Amer. Math. Soc. (N. S.) 6(3), 357–381 (1982).

    MATH  MathSciNet  Google Scholar 

  6. P. Skott, Geometries on Three-Dimensional Manifolds (Mir, Moscow, 1986) [Russian translation].

    Google Scholar 

  7. R. S. Hamilton, “Three-Manifolds with Positive Ricci Curvature,” J. Differential Geometry 17, 255–306 (1982).

    MATH  MathSciNet  Google Scholar 

  8. G. Perelman, “The Entropy Formula for the Ricci Flow and its Geometric Applications,” arXiv.org/abs/math.DG/0211159.

  9. G. Perelman, “Ricci Flow with Surgery on Three-Manifolds,” arXiv.org/abs/math.DG/0303109.

  10. G. Perelman, “Finite Extinction Time for the Solutions to the Rici Flow on Certain Three-Manifolds,” arXiv.org/abs/math.DG/0307245.

  11. J. Milnor, “Towards the Poincaré Conjecture and the Classification of 3-Manifolds,” Notices Amer. Math. Soc. 50(10), 1226–1233 (2003).

    MATH  MathSciNet  Google Scholar 

  12. T. Shioya and T. Yamaguchi, “Volume Collapsed Three-Manifolds with a Lower Curvature Bound,” Math. Ann. 333, 131–155 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  13. J. W. Morgan, “Recent Progress on the Poincaré Conjecture and the Classification of 3-Manifolds,” Bull. Amer. Math. Soc. 42(1), 57–78 (2004).

    Article  Google Scholar 

  14. M. Anderson, “Geometrization of 3-Manifolds via the Ricci Flow,” Notices Amer. Math. Soc. 2004(2), 184–193 (2004).

    Google Scholar 

  15. T. H. Colding and W. P. Minicozzi II, “Estimates for the Extinction Time for the Ricci Flow of Certain 3-Manifolds and a Question of Perelman,” Journ. Amer. Math. Soc. 18(3), 561–569 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  16. B. Kleiner and J. Lott, “Notes on Perelman's Papers,” arXiv.org/abs/math.DG/0605667.

  17. G. Besson, Preuve de le Conjecture de Poincaré en Deformant le Métrique par la Corbure de Ricci, d'Après G. Perel'man (Astérisque, Société Mathématique de France, 2006), Vol. 307.

  18. J. W. Morgan and G. Tian, “Ricci Flow and the Poincaré Conjecture,” arXiv.org/abs/math.DG/0607607.

  19. H.-D. Cao and X.-P. Zhu, “A Complete Proof of the Poincaré and Geometrization Conjectures. — Application of the Hamilton-Perelman Theory of the Ricci Flow,” Asian J. Math. 10(2), 145–492, 2006.

    MathSciNet  Google Scholar 

  20. R. H. Bing, “Necessary and Sufficient Conditions that a 3-Manifold be S 3,” Ann. of Math. 68, 37–65 (1958).

    Article  MathSciNet  Google Scholar 

  21. M. H. A. Newman, “The Engulfing Theorem for Topological Manifolds,” Ann. of Math. 84, 555–571 (1966).

    Article  MathSciNet  Google Scholar 

  22. M. H. Freedman, “The Topology of Four-Dimensional Manifolds,” J. Dif. Geom. 17, 357–453 (1982).

    MATH  Google Scholar 

  23. T. Rado, “Über den Begriff der Riemannschen Fläche,” Acta Litt. Scient. Univ. Szeged 2, 101–121 (1925).

    Google Scholar 

  24. B. V. Kerékjártó, Vorlesungen über Topologie. I. Flächentopologie (Springer, 1923).

  25. E. E. Moise, “Affine Structures on 3-Manifolds. V. The Tringulation Theorem and Hauptvermutung,” Ann. of Math. 56, 96–114 (1952).

    Article  MathSciNet  Google Scholar 

  26. M. W. Davis and T. Januszkiewicz, “Hyperbolization of Polyhedra,” J. Differential Geom. 34(2), 347–388 (1991).

    MATH  MathSciNet  Google Scholar 

  27. R. H. Bing, “Some Aspects of the Topology of 3-Manifolds Related to the Poincaré Conjecture,” in T. L. Saaty (Ed.) Lectures on modern mathematics (John Wiley and Sons, New York, 1964) II, pp. 93–128.

    Google Scholar 

  28. V. N. Berestovskii, “Pathologies in Alexandrov Spaces with Curvature Bounded Above,” Siberian Adv. Math. 12(4), 1–18 (2003).

    Google Scholar 

  29. M. Bestvina, R. J. Daverman, G. A. Venema, and J. J. Walsh, “A 4-Dimensional 1-LCC Shrinking Theorem,” in Geometric Topology and Geometric Group Theory (Milwaukee, Wi, 1997), Top. Appl. 110 (1), 3–20 (2001.

    Google Scholar 

  30. S. S. Cairns, “Homeomophisms Between Topological Manifolds and Analytic Riemannian Manifolds,” Ann. of Math. (2) 41, 796–808 (1940).

    Article  MathSciNet  Google Scholar 

  31. J. H. C. Whitehead, “Manifolds with Transverse Fields in Euclidean Space,” Ann. of Math. 73, 154–212 (1961).

    Article  MathSciNet  Google Scholar 

  32. J. Milnor and J. Stasheff, Characteristic Classes (Mir, Moscow, 1979) [Russian translation].

    MATH  Google Scholar 

  33. L. E. J. Brower, “Zurn Triangulationsproblem,” Nederl. Akad. Wetensch. Proc. 42, 701–706 (1939).

    MathSciNet  Google Scholar 

  34. J. Cerf, Sur les Difféomorphìsmes de la Sphére de Dimension Trois (Γ4 = 0) (Springer-Verlag, Berlin, 1968) 53.

    MATH  Google Scholar 

  35. S. S. Cairns, “Introduction of a Riemannian Geometry on a Triangulable 4-Manifolds,” Ann. of Math. (2) 45(2), 218–219 (1944).

    Article  MathSciNet  Google Scholar 

  36. M. Hirsh and B. Masur, “Smoothing Piecewise Linear Manifolds,” Annals of Math. Studies No. 80 (Princeton University Press, Princeton, 1974).

    Google Scholar 

  37. S. De Michelis and M. H. A. Freedman, “Uncountably Many Exotic ℝ4's in Standard 4-Space,” J. Differential Geom. 35(1), 219–255 (1992).

    MathSciNet  Google Scholar 

  38. S. Donaldson, “Irrationality and h-Cobordism Conjecture,” J. Differential Geom. 2(1), 141–168 (1987).

    MathSciNet  Google Scholar 

  39. A. Dold, Lectures on Algebraic Tpology (Mir, Moscow, 1976) [Russian translation].

    Google Scholar 

  40. D. Fried and K. Uhlenbeck, Instantons and Four-Dimensional Manifolds (Mir, Moscow, 1988) [Russian translation].

    Google Scholar 

  41. J. Milnor and D. Husemoller, Symmetric Bilinear Forms (Nauka, Moscow, 1986) [Russian translation].

    Google Scholar 

  42. J. H. C. Whitehead, “On Simply Connected 4-Dimensional Polyhedra,” Comment. Math. Helv. 22, 48–92 (1949).

    Article  MATH  MathSciNet  Google Scholar 

  43. V. A. Rohlin, “New Results in the Theory of Four-Dimensional Manifolds,” Dokl. Akad. Nauk 84, 221–224 (1952).

    MathSciNet  Google Scholar 

  44. L. Guillou and A. Marin (Eds.) A la Recherche de la Topologie Perdue (Mir, Moscow, 1989) [Russian translation].

    Google Scholar 

  45. Four-Dimensional Riemannian Geometry. Athur Besse's Seminar. 1978/79 (Mir, Moscow, 1985 [Russian translation].

  46. R. Mandelbaum, Four-Dimensional Topology (Mir, Moscow, 1981) [Russian translation].

    MATH  Google Scholar 

  47. M. H. Freedman and F. Quinn, Topology of 4-Manifolds (Princeton University Press, Princeton, New Jersey, 1990).

    MATH  Google Scholar 

  48. S.K. Donaldson S. K. “An Application of Gauge Theory to the Topology of 4-Manifolds,” J. Diff. Geom. 18, 269–316 (1983).

    MATH  MathSciNet  Google Scholar 

  49. M. H. Freedman and F. Luo, “Selected Application of Geometry to Low-Dimensional Topology,” in Marker Lectures in the Mathematical Sciences (The Pennsylvania State University, University Lecture Series I. Amer. Math. Soc. Providence, Rhode Island, 1989).

    Google Scholar 

  50. M. Furuta, “Monopole Equation of the 11/8-Conjecture,” Math. Res. Lett. 8, 279–291 (2001).

    MATH  MathSciNet  Google Scholar 

  51. R. C. Kirby and L. C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations (Princeton University Press and University of Tokyo Press, Princeton, New Jersey, 1977).

    MATH  Google Scholar 

  52. C. T. C. Wall (Ed. A. A. Ranicki A.A.) Surgery on Compact Manifolds. 2nd Ed. Mathemtical Surveys and Monographs (Amer. Math. Soc. Providence, Rhode Island, 1999) 69.

    Google Scholar 

  53. R. Fintushel and R. J. Stern, “Knots, Links, and 4-Manifolds,” Invent. Math. 134(2), 363–400 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  54. W. Massey and J. Stallings, Algebraic Topology: An Introduction (Mir, Moscow, 1972) [Russian translation].

    Google Scholar 

  55. M. Hirsch, Differential Topology (Mir, Moscow, 1979) [Russian translation].

    MATH  Google Scholar 

  56. V. V. Prasolov and A. B. Sosinskii, Knots, Links, Braids and Three-Dimensional Manifolds (MTsNMO, 1997) [in Russian].

  57. R. Baer, “Nothersche Gruppen,” Math. Z. 66, 269–288 (1956).

    Article  MATH  MathSciNet  Google Scholar 

  58. J. Stallings, “How Not to Prove the Poincaré Conjecture,” Ann. of Math. Studies (Princeton Univ. Press, 1966) 60, pp. 83–88.

    MATH  Google Scholar 

  59. W. Jaco, “Heegaard Splittings and Splitting Homomorphisms,” Trans. Amer. Math. Soc. 144, 365–379 (1969).

    Article  MATH  MathSciNet  Google Scholar 

  60. F. Waldhausen, “Heegaard-Zerlegungen der 3-Sphere,” Topology 7, 195–203 (1968).

    Article  MATH  MathSciNet  Google Scholar 

  61. C. D. Papakyriakopolos, “On Dehn's Lemma and Asphericity of Knots,” Ann. of Math. 66, 1–26 (1957).

    Article  MathSciNet  Google Scholar 

  62. D. B. A. Epstein, “Curves on 2-Manifolds and Isotopies,” Acta Math. 115, 83–107 (1966).

    Article  MATH  MathSciNet  Google Scholar 

  63. A. Casson and S. Bleiler, Automorphisms of Surfaces after Nielsen and Thurston (Fazis, Moscow, 1998) [Russian translation].

    Google Scholar 

  64. J. Stallings, “A Topological Proof of Grushko's Theorem on Free Products,” Math. Zeit. 90, 1–8 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  65. J. W. Milnor, “A Unique Factorization Theorem for 3-Manifolds,” Amer. J. Math. 74, 1–7 (1962).

    Article  MathSciNet  Google Scholar 

  66. J. W. Alexander, “The Combinatorial Theory of Complexes,” Ann. of Math. 31, 292–320 (1930).

    Article  MathSciNet  Google Scholar 

  67. J. W. Alexander, “An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected,” Proc. Nat. Acad. Sci. USA 10, 8–10 (1924).

    Article  Google Scholar 

  68. R. H. Bing, “A Homeomorphism Between the 3-Sphere and the Sum of Two Solid Horned Spheres,” Ann. of Math. 56(2), 354–362 (1952).

    Article  MathSciNet  Google Scholar 

  69. H. Kneser, “Geschlossene Flächen in Dreidimensionale Mannigfaltigkeiten,” Jahresber. Deutsch. Math.-Verein 38, 248–260 (1929).

    MATH  Google Scholar 

  70. J. H. C. Whitehead, “On 2-Spheres in 3-Manifolds,” Bull. Amer. Math. Soc. 64, 161–166 (1958).

    MATH  MathSciNet  Google Scholar 

  71. J. H. C. Whitehead, “A Certain Open Manifold whose Group is Unity,” Quart. J. Math. (2), 6, 268–279 (1935).

    Article  Google Scholar 

  72. J. H. C. Whitehead (Ed. I. M. James) The Mathemtical Works of J. H. C. Whitehead. Complexes and Manifolds (Pergamon Press, Oxford-New York-Paris, 1962), Vol. II.

    Google Scholar 

  73. D. R. McMillan (Jr.), “Some Contractible Open 3-Manifolds,” Trans. Amer. Math. Soc. — 102, 373–382 (1962).

    Article  MathSciNet  Google Scholar 

  74. D. R. McMillan (Jr.), Cartesian Products of Contractible Open Manifolds,” Bull. Amer. Math. Soc. 67, 510–514 (1961).

    MATH  MathSciNet  Google Scholar 

  75. D. R. McMillan (Jr.) and E. C. Zeeman, “On Contractible Open Manifolds,” Proc. Camb. Phil. Soc. 58, 221–224 (1962).

    MATH  MathSciNet  Google Scholar 

  76. J. M. Kister and D. R. McMillan (Jr.) “Locally Euclidean Factors of E 4 which Cannot be Embedded in E 3,” Ann. of Math. 76, 541–546 (1962).

    Article  MathSciNet  Google Scholar 

  77. J. Stallings, “On the Loop Theorem,” Ann. of Math. 72, 12–19 (1960).

    Article  MathSciNet  Google Scholar 

  78. D. B. A. Epstein, “Projective Planes in 3-Manifolds,” Proc. London Math. Soc. (3) 11, 469–484 (1961).

    Article  MATH  MathSciNet  Google Scholar 

  79. C. B. Thomas, “Free Actions by Finite Groups on S 3,” in Proceedings of Sympos. Pure Math., Algebraic and Geometric Topology, Stanford Univ., Stanford, Calif., 1976, Proc. Sympos. Pure. Math., XXXII (Amer. Math. Soc., Providence, R. I., 1978), Part 1, pp. 125–130.

    Google Scholar 

  80. A. Hatcher, “A Proof of the Smale Conjecture, Diff (S 3) ≅ O(4),” Ann. of Math. 117, 553–607 (1983).

    Article  MathSciNet  Google Scholar 

  81. H. Hopf, Zum Clifford-Kleinschen Raumproblem,” Math. Ann. 95, 313–319 (1925–1926).

    Article  MathSciNet  MATH  Google Scholar 

  82. H. Seifert and W. Threlfall, “Topologische Untersuchung der Diskontinuitätsbereïche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes,” Math. Ann. 104, 1–70 (1930–1931).

    MathSciNet  MATH  Google Scholar 

  83. J. Wolf, Spaces of Constant Curvature (Nauka, Moscow, 1982) [Russian translation].

    MATH  Google Scholar 

  84. H. Poincaré, Selected Works. Second Supplement to “Analysis Situś.” New Methods of Celestial Mechanics. Topology. Number Theory (Nauka, Moscow, 1972), Vol. II, pp. 594–622 [Russian translation].

    Google Scholar 

  85. B. Evans and L. Moser, “Solvable Fundamental Groups of Compact 3-Manifolds,” Trans. Amer. Math. Soc. 168, 189–210.

  86. J. W. Alexander, “Note on Two Three-Dimensional Manifolds with the Same Group,” Trans. Amer. Math. Soc. 20, 339–342 (1919).

    Article  MathSciNet  MATH  Google Scholar 

  87. J. Hass, J. H. Rubinstein, and P. Scott, “Covering Spaces of 3-Manifolds,” J. Differential Geom. 30, 817–832 (1989).

    MATH  MathSciNet  Google Scholar 

  88. F. Waldhausen, “On Irreducible 3-Manifolds which are Sufficiently Large,” Ann. of Math. (2) 87, 56–88 (1968).

    Article  MathSciNet  Google Scholar 

  89. W. Jaco, Lectures on 3-Manifold Topology, (American Mathematical Society, Providence, R. I., 1980), CBMS Regional Conference Series in Mathematics 43.

    Google Scholar 

  90. F. Waldhausen, “The Word Problem in Fundamental Groups of Sufficiently Large Irreducible 3-Manifolds,” Ann. of Math. (2) 88, 272–280 (1968).

    Article  MathSciNet  Google Scholar 

  91. N. M. Dunfield and W. P. Thurston, “The Virtual Haken Conjecture: Experiments and Examples,” Geometry and Topology 7, 399–441 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  92. S. Mateev, Algorithmic Topology and Classification of 3-Manifolds (Springer-Verlag: Berlin, Heidelberg, New York, 2003), Algorithms and Computation in Mathematics 9.

    Google Scholar 

  93. W. Haken, “Über das Homöomorphieproblem der 3-Mannigfaltigkeiten. I,” Math. Z. 80, 89–120 (1962).

    Article  MATH  MathSciNet  Google Scholar 

  94. G. Hemion, “On the Classification of Knots and 3-Dimensional Spaces,” Acta Math. 142(1–2), 123–155 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  95. F. Waldhausen, “Recent Results on Sufficiently Large 3-Manifolds.” in Proceedings of Sympos. Pure Math., Algebraic and geometric topology, Stanford Univ., Stanford, Calif., 1976, Proc. Sympos. pure Math., XXXII (Amer. Math. Soc., Providence, R. I., 1978), Part 2, pp. 21–38.-

    Google Scholar 

  96. K. Johanson, “Classification Problems in Low-Dimensional Topology,” in Geometric and Algebraic Topology (Banach Center Publ., Warsaw, 1986), Vol. 18, pp. 37–59.

    Google Scholar 

  97. G. Hemion, The Classification of Knots and 3-Dimensional Spaces (Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992).

    MATH  Google Scholar 

  98. M. Bestvina and M. Handel, “Train-Tracks for Surface Homeomorphisms,” Topology 34(1), 109–140 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  99. H. Seifert, “Topologie dreidimensionaler gefaserter Räume,” Acta Math. 60, 147–238 (1933).

    Article  MATH  MathSciNet  Google Scholar 

  100. H. Seifert and W. Threlfall, A Textbook of Topology (Academic Press, 1980), Pure and Applied Mathematics 89.

  101. P. Orlik, Seifert Manifolds (Springer, Berlin, 1972) Lect. Notes in Math. 291.

    MATH  Google Scholar 

  102. A. D. Alexandrov, “A Theorem on Triangles in Metric Space and its Applications,” Tr. Mat. Inst. im. V. A. Steklova, Ross. Akad. Nauk 38, 5–23 (1951).

    Google Scholar 

  103. A. D. Alexandrov, “Über eine Verallgemeinerung der Riemannschen Geometrie,” Schriftenr. Inst. Math. Deutsch. Acad. Wiss., Hf. 1, 33–84 (1957).

    Google Scholar 

  104. A. D. Alexandrov, V. N. Berestovskii, and I. G. Nikolaev, “Generalized Riemannian Spaces,” Usp. Mat. Nauk 41,(3), 3–44 (1986).

    Google Scholar 

  105. M. R. Bridson and A. Haelliger, Metric Spaces of Non-Positive Curvature (Springer-Verlag, 1999).

  106. V. N. Berestovskii, “On Spaces with Bounded Curvature,” Dokl. Akad. Nauk 258(2), 269–271 (1981).

    MathSciNet  Google Scholar 

  107. V. N. Berestovskii, “Spaces with Bounded Curvature and Distance Geometry,” Sib. Matem. Zhurn. 27(1), 11–25 (1986).

    MathSciNet  Google Scholar 

  108. F. Bruhat and J. Tits, “Groupes Réductifs sur un Corps Local. I. Données Radicielles Valuées,” Inst. Hautes Études Sci. Publ. Math. 41, 5–251 (1972).

    Article  MATH  MathSciNet  Google Scholar 

  109. S. B. Alexander and R. L. Bishop, “The Hadamard-Cartan Theorem in Locally Convex Spaces,” L'Enseign. Math. 36, 309–320 (1990).

    MathSciNet  MATH  Google Scholar 

  110. Yu. Burago, M. Gromov, and G. Perelman, “A. D. Alexandrov's Spaces with Curvature Bounded Below,” Usp. Mat. Nauk 47(2), 3–51 (1992).

    MathSciNet  Google Scholar 

  111. V. N. Berestovskii, “Borsuk's Problem on Metrization of Polyhedron,” Dokl. Akad. Nauk 268(2), 273–277 (1983).

    MathSciNet  Google Scholar 

  112. V. N. Berestovskii, “Manifolds with Inner Metric of Unilaterally Bounded Alexandrov Curvature,” Matem. Fizika. Analiz. Geometriya 1(1), 41–59 (1994).

    MathSciNet  Google Scholar 

  113. K. Borsuk, Retract Theory (Mir, Moscow, 1971) [Russian translation].

    Google Scholar 

  114. F. D. Ancel and C. R. Giulbault, “Interiors of Compact Contractible n-Manifolds are Hyperbolic (n ≥ 5),” J. Differential Geom. 45, 1–32 (1997).

    MATH  MathSciNet  Google Scholar 

  115. I. G. Nikolaev, “The Tangent Cone of an Aleksandrov Space of Curvature ≤ K,” Manuscripta Math. 86, 683–689 (1995).

    Article  MathSciNet  Google Scholar 

  116. A. D. Alexandrov and V. N. Berestovskii, “Riemannian spaces, Generalized,” in M. Hazewinkel (Managing Ed.) Encyclopaedia of Mathematics (Kluwer Academic Publishers, Dordrecht-Boston-London, 1992), Vol. 8, pp. 150–152.

    Google Scholar 

  117. V. A. Rohlin, “Any Three-Dimensional Manifold is a Boundary of Four-Dimensional Manifold,” Dokl. Akad. Nauk 81, 355–357 (1951).

    MathSciNet  Google Scholar 

  118. T. Matumoto, “Variétés Simpliciales d'Homologie et Variétés Topologiques Métrisables,” These de doctorat d'etat, Université de Paris XI (Centre d'Orsay, Octobre 1976).

  119. D. Galewski and R. Stern, “Classification of Simplicial Triangulations of Topological Manifolds,” Ann. Math. 11, 1–34 (1980).

    Article  MathSciNet  Google Scholar 

  120. J. W. Cannon, “The Recognition Problem: What is a Topological Manifold?,” Bull. Amer. Math. Soc. 84, 832–866 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  121. R. H. Bing, “A Decomposition of E 3 into Points and Tame Arcs Such That the Decomposition Space is Topologically Different from E 3,” Ann. of Math. (2) 65, 484–500 (1957).

    Article  MathSciNet  Google Scholar 

  122. R. D. Edwards, “The Topology of Manifolds and Cell-Like Maps,” in Proceedings of the International Congress of Mathematicians, Helsinki, 1978 (Acad. Sci. Fennica, Helsinki, 1980), pp. 111–127.

    Google Scholar 

  123. R. J. Daverman, Decompositions of Manifolds (Academic Press, Inc., Orlando, Fl., 1986), Pure and Applied Mathematics 124.

    MATH  Google Scholar 

  124. F. Quinn, “An Obstruction to the Resolution of Homology Manifolds,” Michigan Math. J. 34, 285–291 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  125. R. Bryant, S. Ferry, W. Mio, and S. Weinberger, “Topology of Homology Manifolds,” Ann. Math. (2) 143(3), 435–483 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  126. G. Busemann, Geometry of Geodesics (Fizmatgiz, Moscow, 1962) [Russian translation].

    MATH  Google Scholar 

  127. V. N. Berestovskii, “The Busemann Spaces of Alexandrov Curvature Bounded from Above,” Algebra i Analiz 14(5), 3–18 (2002).

    MATH  MathSciNet  Google Scholar 

  128. W. Jakobsche, “The Bing-Borsuk Conjecture is Stronger Than the Poincare Conjecture,” Fund. Math. 106(2), 127–134 (1980).

    MATH  MathSciNet  Google Scholar 

  129. V. N. Berestovskii, “On the Problem of Finite Dimensionality of Busemann G-Space,” Sib. Mat. Zhurn. 18(1), 219–221 (1977).

    MathSciNet  Google Scholar 

  130. B. Krakus, “Any 3-Dimensional G-Space is a Manifold,” Bull. Acad. Pol. Sci. Sér. Math. Astronom. Phys. 16, 285–291 (1968).

    MathSciNet  Google Scholar 

  131. P. Thurston, “4-Dimensional Busemann G-Spaces are 4-Manifolds,” Differential Geom. and Appl. 6(3), 245–270 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  132. R. J. Daverman and T. L. Thickstun, “The 3-Manifolds Recognition Problem,” Trans. Amer. Math. Soc. 358, 5257–5270 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  133. A. Kavichcholi, D. Repovsh, and T. Tikstun, “Geometric Topology of Generalized 3-Manifolds. Fundam. Prikl. Mat. 11(4), 71–84 (2005).

    MathSciNet  Google Scholar 

  134. W. H. Meeks III and S. T. Yau, “Topology of Three Dimensional Manifolds and the Embedding Problems in Minimal Surface Theory,” Ann. Math. Soc. 112, 441–484 (1980).

    Article  MathSciNet  Google Scholar 

  135. W. H. Meeks III and S. T. Yau, “The Classical Plateu Problem and the Topology of Three-Dimensional Manifolds. The Embedding of the Solution Given by Douglas-Morrey and an Analytic Proof of Dehn's Lemma,” Topology 21(4), 409–442 (1982).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Omsk Branch of Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, ul. Pevtsova 13, Omsk, 644099, Russia

    V. N. Berestovskii

Authors
  1. V. N. Berestovskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. N. Berestovskii.

Additional information

Original Russian Text © V.N. Berestovskii, 2007, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2007, No. 9, pp. 3–41.

About this article

Cite this article

Berestovskii, V.N. Poincaré conjecture and related statements. Russ Math. 51, 1–36 (2007). https://doi.org/10.3103/S1066369X07090010

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X07090010

Keywords

Search

Navigation