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DOI: https://doi.org/10.4213/rm334

Аннотация: В работе рассматривается параллель между локальной топологией и ее глобальным аналогом, асимптотической топологией. В асимптотическом случае вводятся понятия размерности, когомологической размерности, абсолютных экстензоров и доказываются некоторые базисные факты о них. Рассматривается функтор взятия короны Хигсона, устанавливающий связь между макро- и микро-топологией. Обсуждается связь между проблемами общей асимптотической топологии и гипотезой Новикова о высших сигнатурах. Приводятся некоторые новые результаты в направлении гипотезы Новикова и других родственных гипотез.
Библиография: 27 названий.

Поступила в редакцию: 29.12.1999

Англоязычная версия:
Russian Mathematical Surveys, 2000, Volume 55, Issue 6, Pages 1085–1129
DOI: https://doi.org/10.1070/rm2000v055n06ABEH000334

Реферативные базы данных:

Образец цитирования: А. Н. Дранишников, “Асимптотическая топология”, УМН, 55:6(336) (2000), 71–116; Russian Math. Surveys, 55:6 (2000), 1085–1129

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https://doi.org/10.4213/rm334

https://www.mathnet.ru/rus/rm/v55/i6/p71

Эта публикация цитируется в следующих статьяx:

1. Bell G., Dranishnikov A., “On asymptotic dimension of groups”, Algebr. Geom. Topol., 1 (2001), 57–71      
2. Daniele Guido, Tommaso Isola, “An asymptotic dimension for metric spaces, and the 0-th Novikov–Shubin invariant”, Pacific J Math, 204:1 (2002), 43        
3. Malešič J., Repovš D., Semenov P.V., “On controlled extensions of functions”, J. Math. Anal. Appl., 285:1 (2003), 62–73              
4. Bolotov D.V., “Macroscopic dimension of 3-manifolds”, Math. Phys. Anal. Geom., 6:3 (2003), 291–299              
5. Dranishnikov A.N., “On hypersphericity of manifolds with finite asymptotic dimension”, Trans. Amer. Math. Soc., 355:1 (2003), 155–167            
6. А. Н. Дранишников, “Липшицевы когомологии, гипотеза Новикова и экспандеры”, Геометрическая топология и теория множеств, Сборник статей. К 100-летию со дня рождения профессора Людмилы Всеволодовны Келдыш, Труды МИАН, 247, Наука, МАИК «Наука/Интерпериодика», М., 2004, 59–73      ; A. N. Dranishnikov, “Lipschitz Cohomology, Novikov Conjecture, and Expanders”, Proc. Steklov Inst. Math., 247 (2004), 50–63
7. Carlsson G., Goldfarb B., “The integral $K$-theoretic Novikov conjecture for groups with finite asymptotic dimension”, Invent. Math., 157:2 (2004), 405–418            
8. Bell G.C., Dranishnikov A.N., Keesling J.E., “On a formula for the asymptotic dimension of free products”, Fund. Math., 183:1 (2004), 39–45            
9. Dranishnikov A., Zarichnyi M., “Universal spaces for asymptotic dimension”, Topology Appl., 140:2-3 (2004), 203–225            
10. Carlsson G., Goldfarb B., “On homological coherence of discrete groups”, J. Algebra, 276:2 (2004), 502–514            
11. Bell G., Dranishnikov A., “On asymptotic dimension of groups acting on trees”, Geom. Dedicata, 103:1 (2004), 89–101              
12. Sawicki M., “Absolute extensors and absolute neighborhood extensors in asymptotic categories”, Topology Appl., 150:1-3 (2005), 59–78            
13. Protasov I.V., “Coronas of balleans”, Topology Appl., 149:1-3 (2005), 149–160            
14. Bell G.C., “Asymptotic properties of groups acting on complexes”, Proc. Amer. Math. Soc., 133:2 (2005), 387–396            
15. Dranishnikov A.N., “On hypereuclidean manifolds”, Geom. Dedicata, 117:1 (2006), 215–231            
16. Dranishnikov A.N., “Groups with a polynomial dimension growth”, Geom. Dedicata, 119:1 (2006), 1–15            
17. Bell, GC, “A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory”, Transactions of the American Mathematical Society, 358:11 (2006), 4749            
18. Dranishnikov, A, “Asymptotic dimension of discrete groups”, Fundamenta Mathematicae, 189:1 (2006), 27            
19. Kallipoliti, M, “Simply connected homogeneous continua are not separated by arcs”, Topology and Its Applications, 154:17 (2007), 3039            
20. Brodskiy, N, “Dimension zero at all scales”, Topology and Its Applications, 154:14 (2007), 2729              
21. Kunzi, HPA, “Uniform structures in the beginning of the third millenium”, Topology and Its Applications, 154:14 (2007), 2745              
22. Filah, M, “Slowly oscillating functions in semigroup compactifications and convolution algebras”, Journal of Functional Analysis, 250:1 (2007), 144          
23. Dranishnikov, AN, “On asymptotic Assouad-Nagata dimension”, Topology and Its Applications, 154:4 (2007), 934            
24. Filali, M, “Asymptotic oscillations”, Topology and Its Applications, 154:3 (2007), 561            
25. Dranishnikov A.N., Ferry S.C., Weinberger S.A., “An etale approach to the Novikov conjecture”, Comm. Pure Appl. Math., 61:2 (2008), 139–155            
26. Gentimis, T, “Asymptotic dimension of finitely presented groups”, Proceedings of the American Mathematical Society, 136:12 (2008), 4103            
27. Cuchillo-Ibáñez E., Dydak J., Koyama A., Morón M.A., “$C_0$-coarse geometry of complements of Z-sets in the Hilbert cube”, Trans. Amer. Math. Soc., 360:10 (2008), 5229–5246            
28. Bell, G, “Asymptotic dimension”, Topology and Its Applications, 155:12 (2008), 1265              
29. N. Brodskiy, J. Dydak, “Coarse dimensions and partitions of unity”, RACSAM, 102:1 (2008), 1            
30. Ceccherini-Silberstein T.G., Samet-Vaillant A.Y., “Asymptotic invariants of finitely generated algebras. A generalization of Gromov's quasi-isometric viewpoint”, J. Math. Sci. (New York), 156:1 (2009), 56–108          
31. Dranishnikov A., “Cohomological approach to asymptotic dimension”, Geom. Dedicata, 141:1 (2009), 59–86            
32. N. Brodskiy, J. Dydak, J. Higes, A. Mitra, “Assouad-Nagata dimension via Lipschitz extensions”, Isr J Math, 171:1 (2009), 405            
33. Protasov, IV, “Small subsets of groups”, Topology and Its Applications, 156:16 (2009), 2651              
34. Protasov I.V., “Counting Omega-ideals”, Algebra Universalis, 62:4 (2009), 339–343            
35. Paul D Mitchener, “The general notion of descent in coarse geometry”, Algebr Geom Topol, 10:4 (2010), 2419          
36. Chinen N., Hosaka T., “Asymptotic Dimension of Proper CAT(0) Spaces that are Homeomorphic to the Plane”, Canadian Mathematical Bulletin-Bulletin Canadien de Mathematiques, 53:4 (2010), 629–638            
37. Radul T., “On transfinite extension of asymptotic dimension”, Topology and Its Applications, 157:14 (2010), 2292–2296            
38. Protasov I.V., Tsvietkova A., “Decomposition of Cellular Balleans”, Topology Proceedings, 36 (2010), 77–83      
39. Dušan Repovš, Mykhailo Zarichnyi, “Convex hyperspaces of probability measures and extensors in the asymptotic category”, Topology and its Applications, 2011          
40. Salvador Hernandez, Igor V. Protasov, “Balleans of topological groups”, J Math Sci, 2011        
41. I.V. Protasov, “Weak P-points in coronas of G-spaces”, Topology and its Applications, 2011          
42. Cencelj M., Dydak J., Smrekar J., Vavpetic A., “Sublinear Higson Corona and Lipschitz Extensions”, Houston J Math, 37:4 (2011), 1307–1322      
43. Hambleton I., Savin L., “Coarse Geometry and P. A. Smith Theory”, Homology Homotopy Appl, 13:2 (2011), 73–102            
44. Repovs D., Zarichnyi M., “On the Asymptotic Extension Dimension”, Ukrainian Math J, 62:11 (2011), 1766–1774          
45. Papasoglu P., “Splittings and the Asymptotic Topology of the Lamplighter Group”, Trans. Am. Math. Soc., 364:7 (2012), 3861–3873            
46. Jesús.P.. Moreno-Damas, “Dugundji's canonical covers, asymptotic and covering dimension”, Topology and its Applications, 172 (2014), 95            
47. Jacek Kucab, Mykhailo Zarichnyi, “Subpower Higson corona of a metric space”, Algebra Discrete Math., 17:2 (2014), 280–287    
48. Dranishnikov A., Zarichnyi M., “Asymptotic Dimension, Decomposition Complexity, and Haver's Property C”, Topology Appl., 169:SI (2014), 99–107        
49. Takamitsu Yamauchi, “Asymptotic property C of the countable direct sum of the integers”, Topology and its Applications, 184 (2015), 50            
50. Dydak J., “Coarse Amenability and Discreteness”, J. Aust. Math. Soc., 100:1 (2016), 65–77          
51. Dydak J., Virk Z., “Preserving Coarse Properties”, Rev. Mat. Complut., 29:1 (2016), 191–206          
52. Carlsson G., Goldfarb B., “On modules over infinite group rings”, Int. J. Algebr. Comput., 26:3 (2016), 451–466            
53. Dydak J., Mitra A.J., “Large scale absolute extensors”, Topology Appl., 214 (2016), 51–65          
54. Weighill T., “On spaces with connected Higson coronas”, Topology Appl., 209 (2016), 301–315          
55. Kasprowski D., “The asymptotic dimension of quotients by finite groups”, Proc. Amer. Math. Soc., 145:6 (2017), 2383–2389          
56. Yamauchi T., “Hereditarily infinite-dimensional property for asymptotic dimension and graphs with large girth”, Fundam. Math., 236:2 (2017), 187–192          
57. Austin K., Virk Z., “Higson compactification and dimension raising”, Topology Appl., 215 (2017), 45–57          
58. Yamauchi T., “Straight Finite Decomposition Complexity Implies Property a For Coarse Spaces”, Topology Appl., 231 (2017), 329–336          
59. Bell G., Moran D., Nagorko A., “Coarse Property C and Decomposition Complexity”, Topology Appl., 227 (2017), 30–50          
60. Dranishnikov A., Zarichnyi M., “Remarks on Straight Finite Decomposition Complexity”, Topology Appl., 227 (2017), 102–110          
61. Dranishnikov A.N., “Dimension of Compact Metric Spaces”, Indag. Math.-New Ser., 29:1, SI (2018), 429–449          
62. Bell G.C., Nagorko A., “On the Stability of Asymptotic Property C For Products and Some Group Extensions”, Algebr. Geom. Topol., 18:1 (2018), 221–245          
63. Wu Ya., Zhu J., “Classification of Metric Spaces With Infinite Asymptotic Dimension”, Topology Appl., 238 (2018), 90–101          
64. Dydak J., Weighill T., “Monotone-Light Factorizations in Coarse Geometry”, Topology Appl., 239 (2018), 160–180          
65. Beckhardt S., Goldfarb B., “Extension Properties of Asymptotic Property C and Finite Decomposition Complexity”, Topology Appl., 239 (2018), 181–190          
66. Protasov I., “Varieties of Coarse Spaces”, Axioms, 7:2 (2018), 32      
67. Gromov M., “Metric Inequalities With Scalar Curvature”, Geom. Funct. Anal., 28:3 (2018), 645–726          
68. Protasov I., Protasova K., “On Hyperballeans of Bounded Geometry”, Eur. J. Math., 4:4 (2018), 1515–1520        
69. Alvarez Lopez J.A., Candel A., “Generic Coarse Geometry of Leaves Preface”: Lopez, JAA Candel, A, Generic Coarse Geometry of Leaves, Lect. Notes Math., Lecture Notes in Mathematics, 2223, Springer International Publishing Ag, 2018, VII+    
70. Spakula J., Tikuisis A., “Relative Commutant Pictures of Roe Algebras”, Commun. Math. Phys., 365:3 (2019), 1019–1048          
71. Bell G., Glodkowski D., Nagorko A., “Decomposition Theorems For Asymptotic Property C and Property a”, Topology Appl., 256 (2019), 248–261          
72. Banakh T. Protasov I., “Functional Boundedness of Balleans: Coarse Versions of Compactness”, Axioms, 8:1 (2019), 33      
73. Hartmann E., “a Totally Bounded Uniformity on Coarse Metric Spaces”, Topology Appl., 263 (2019), 350–371    
74. Kucab J., Zarichnyi M., “on Convergence in the Subpower Higson Corona of Metric Spaces”, Topology Appl., 272 (2020), 107068    
75. Orzechowski K., “Apd Profiles and Transfinite Asymptotic Dimension”, Topology Appl., 283 (2020), 107394      
76. Dydak J., “Matrix Algebra of Sets and Variants of Decomposition Complexity”, Rev. Mat. Complut., 33:2 (2020), 373–388      
77. Dikranjan D., Zava N., “The Impact of Pontryagin and Bohr Functors on Large-Scale Properties of Locally Compact Abelian Groups”, Topology Appl., 275 (2020), 107025      
78. Mitchener P.D., Norouzizadeh B., Schick T., “Coarse Homotopy Groups”, Math. Nachr., 293:8 (2020), 1515–1533      
79. Yamauchi T., “Connectedness Properties of the Higson Corona of the Half Line”, Topology Appl., 301:SI (2021), 107542    
80. Weighill T., “Lifting Coarse Homotopies”, Adv. Geom., 21:4 (2021), 577–590    
81. Wu Ya., Zhu J., “The Relationship Between Asymptotic Decomposition Properties”, Topology Appl., 292 (2021), 107623    
82. Yamauchi T., “Transfinite Asymptotic Dimension and Apd Profiles”, Topology Appl., 295 (2021), 107675    
83. Wu Ya., Zhu J., “A Metric Space With Transfinite Asymptotic Dimension 2 Omega+1”, Chin. Ann. Math. Ser. B, 42:3 (2021), 357–366    
84. Zhu J., Wu Ya., “Metric Spaces With Asymptotic Property C and Finite Decomposition Complexity”, J. Nonlinear Funct. Anal., 2021, 15    
85. Weighill T., Yamauchi T., Zava N., “Coarse Infinite-Dimensionality of Hyperspaces of Finite Subsets”, Eur. J. Math., 8:1 (2022), 335–355    

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