Matematicheskii Sbornik. Novaya Seriya
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

 Mat. Sb. (N.S.), 1986, Volume 129(171), Number 1, Pages 121–139 (Mi msb1810)

Universal Menger compacta and universal mappings

A. N. Dranishnikov

Abstract: For any positive integer $n$ the author constructs a continuous mapping $f_n\colon M_n\to M_n$ of the $n$-dimensional Menger compactum onto itself that is universal in the class of mappings between $n$-dimensional compacta, i.e., for any continuous mapping $g\colon X\to Y$ between $n$-dimensional compacta there exist imbeddings of $X$ and $Y$ in $M_n$ such that the restriction of $f_n$ to $X$ is homeomorphic to $g$. The mapping $f_n$ plays the same role in the theory of Menger $n$-dimensional manifolds as the projection $\pi\colon Q\times Q\to Q$ plays in the theory of $Q$-manifolds ($Q$ is the Hilbert cube). It can be used to carry over the classical theorems in the theory of $Q$-manifolds to the theory of $M_n$-manifolds:
Stabilization theorem. {\it For any $M_n$-manifold $X$ and any imbedding of $X$ in $M_n$ the space $f_n^{-1}(X)$ is homeomorphic to $X$.}
Triangulation theorem. {\it For any $M_n$-manifold $X$ there exists an $n$-dimensional polyhedron $K$ such that the space $f_n^{-1}(K)$ is homeomorphic to $X$ for every imbedding of $K$ in $M_n$.}
Bibliography: 20 titles.

Full text: PDF file (1238 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1987, 57:1, 131–149

Bibliographic databases:

UDC: 515.12
MSC: Primary 54C25, 54C55, 54E45; Secondary 54F45, 54C20

Citation: A. N. Dranishnikov, “Universal Menger compacta and universal mappings”, Mat. Sb. (N.S.), 129(171):1 (1986), 121–139; Math. USSR-Sb., 57:1 (1987), 131–149

Citation in format AMSBIB
\Bibitem{Dra86} \by A.~N.~Dranishnikov \paper Universal Menger compacta and universal mappings \jour Mat. Sb. (N.S.) \yr 1986 \vol 129(171) \issue 1 \pages 121--139 \mathnet{http://mi.mathnet.ru/msb1810} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=830099} \zmath{https://zbmath.org/?q=an:0622.54026} \transl \jour Math. USSR-Sb. \yr 1987 \vol 57 \issue 1 \pages 131--149 \crossref{https://doi.org/10.1070/SM1987v057n01ABEH003059} 

• http://mi.mathnet.ru/eng/msb1810
• http://mi.mathnet.ru/eng/msb/v171/i1/p121

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. N. Dranishnikov, E. V. Shchepin, “Cell-like maps. The problem of raising dimension”, Russian Math. Surveys, 41:6 (1986), 59–111
2. Dranishnikov A., “On Resolutions of Lcn-Compacta”, Lect. Notes Math., 1283 (1987), 48–59
3. A. N. Dranishnikov, “On free actions of zero-dimensional compact groups”, Math. USSR-Izv., 32:1 (1989), 217–232
4. A. Ch. Chigogidze, “The theory of $n$-shapes”, Russian Math. Surveys, 44:5 (1989), 145–174
5. A. Ch. Chigogidze, “$n$-shapes and $n$-cohomotopy groups of compacta”, Math. USSR-Sb., 66:2 (1990), 329–342
6. Chigogidze A., “On Uvn Equivalent Compacta”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1989, no. 3, 33–35
7. Valov V., “Linear Topological Classifications of Certain Function-Spaces”, Trans. Am. Math. Soc., 327:2 (1991), 583–600
8. L. V. Shirokov, “On $\operatorname{AE}(n)$-bicompacta”, Russian Acad. Sci. Izv. Math., 41:3 (1993), 557–566
9. Mitchell W. Repovs D. Scepin E., “On 1-Cycles and the Finite Dimensionality of Homology 4-Manifolds”, Topology, 31:3 (1992), 605–623
10. Chigogidze A., “Uvn-Equivalence and N-Equivalence”, Topology Appl., 45:3 (1992), 283–291
11. Kawamura K., “A Characterization of Lc(N) Compacta in Terms of Gromov-Hausdorff Convergence”, Can. Math. Bul.-Bul. Can. Math., 37:4 (1994), 505–513
12. Chigogidze A., Valov V., “Set-Valued Maps and Ae(0)-Spaces”, Topology Appl., 55:1 (1994), 1–15
13. Kazuhiro Kawamura, “An inverse system approach to Menger manifolds”, Topology and its Applications, 61:3 (1995), 281
14. Rolando Jimenez, Leonard R Rubin, “An addition theorem for n-fundamental dimension in metric compacta”, Topology and its Applications, 62:3 (1995), 281
15. Zarichnyi M., “Universal Map of SIGMA Onto SIGMA and Absorbing Sets in the Classes of Absolute Borelian and Projective Finite-Dimensional Spaces”, Topology Appl., 67:3 (1995), 221–230
16. Iwamoto Y., “Menger Manifolds Homeomorphic to their N-Homotopy Kernels”, Proc. Amer. Math. Soc., 123:3 (1995), 945–953
17. M. M. Zarichnyi, “Universal $n$-soft maps of $n$-dimensional spaces in absolute Borel and projective classes”, Math. Notes, 60:6 (1996), 638–641
18. A. Chigogidze, K. Kawamura, R.B. Sher, “Finiteness results in n-homotopy theory”, Topology and its Applications, 74:1-3 (1996), 3
19. Chigogidze A., Kawamura K., Tymchatyn E., “Nobeling Spaces and Pseudo-Interiors of Menger Compacta”, Topology Appl., 68:1 (1996), 33–65
20. M. M. Zarichnyi, “Absorbing sets for $n$-dimensional spaces in absolute Borel and projective classes”, Sb. Math., 188:3 (1997), 435–447
21. Chigogidze A., “Cohomological Dimension of Tychonov Spaces”, Topology Appl., 79:3 (1997), 197–228
22. Gutev V., “Continuous Selections for Continuous Set-Valued Mappings and Finite-Dimensional Sets”, Set-Valued Anal., 6:2 (1998), 149–170
23. Iwamoto Y. Sakai K., “Strong N-Shape Theory”, Topology Appl., 122:1-2 (2002), 253–267
24. Chigogidze A. Karasev A., “Topological Model Categories Generated by Finite Complexes”, Mon.heft. Math., 139:2 (2003), 129–150
25. B. A. Pasynkov, “The Subset Theorem in Dimension Theory and Open Maps Raising Dimension”, Proc. Steklov Inst. Math., 247 (2004), 184–194
26. Jerzy Dydak, Rolando Jimenez, “Movability in the sense of n-shape”, Topology and its Applications, 146-147 (2005), 51
27. Repovs D. Zarichnyi M., “Topology of Manifolds Modeled on Countable Direct Limits of Menger Compacta”, Topology Appl., 153:17 (2006), 3230–3240
28. N. Brodsky, A. Chigogidze, E.V. Ščepin, “Sections of Serre fibrations with 2-manifold fibers”, Topology and its Applications, 155:8 (2008), 773
29. Ageev, SM, “Preserving Z-sets by Dranishnikov's resolution”, Topology and Its Applications, 156:13 (2009), 2175
30. Alkins R., Valov V., “Functional Extenders and Set-Valued Retractions”, J. Math. Anal. Appl., 399:1 (2013), 306–314
•  Number of views: This page: 337 Full text: 121 References: 29 First page: 1