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

This article is cited in 30 scientific papers (total in 30 papers)

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.

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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
Received: 22.11.1984

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}


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