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Mat. Sb., 1994, Volume 185, Number 4, Pages 81–90 (Mi msb891)  

This article is cited in 27 scientific papers (total in 27 papers)

The Eilenberg–Borsuk theorem for mappings into an arbitrary complex

A. N. Dranishnikov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The classical Eilenberg–Borsuk theorem on extension of partial mappings into a sphere is generalized to the case of an arbitrary complex $K$. It is formulated in terms of extraordinary dimension theory, which is developed in the present paper. When $K = K(G,  k)$ is an Eilenberg–MacLane complex, the result can be expressed in terms of cohomological dimension theory. For partial mappings $\varphi\colon A\to K(G,  k)$ of an $n$-manifold $M$, the following is obtained:
Theorem. If $k<n-2$, then there exists a compactum $X\subset M$ of dimension $n-k-1$, such that the mapping $\varphi$ extends to $M-X$ and for every abelian group $\pi$ with $\pi\otimes G=0$ the cohomological dimension of $X$ with coefficients in $\pi$ does not exceed $n-k-2$.
Thus, in comparison with the classical Eilenberg–Borsuk theorem, there is obtained an additional condition as to the cohomological dimension of $X$.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 81:2, 467–475

Bibliographic databases:

UDC: 515.1
MSC: Primary 54C20, 55S36; Secondary 54F45, 55M10
Received: 22.10.1992

Citation: A. N. Dranishnikov, “The Eilenberg–Borsuk theorem for mappings into an arbitrary complex”, Mat. Sb., 185:4 (1994), 81–90; Russian Acad. Sci. Sb. Math., 81:2 (1995), 467–475

Citation in format AMSBIB
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\by A.~N.~Dranishnikov
\paper The Eilenberg--Borsuk theorem for mappings into an~arbitrary complex
\jour Mat. Sb.
\yr 1994
\vol 185
\issue 4
\pages 81--90
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1272187}
\zmath{https://zbmath.org/?q=an:0832.55001}
\transl
\jour Russian Acad. Sci. Sb. Math.
\yr 1995
\vol 81
\issue 2
\pages 467--475
\crossref{https://doi.org/10.1070/SM1995v081n02ABEH003546}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995RB51300009}


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    This publication is cited in the following articles:
    1. S. M. Ageev, S. A. Bogatyi, “Obstructions to the extension of partial maps”, Math. Notes, 62:6 (1997), 675–682  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. A.N. Dranishnikov, V.V. Uspenskij, “Light maps and extensional dimension”, Topology and its Applications, 80:1-2 (1997), 91  crossref  mathscinet  zmath
    3. Chigogidze A., “Cohomological Dimension of Tychonov Spaces”, Topology Appl., 79:3 (1997), 197–228  crossref  mathscinet  zmath  isi
    4. A. N. Dranishnikov, “Extension theory for maps of compact spaces”, Russian Math. Surveys, 53:5 (1998), 929–935  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. E. V. Shchepin, “Arithmetic of dimension theory”, Russian Math. Surveys, 53:5 (1998), 975–1069  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. Chigogidze A., Zarichnyi M., “On Absolute Extensors Module a Complex”, Topology Appl., 86:2 (1998), 169–178  crossref  mathscinet  zmath  isi
    7. Chigogidze A., Fedorchuk V., “On Some Dimensional Properties of 4-Manifolds”, Topology Appl., 107:1-2 (2000), 67–78  crossref  mathscinet  zmath  isi
    8. Dranishnikov A., Dydak J., “Extension Theory of Separable Metrizable Spaces with Applications to Dimension Theory”, Trans. Am. Math. Soc., 353:1 (2000), 133–156  crossref  mathscinet  isi
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    13. Chigogidze A. Karasev A., “Topological Model Categories Generated by Finite Complexes”, Mon.heft. Math., 139:2 (2003), 129–150  crossref  mathscinet  zmath  isi
    14. Dydak J., “Extension Dimension for Paracompact Spaces”, Topology Appl., 140:2-3 (2004), 227–243  crossref  mathscinet  zmath  isi
    15. Turygin YA., “Approximation of K-Dimensional Maps”, Topology Appl., 139:1-3 (2004), 227–235  crossref  mathscinet  zmath  isi
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    17. Ivansic I., Rubin L., “Limit Theorem for Semi-Sequences in Extension Theory”, Houst. J. Math., 31:3 (2005), 787–807  mathscinet  zmath  isi
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    20. Karasev A., “On Two Problems in Extension Theory”, Topology Appl., 153:10 (2006), 1609–1613  crossref  mathscinet  zmath  isi
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    22. V. V. Fedorchuk, “Dimension scales of bicompacta”, Siberian Math. J., 49:3 (2008), 549–561  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    23. Ivansic I., Rubin L.R., “Extension Theory and the Psi(Infinity) Operator”, Publ. Math.-Debr., 73:3-4 (2008), 265–280  mathscinet  zmath  isi
    24. Ivansic I., Rubin L.R., “Extension Dimension of a Wide Class of Spaces”, J. Math. Soc. Jpn., 61:4 (2009), 1097–1110  crossref  mathscinet  zmath  isi
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    26. Miyata T., “Approximate Extension Property of Mappings”, Topology Appl., 159:3 (2012), 921–932  crossref  mathscinet  zmath  isi
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  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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