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

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

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
\Bibitem{Dra94} \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 \mathnet{http://mi.mathnet.ru/msb891} \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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2. A.N. Dranishnikov, V.V. Uspenskij, “Light maps and extensional dimension”, Topology and its Applications, 80:1-2 (1997), 91
3. Chigogidze A., “Cohomological Dimension of Tychonov Spaces”, Topology Appl., 79:3 (1997), 197–228
4. A. N. Dranishnikov, “Extension theory for maps of compact spaces”, Russian Math. Surveys, 53:5 (1998), 929–935
5. E. V. Shchepin, “Arithmetic of dimension theory”, Russian Math. Surveys, 53:5 (1998), 975–1069
6. Chigogidze A., Zarichnyi M., “On Absolute Extensors Module a Complex”, Topology Appl., 86:2 (1998), 169–178
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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
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12. V. V. Fedorchuk, “Fully closed mappings and their applications”, J. Math. Sci., 136:5 (2006), 4201–4292
13. Chigogidze A. Karasev A., “Topological Model Categories Generated by Finite Complexes”, Mon.heft. Math., 139:2 (2003), 129–150
14. Dydak J., “Extension Dimension for Paracompact Spaces”, Topology Appl., 140:2-3 (2004), 227–243
15. Turygin YA., “Approximation of K-Dimensional Maps”, Topology Appl., 139:1-3 (2004), 227–235
16. Chigogidze A., “Extraordinary Dimension Theories Generated by Complexes”, Topology Appl., 138:1-3 (2004), 1–20
17. Ivansic I., Rubin L., “Limit Theorem for Semi-Sequences in Extension Theory”, Houst. J. Math., 31:3 (2005), 787–807
18. Alex Karasev, Vesko Valov, “Extension dimension and quasi-finite CW-complexes”, Topology and its Applications, 153:17 (2006), 3241
19. Alex Chigogidze, Vesko Valov, “Extraordinary dimension of maps”, Topology and its Applications, 153:10 (2006), 1586
20. Karasev A., “On Two Problems in Extension Theory”, Topology Appl., 153:10 (2006), 1609–1613
21. Karasev A., Valov V., “Universal Absolute Extensors in Extension Theory”, Proc. Amer. Math. Soc., 134:8 (2006), 2473–2478
22. V. V. Fedorchuk, “Dimension scales of bicompacta”, Siberian Math. J., 49:3 (2008), 549–561
23. Ivansic I., Rubin L.R., “Extension Theory and the Psi(Infinity) Operator”, Publ. Math.-Debr., 73:3-4 (2008), 265–280
24. Ivansic I., Rubin L.R., “Extension Dimension of a Wide Class of Spaces”, J. Math. Soc. Jpn., 61:4 (2009), 1097–1110
25. Valov V., “Parametric Bing and Krasinkiewicz Maps: Revisited”, Proc. Amer. Math. Soc., 139:2 (2011), 747–756
26. Miyata T., “Approximate Extension Property of Mappings”, Topology Appl., 159:3 (2012), 921–932
27. Ivansic I., Rubin L.R., “Pseudo-Compactness of Direct Limits”, Topology Appl., 160:2 (2013), 360–367
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