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Uspekhi Mat. Nauk, 2009, Volume 64, Issue 3(387), Pages 73–166 (Mi umn9284)  

This article is cited in 7 scientific papers (total in 7 papers)

Steenrod homotopy

S. A. Melikhov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Steenrod homotopy theory is a natural framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; or from a different viewpoint, it studies the topology of the $\lim^1$ functor (for inverse sequences of groups). This paper is primarily concerned with the case of compacta, in which Steenrod homotopy coincides with strong shape. An attempt is made to simplify the foundations of the theory and to clarify and improve some of its major results. With geometric tools such as Milnor's telescope compactification, comanifolds (=mock bundles), and the Pontryagin–Thom construction, new simple proofs are obtained for results by Barratt–Milnor, Geoghegan–Krasinkiewicz, Dydak, Dydak–Segal, Krasinkiewicz–Minc, Cathey, Mittag-Leffler–Bourbaki, Fox, Eda–Kawamura, Edwards–Geoghegan, Jussila, and for three unpublished results by Shchepin. An error in Lisitsa's proof of the ‘Hurewicz theorem in Steenrod homotopy’ is corrected. It is shown that over compacta, R. H. Fox's overlayings are equivalent to I. M. James' uniform covering maps. Other results include:
$\bullet$ A morphism between inverse sequences of countable (possibly non-Abelian) groups that induces isomorphisms on $\lim$ and $\lim^1$ is invertible in the pro-category. This implies the ‘Whitehead theorem in Steenrod homotopy’, thereby answering two questions of Koyama.
$\bullet$ If $X$ is an $LC_{n-1}$-compactum, $n\ge 1$, then its $n$-dimensional Steenrod homotopy classes are representable by maps $S^n\to\nobreak X$, provided that $X$ is simply connected. The assumption of simple connectedness cannot be dropped, by a well-known result of Dydak and Zdravkovska.
$\bullet$ A connected compactum is Steenrod connected (=pointed 1-movable), if and only if every uniform covering space of it has countably many uniform connected components.
Bibliography: 117 titles.

DOI: https://doi.org/10.4213/rm9284

Full text: PDF file (1477 kB)
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English version:
Russian Mathematical Surveys, 2009, 64:3, 469–551

Bibliographic databases:

UDC: 515.142.26
MSC: 55D**, 55N**
Received: 17.03.2009

Citation: S. A. Melikhov, “Steenrod homotopy”, Uspekhi Mat. Nauk, 64:3(387) (2009), 73–166; Russian Math. Surveys, 64:3 (2009), 469–551

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. S. Bogatyi, O. Frolkina, “On multiplicity of maps”, Topology Appl., 159:7 (2012), 1778–1786  crossref  mathscinet  isi  elib  scopus
    2. S. A. Melikhov, J. Zaja̧c, “Contractible polyhedra in products of trees and absolute retracts in products of dendrites”, Proc. Amer. Math. Soc., 141:7 (2013), 2519–2535  crossref  mathscinet  zmath  isi  elib  scopus
    3. A. R. Alimov, “Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces”, Izv. Math., 78:4 (2014), 641–655  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. J. Brazas, P. Fabel, “Spanier groups and the first shape group”, Rocky Mountain J. Math., 44:5 (2014), 1415–1444  crossref  mathscinet  zmath  isi  scopus
    5. A. R. Alimov, I. G. Tsar'kov, “Connectedness and other geometric properties of suns and Chebyshev sets”, J. Math. Sci., 217:6 (2016), 683–730  mathnet  crossref  mathscinet
    6. A. R. Alimov, I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Russian Math. Surveys, 71:1 (2016), 1–77  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. Dydak J., “Overlays and group actions”, Topology Appl., 207 (2016), 22–32  crossref  mathscinet  zmath  isi  elib  scopus
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