Steenrod homotopy

S. A. Melikhov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 64 (2009) no. 3, pp. 469-551

Résumé

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Krasinkiewicz–Minc, Cathey, Mittag-Leffler–Bourbaki, Fox, Eda–Kawamura, Edwards–Geoghegan, Jussila, and for three unpublished results by Shchepin.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, $n\geqslant 1$, 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.