Menger (Nöbeling) Manifolds versus Hilbert Cube (Space) Manifolds - A Categorical Comparison

Chigogidze, Alex; Fedorchuk, V. V.

doi:10.1007/s006050070039

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“…Algebraic models for n-types (the so called cat n -groups) were found in [24]. Approximately at the same time n-homotopies (and subsequently even n-shapes [7], [6]) begun to play a substantial role in the revitalized theory of Menger manifolds [2], [5] (see [8] for a discussion of categorical connections between n-homotopies and homotopies via the theories of manifolds modeled on Menger and Hilbert cubes respectively). The following theorem has been proved in [16].…”

Section: Theorem Bmentioning

confidence: 99%

Topological Model Categories Generated by Finite Complexes

Chigogidze

Karasev

2003

Monatshefte f�r Mathematik

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Abstract. Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov's notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them -describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups -is obtained by letting L = {point}. The other -describing n-homotopy equivalences between at most (n + 1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ≤ n -by letting L = S n+1 , n ≥ 0.

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