Infinite Dimensional Topology and Shape Theory

Chigogidze, Alex

doi:10.1016/b978-044482432-5/50008-1

Cited by 21 publications

(33 citation statements)

References 106 publications

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“…We also present detailed description of L-homotopy groups also introduced in [4]. Our main result is as follows.…”

Section: Theorem Dmentioning

confidence: 97%

“…It is easy to verify [4,Proposition 2.7] that extension dimension of a locally compact polyhedron is identical with its standard dimension. However the concept of [L]-homotopy differs from the classical concepts of homotopy or n-homotopy even for maps between finite polyhedra -this can be easily understood analyzing the identity map of the complex indicated in example 2.3(viii).…”

Section: [L]-homotopies and [L]-complexesmentioning

confidence: 99%

“…Proof of this statement, with minor adjustments, follows Dranishnikov's construction [12]. Lemmas 2.6 and 2.7 from [4] (see also [12, Lemmas 2.2 and 2.3]) allow us to construct inductively an inverse sequence S = {X n , p n+1 n }, consisting of locally compact polyhedra X n (with certain triangulations whose meshes converge to zero) and [L]-invertible, approximately [L]-soft, proper, simplicial bonding maps p n+1 n : X n+1 → X n so that X 1 is the given polyhedron X considered with the given triangulation τ . As in [12, Lemma 2.3], we may assume that S is L-resolvable inverse sequence.…”

Section: [L]-homotopies and [L]-complexesmentioning

confidence: 99%

“…Below, for each finite CW-complex L we consider the concept of [L]-homotopy introduced in [4]. We also present detailed description of L-homotopy groups also introduced in [4].…”

Section: Theorem Dmentioning

confidence: 99%

“…Definition 3.1 (Definition 2.2, [4]). We say that a space X is an absolute (neighborhood) extensor modulo L, or shortly that ≃ G.…”

Section: [L]-homotopies and [L]-complexesmentioning

confidence: 99%

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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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“…We also present detailed description of L-homotopy groups also introduced in [4]. Our main result is as follows.…”

Section: Theorem Dmentioning

confidence: 97%

Section: [L]-homotopies and [L]-complexesmentioning

confidence: 99%

Section: [L]-homotopies and [L]-complexesmentioning

confidence: 99%

“…Below, for each finite CW-complex L we consider the concept of [L]-homotopy introduced in [4]. We also present detailed description of L-homotopy groups also introduced in [4].…”

Section: Theorem Dmentioning

confidence: 99%

“…Definition 3.1 (Definition 2.2, [4]). We say that a space X is an absolute (neighborhood) extensor modulo L, or shortly that ≃ G.…”

Section: [L]-homotopies and [L]-complexesmentioning

confidence: 99%

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