2002/05/01 by Alex Chigogidze, A. Chigogidze, Chigogidze, A. +2
Mathematics · Medicine · #18D15 #55U40 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Intracranial Aneurysms: Treatment and Complications #math.AT #math.CT #msc:18D15 #msc:55U40
paper · pdf · doi:10.48550/arxiv.math/0205014
24 pages
openalex publication_date 2002/05/01 · arxiv created 2002/07/31 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Our main result states that for each finite complex L the category \bf 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 = \\rm point\. The other -- describing n-homomotopy equivalences between at most (n+1)-dimensional CW-complexes as maps inducing isomorophisms of k-dimensional homotopy groups with k ≤ n -- by letting L = Sn+1, n ≥ 0.