Topological model categories generated by finite complexes
Abstract
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 CWcomplexes as maps inducing isomorphisms of all homotopy groups  is obtained by letting $L = \{{\rm point}\}$. The other  describing nhomomotopy equivalences between at most $(n+1)$dimensional CWcomplexes as maps inducing isomorophisms of kdimensional homotopy groups with $k \leq n$  by letting $L = S^{n+1}$, $n \geq 0$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2002
 DOI:
 10.48550/arXiv.math/0205014
 arXiv:
 arXiv:math/0205014
 Bibcode:
 2002math......5014C
 Keywords:

 Algebraic Topology;
 Category Theory;
 55U40;
 18D15
 EPrint:
 24 pages