Topological Model Categories Generated By Finite Complexes

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
A. "Alex" Chigogidze, Professor and Department Chair (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site:

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 = Sn+1, n = 0.

Additional Information

arXiv:math/0205014v2 [math.AT] 31 Jul 2002
Language: English
Date: 2002
Model category, [L]–homotopy, [L]-com

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