  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 $k \leq n$ -- by letting $L = S^{n+1}$, $n
\geq 0$.

Algebraic Topology;Category Theory