Miller spaces and spherical resolvability of finite complexes
Volume 178 / 2003
Abstract
Let ${\mathcal A}$ be a fixed collection of spaces, and suppose $K$ is a nilpotent space that can be built from spaces in ${\mathcal A}$ by a succession of cofiber sequences. We show that, under mild conditions on the collection ${\mathcal A}$, it is possible to construct $K$ from spaces in ${\mathcal A}$ using, instead, homotopy (inverse) limits and extensions by fibrations. One consequence is that if $K$ is a nilpotent finite complex, then ${\mit \Omega }K$ can be built from finite wedges of spheres using homotopy limits and extensions by fibrations. This is applied to show that if ${\rm map}_*(X,S^n)$ is weakly contractible for all sufficiently large $n$, then ${\rm map}_*(X,K)$ is weakly contractible for any nilpotent finite complex $K$.
