Miller Spaces and Spherical Resolvability of Finite Complexes

We show that if $K$ is a nilpotent finite complex, then $\mathrm{\Omega }K$ can be built from spheres using fibrations and homotopy (inverse) limits. This is applied to show that if ${\mathrm{m}\mathrm{a}\mathrm{p}}_{\ast }(X,S^n)$ is weakly contractible for all $n$, then ${\mathrm{m}\mathrm{a}\mathrm{p}}_{\ast }(X,K)$ is weakly contractible for any nilpotent finite complex $K$.