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We show that if $K$ is a nilpotent finite complex, then $\Omega K$ can bebuilt from spheres using fibrations and homotopy (inverse) limits. This isapplied to show that if ${\mathrm {map}}_*(X,S^n)$ is weakly contractible forall $n$, then ${\mathrm {map}}_*(X,K)$ is weakly contractible for any nilpotentfinite complex $K$.

Miller spaces and spherical resolvability of finite complexes