Finite-dimensional spaces in resolving classes

Using the theory of resolving classes, we show that if $X$ is a CW complex of finite type such that ${\text{map}}_{\ast }(X,S^{2n+1})\sim \ast$ for all sufficiently large $n$, then ${\text{map}}_{\ast }(X,K)\sim \ast$ for every simply-connected finite-dimensional CW complex $K$; and under mild hypotheses on ${\pi }_1(X)$, the same conclusion holds for $\mathit{\text{all}}$ finite-dimensional complexes $K$. Since it is comparatively easy to prove the former condition for $X=B\text{ZZ}\mathrm{/}p$ (we give a proof in an appendix), this result can be applied to give a new, more elementary proof of the Sullivan conjecture.