On some dimensional properties of 4-manifolds

It is shown, under the assumption of Jensen's principle $\lozenge$, that if for a complex L with $[L] \geq [S^{4}]$ there exists a metrizable compactum whose extension dimension is L, then there exists a differentiable, countably compact, perfectly normal and hereditarily separable 4-manifold whose extension dimension is also [L].