We look at complete, locally conformally flat (lcf) metrics defined on a domain $\Omega\subset S^n$ . There is a lot of information about the singular set $\partial\Omega$ contained in the positivity of $\sigma_k$ , and, in particular, we obtain a bound for the Hausdorff dimension of $\partial\Omega$ in relation to Schoen and Yau's work [18] for the scalar curvature. On the other hand, since some locally conformally flat manifolds can be embedded into $S^n$ , this dimension bound implies several topological corollaries.