  We study the regularity of the free boundary in the obstacle problem for the
fractional Laplacian under the assumption that the obstacle $\varphi$ satisfies
$\Delta \varphi\leq 0$ near the contact region. Our main result establishes
that the free boundary consists of a set of regular points, which is known to
be a $(n-1)$-dimensional $C^{1,\alpha}$ manifold by the results in \cite{CSS},
and a set of singular points, which we prove to be contained in a union of
$k$-dimensional $C^1$-submanifold, $k=0,\ldots,n-1$.
  Such a complete result on the structure of the free boundary was known only
in the case of the classical Laplacian \cite{C-obst1,C-obst2}, and it is new
even for the Signorini problem (which corresponds to the particular case of the
$\frac12$-fractional Laplacian). A key ingredient behind our results is the
validity of a new non-degeneracy condition $\sup_{B_r(x_0)}(u-\varphi)\geq
c\,r^2$, valid at all free boundary points $x_0$.
