Alexandrov spaces which are not limits of Riemannian manifolds Are there important/ interesting/ natural examples of compact Alexandrov spaces with curvature bounded from below which are not Gromov-Hausdorff limits of smooth compact Riemannian manifolds with uniformly bounded from below sectional curvature? (The condition of compactness might be relaxed somehow.)
EDIT: The dimension of manifolds in the approximating sequence is supposed to be bounded from above.
 A: *

*There is an Alexandrov space, say $A$, with curvature $\ge 1$ which can not obtained as a limit of Riemannian manifolds with curvature $\ge \kappa$, if $\kappa>\tfrac14$; see "Metric constraints on exotic spheres via Alexandrov geometry." by Grove and Wilhelm.

*There is an Alexandrov space, say $A$, such that if it can appear as a limit of Riemannian manifolds $M_n$ with uniformly bounded curvature then $\dim M_n\ge \dim A+8$ for all large $n$; see "Regularity of limits of noncollapsing sequences of manifolds" by Kapovitch. It is expected that in this formula one can exchange $8$ to $\infty$.
A: I believe (I am not an expert in the subject) a simple example (though using a highly non-trivial theorem) is given by considering the following example:
Let
$$\mathbb{S}^3\subset \mathbb{R}^4$$
be the round sphere of curvature $1$ and consider the isometry 
$\phi:\mathbb{S}^3\to \mathbb{S}^3$ induced by
$$
(x_1, x_2, x_3, x_4)\mapsto (x_1, -x_2, -x_3, -x_4)
$$
and let 
$$
X=\mathbb{S}^3/\phi.
$$
be the obvious quotient space with the natural quotient metric.
This quotient has curvature $\geq 1$, but is not a topological manifold (as small balls about $(\pm 1 , 0, 0, 0)$ look like cones over $\mathbb{RP}^2$).
Hence, by Perelman's stability theorem, $X$ cannot be the GH limit of smooth manifolds of curvature $\geq k$ for any $k\leq 1$.
