For Alexandrov manifold in the title we mean 3-dim Alexandrov apace which is also a topological. manifold. 
Shioya-Yamaguchi posted a conjecture on their paper "Collapsing 3-manifold with lower sectional curvature bound"



> 
> Any three-dimensional compact, simply connected, nonnegatively curved Alexandrov space without boundary which is a topological manifold is homeomorphic to a sphere.
> 
> 
> 


It seems for me that Riemannian geometric tools do not apply here. So any progress in this directoin or it has been proved somewhere? 



3-dim 1-connected Alexandrov manifold with curvature $\ge 0$ Heomomorphic to sphere?
The Poincare conjecture in dimension 3 tells us that any 3-dimensional compact simply connected topological manifold is diffeomorphic to the 3-sphere. Surely you don't want to assume that your space is a manifold? Is its dimension as a manifold perhaps larger than 3?


