The Poincaré conjecture for three-dimensional Alexandrov spaces with nonnegative curvature

Let $A$ and $A'$ be nonnegatively curved Alexandrov spaces with isometric boundaries, and let $A\cup_\partial A'$ denote their gluing along a boundary isometry. Let $D^3$, $K_1(P^2)$, $B(\mathrm{pt})$, $B(S_2)$, and $B(S_4)$ be the listed nonnegatively curved Alexandrov spaces. Poincaré conjecture. Every simply connected three-dimensional closed Alexandrov space with nonnegative curvature is homeomorphic to an isometric gluing

$$A\cup_\partial A'$$

for $A$ and $A'$ chosen from

$$D^3,\ K_1(P^2),\ B(\mathrm{pt}),\ B(S_2),\ B(S_4).$$

This conjecture is proposed as a topological classification of collapsing three-dimensional closed Alexandrov spaces with nonnegative curvature. The supplied text gives no resolution status.