Let $\Psi(i)\to 0$ as $i\to \infty$. Let $A_i$ be a sequence of n dimensional Alexandrov spaces with curvature $\geqslant k$, that Gromov Hausdorff converge to an m dimensional closed Riemannian manifold $A$. Here we consider the collapsing case, i.e. $n>m$. By Yamaguchi's almost Lipschitz submersion theorem, there is a $\Psi(i)$-almost Lipschitz submersion $f_i:A_i\to A$.
Next we discuss whether $f_i$ is a locally fiber bundle. For any point $p\in A$, there exist $a_1,...,a_m\in A$ near p, such that $$ \tilde{\angle}_k a_j p a_l >\frac{\pi}{2} \text{ for all } j\neq l, $$ and there is a neighborhood $U\ni p$ such that $G(x)=(d(a_1,x),...,d(a_m,x)): U \to \mathbb{R}^m$ is a homeomorphism. For $x_i\in f_i^{-1}(U)$, we know $$ d(f_i^{-1}(a_j),x_i)=d(a_j, f_i(x_i))+\Psi(i), \quad d(f_i^{-1}(a_j), f_i^{-1}(a_l))=d(a_j,a_l)+\Psi(i). $$ So $$ G_i(x_i)=(d(f_i^{-1}(a_1),x_i),...,d(f_i^{-1}(a_m),x_i)) $$ have no critical point in $f_i^{-1}(U)$.
By the Morse theorem for Alexandrov spaces proved by Perelman, $G_i:f_i^{-1}(U)\to \mathbb{R}^m$ is a locally trivial fiber bundle. We don't know whether its fiber coincide with $f_i^{-1}(p)$ since for $x,y\in f_i^{-1}(p)$, we don't have $G_i(x)=G_i(y)$. Is there a locally trivial fiber bundle from $A_i\to A$? Are there counter examples or illuminating examples?
