Tadashi Fujioka

Suppose a sequence ${M}_{j}$ of Alexandrov spaces collapses to a space $X$ with only weak singularities. Yamaguchi constructed a map ${f}_{j}\phantom{\rule{.17em}{0ex}}:\phantom{\rule{.17em}{0ex}}{M}_{j}\to X$ called an almost Lipschitz submersion for large $j$. We prove that if ${M}_{j}$ has a uniform positive lower bound for the volumes of spaces of directions, which is sufficiently large compared to the weakness of singularities of $X$, then ${f}_{j}$ is a locally trivial fibration. Moreover, we show some properties on the intrinsic metric and the volume of the fibers of ${f}_{j}$.