A fibration theorem for collapsing sequences of Alexandrov spaces

Abstract

Suppose a sequence [Formula: see text] of Alexandrov spaces collapses to a space [Formula: see text] with only weak singularities. Yamaguchi constructed a map [Formula: see text] called an almost Lipschitz submersion for large [Formula: see text]. We prove that if [Formula: see text] has a uniform positive lower bound for the volumes of spaces of directions, which is sufficiently large compared to the weakness of singularities of [Formula: see text], then [Formula: see text] is a locally trivial fibration. Moreover, we show some properties on the intrinsic metric and the volume of the fibers of [Formula: see text].