Lipschitz Homotopy Convergence of Alexandrov Spaces

Abstract

We introduce the notion of good coverings of metric spaces, and prove that if a metric space admits a good covering, then it has the same locally Lipschitz homotopy type as the nerve complex of the covering. As an application, we obtain a Lipschitz homotopy stability result for a moduli space of compact Alexandrov spaces without collapsing.

Application of good coverings to collapsing Alexandrov spaces

“…It is also worth mentioning that the strict concavity of the function h of Theorem 3.11 is liftable under the noncollapsing convergence of Alexandrov spaces. Therefore, the goodness of a covering is also liftable in the noncollapsing case, which yields the finiteness of Lipschitz homotopy types of noncollapsing Alexandrov spaces ( [13]). In this paper, we study the relation between good coverings and collapsing.…”

“…It is of course good in the topological sense and has more geometric properties. Using such coverings, Mitsuishi-Yamaguchi showed in [13] the Lipschitz homotopy finiteness of the class of noncollapsing Alexandrov spaces. In this paper, we study the relation between good coverings and collapsing.…”