Application of good coverings to collapsing Alexandrov spaces

TL;DR

This paper improves the understanding of the topology of collapsing Alexandrov spaces by constructing homotopy and cohomology spectral sequences using good coverings, extending previous results to extremal subsets.

Contribution

It refines Perelman's results on homotopy sequences for collapsing Alexandrov spaces and extends the framework to extremal subsets using good coverings.

Findings

Constructed an infinite exact sequence of homotopy groups.

Developed a spectral sequence of cohomology groups for the pair (M,X,F).

Extended results to primitive extremal subsets.

Abstract

Let $M$ be an Alexandrov space collapsing to an Alexandrov space $X$ of lower dimension. Suppose $X$ has no proper extremal subsets and let $F$ denote a regular fiber. We slightly improve the result of Perelman to construct an infinitely long exact sequence of homotopy groups and a spectral sequence of cohomology groups for the pair $(M,X,F)$. The proof is an application of the good coverings of Alexandrov spaces introduced by Mitsuishi-Yamaguchi. We also extend this result to each primitive extremal subset of $X$.