#### Vol. 316, No. 2, 2022

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Application of good coverings to collapsing Alexandrov spaces

Vol. 316 (2022), No. 2, 335–365
##### 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 $\left(M,X,F\right)$. The proof is an application of the good coverings of Alexandrov spaces introduced by Mitsuishi and Yamaguchi. We also extend this result to each primitive extremal subset of $X$.

##### Keywords
Alexandrov spaces, collapse, good coverings, extremal subsets
##### Mathematical Subject Classification
Primary: 53C20, 53C23