Vol. 316, No. 2, 2022

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Application of good coverings to collapsing Alexandrov spaces

Tadashi Fujioka

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 (M,X,F). 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
Milestones
Received: 3 November 2020
Revised: 26 October 2021
Accepted: 20 November 2021
Published: 6 April 2022
Authors
Tadashi Fujioka
Department of Mathematics
Kyoto University
Kyoto
Japan