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Abstract
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Let
be an Alexandrov space collapsing to an Alexandrov space
of lower dimension.
Suppose
has no proper
extremal subsets and let
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
.
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 .
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Keywords
Alexandrov spaces, collapse, good coverings, extremal
subsets
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Mathematical Subject Classification
Primary: 53C20, 53C23
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Milestones
Received: 3 November 2020
Revised: 26 October 2021
Accepted: 20 November 2021
Published: 6 April 2022
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