Good coverings of Alexandrov spaces

In the present paper, we define a notion of good coverings of Alexandrov spaces with curvature bounded below, and we prove that every Alexandrov space admits such a good covering and that it has the same homotopy type as the nerve of the good covering. We also prove a kind of stability of the isomor...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2019-12, Vol.372 (11), p.8107-8130
Main Authors: Mitsuishi, Ayato, Yamaguchi, Takao
Format: Article
Language:English
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Summary:In the present paper, we define a notion of good coverings of Alexandrov spaces with curvature bounded below, and we prove that every Alexandrov space admits such a good covering and that it has the same homotopy type as the nerve of the good covering. We also prove a kind of stability of the isomorphism classes of the nerves of good coverings in the noncollapsing case. In the proof, we need a version of Perelman's fibration theorem, which is also proved in this paper.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/7849