Application of good coverings to collapsing Alexandrov spaces

Tadashi Fujioka

Vol. 316 (2022), No. 2, 335–365

DOI: 10.2140/pjm.2022.316.335

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