Application of good coverings to collapsing Alexandrov spaces

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-Yamaguchi. We also extend this result to each primitive extremal subset of $X$.

minor changes, to appear in Pacific. J. Math

References in corpus (3)

• Margulis lemma and Hurewicz fibration Theorem on Alexandrov spaces
• A fibration theorem for collapsing sequences of Alexandrov spaces
• Homotopy lifting property of an $e^{\epsilon }$-Lipschitz and co-Lipschitz map

Cited by in corpus (2)

• Collapsing to Alexandrov spaces with isolated mild singularities
• Euler characteristics of collapsing Alexandrov spaces