Globalization and asphericity

Stephanie Alexander, Vitali Kapovitch, Anton Petrunin

Research output: Chapter in Book/Report/Conference proceedingChapter

4 Scopus citations


In this chapter we introduce locally $$\mathrm{CAT}^{}(0)$$ spaces and prove the globalization theorem that provides a sufficient condition for locally $$\mathrm{CAT}^{}(0)$$ spaces to be globally $$\mathrm{CAT}^{}(0)$$. The theorem implies in particular that the universal metric cover of a proper length, locally $$\mathrm{CAT}^{}(0)$$ space is a proper length $$\mathrm{CAT}^{}(0)$$ space. It follows that any proper length, locally $$\mathrm{CAT}^{}(0)$$ space is aspherical; that is, its universal cover is contractible.

Original languageEnglish (US)
Title of host publicationSpringerBriefs in Mathematics
PublisherSpringer Science and Business Media B.V.
Number of pages16
StatePublished - 2019

Publication series

NameSpringerBriefs in Mathematics
ISSN (Print)2191-8198
ISSN (Electronic)2191-8201

All Science Journal Classification (ASJC) codes

  • General Mathematics


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