TY - CHAP

T1 - Globalization and asphericity

AU - Alexander, Stephanie

AU - Kapovitch, Vitali

AU - Petrunin, Anton

N1 - Publisher Copyright:
© 2019, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

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U2 - 10.1007/978-3-030-05312-3_3

DO - 10.1007/978-3-030-05312-3_3

M3 - Chapter

AN - SCOPUS:85101069405

T3 - SpringerBriefs in Mathematics

SP - 33

EP - 48

BT - SpringerBriefs in Mathematics

PB - Springer Science and Business Media B.V.

ER -