Globalization and asphericity

Abstract

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.