Topological realizations and fundamental groups of higher-rank graphs. Issue 1 (10th June 2015)

• Record Type: Journal Article
• Authors: Kaliszewski, S.; Kumjian, Alex; Quigg, John; Sims, Aidan
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 59:Issue 1(2016)
• Page Start: 143
• Page End: 168
• Publication Date: 2015-06-10
• Subjects: k-graph, -- fundamental group, -- CW-complex, -- functor, -- covering, -- projective limit; Primary 05C20, -- Secondary 14H30, -- 18D99; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091515000061
• Languages: English
• ISSNs: 0013-0915

Abstract

Abstract: We investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor and that for each higher-rank graph Λ, this functor determines a category equivalence between the category of coverings of Λ and the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions for k -graphs: projective limits and crossed products by finitely generated free abelian groups.