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Topological realizations and fundamental groups of higher-rank graphs

Part of: Categories with structure Graph theory Curves

Published online by Cambridge University Press:  10 June 2015

S. Kaliszewski ,
Alex Kumjian ,
John Quigg  and
Aidan Sims
S. Kaliszewski
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA (kaliszewski@asu.edu; quigg@asu.edu)
Alex Kumjian
Affiliation:
Department of Mathematics (084), University of Nevada, Reno, NV 89557-0084, USA (alex@unr.edu)
John Quigg
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA (kaliszewski@asu.edu; quigg@asu.edu)
Aidan Sims
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, New South Wales 2522, Australia (asims@uow.edu.au)
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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.

Keywords

k-graphfundamental groupCW-complexfunctorcoveringprojective limit

MSC classification

Secondary: 05C20: Directed graphs (digraphs), tournaments 14H30: Coverings, fundamental group 18D99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 1 , February 2016 , pp. 143 - 168
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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