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Topological realizations and fundamental groups of higher-rank graphs

Journal Article


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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.

UOW Authors


  •   Kaliszewski, S (external author)
  •   Kumjian, Alexander (external author)
  •   Quigg, John C. (external author)
  •   Sims, Aidan

Publication Date


  • 2015

Citation


  • Kaliszewski, S., Kumjian, A., Quigg, J. C. & Sims, A. (2015). Topological realizations and fundamental groups of higher-rank graphs. Proceedings of the Edinburgh Mathematical Society, 59 (1), 143-168.

Scopus Eid


  • 2-s2.0-84930798836

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=5578&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/4557

Number Of Pages


  • 25

Start Page


  • 143

End Page


  • 168

Volume


  • 59

Issue


  • 1

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.

UOW Authors


  •   Kaliszewski, S (external author)
  •   Kumjian, Alexander (external author)
  •   Quigg, John C. (external author)
  •   Sims, Aidan

Publication Date


  • 2015

Citation


  • Kaliszewski, S., Kumjian, A., Quigg, J. C. & Sims, A. (2015). Topological realizations and fundamental groups of higher-rank graphs. Proceedings of the Edinburgh Mathematical Society, 59 (1), 143-168.

Scopus Eid


  • 2-s2.0-84930798836

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=5578&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/4557

Number Of Pages


  • 25

Start Page


  • 143

End Page


  • 168

Volume


  • 59

Issue


  • 1