[Paper Review] Topological realizations and fundamental groups of higher-rank graphs

S. Kaliszewski, Alex Kumjian+2 more|arXiv (Cornell University)|May 13, 2012

Advanced Operator Algebra Research17 references3 citations

TL;DR

This paper establishes a topological realization of higher-rank graphs as CW-complexes, proving that the fundamental group of the graph coincides with that of its topological realization. It further shows that the assignment of a k-graph to its realization is a functor that induces a category equivalence between algebraic coverings of the graph and topological coverings of the space, preserving universal coverings and linking to projective limits and crossed products.

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.

Motivation & Objective

• To define a topological realization of a k-graph as a CW-complex that captures its combinatorial structure.
• To prove that the fundamental group of a k-graph is isomorphic to the fundamental group of its topological realization.
• To show that the assignment of a k-graph to its topological realization is a functor between categories of k-graphs and topological spaces.
• To establish a category equivalence between algebraic coverings of a k-graph and topological coverings of its realization, preserving universal coverings.
• To relate the topological realization to standard constructions such as projective limits of k-graphs and crossed products by Z^l actions.

Proposed method

• Construct the topological realization $X_\Lambda$ of a k-graph $\Lambda$ by gluing open cells into the interiors of commuting cubes in the category.
• Define a continuous map $\widetilde{\varphi}: X_\Lambda \to X_\Gamma$ for each k-graph morphism $\varphi: \Lambda \to \Gamma$, making the assignment a functor.
• Prove that the fundamental group of $X_\Lambda$ is isomorphic to the fundamental group of $\Lambda$ via explicit homotopy-theoretic arguments.
• Demonstrate that the functor preserves coverings, inducing a category equivalence between algebraic coverings of $\Lambda$ and topological coverings of $X_\Lambda$, with universal coverings mapping to universal coverings.
• Show that for a projective limit of finite-to-one coverings $\varprojlim(\Lambda_n, p_n)$, the topological realization $X_{\varprojlim(\Lambda_n, p_n)}$ is homeomorphic to $\varprojlim(X_{\Lambda_n}, \widetilde{p}_n)$, preserving inverse limits of fundamental groups.
• Prove that the topological realization of a crossed product k-graph $\Lambda \times_\alpha \mathbb{Z}^l$ is homeomorphic to the mapping torus $M(\widetilde{\alpha})$ of the induced homeomorphism $\widetilde{\alpha}$ on $X_\Lambda$.

Experimental results

Research questions

• RQ1 Does the fundamental group of a k-graph coincide with the fundamental group of its topological realization as a CW-complex?
• RQ2 Is the assignment $\Lambda \mapsto X_\Lambda$ a functor from the category of k-graphs to the category of topological spaces?
• RQ3 Does this functor induce a category equivalence between algebraic coverings of $\Lambda$ and topological coverings of $X_\Lambda$?
• RQ4 How does the topological realization interact with projective limits of k-graphs?
• RQ5 What is the topological realization of a crossed product k-graph $\Lambda \times_\alpha \mathbb{Z}^l$ in terms of standard topological constructions?

Key findings

• The fundamental group of a k-graph $\Lambda$ is isomorphic to the fundamental group of its topological realization $X_\Lambda$, i.e., $\pi_1(\Lambda) \cong \pi_1(X_\Lambda)$.
• The assignment $\Lambda \mapsto X_\Lambda$ defines a functor from the category of k-graphs to the category of topological spaces, sending morphisms to continuous maps.
• This functor induces a category equivalence between the category of algebraic coverings of $\Lambda$ and the category of topological coverings of $X_\Lambda$, with universal coverings mapping to universal coverings.
• For a projective limit $\varprojlim(\Lambda_n, p_n)$ of finite-to-one coverings of k-graphs, the topological realization satisfies $X_{\varprojlim(\Lambda_n, p_n)} \cong \varprojlim(X_{\Lambda_n}, \widetilde{p}_n)$, and $\pi_1(X_{\varprojlim(\Lambda_n, p_n)}) \cong \varprojlim(\pi_1(\Lambda_n), (p_n)^*)$.
• The topological realization of a crossed product k-graph $\Lambda \times_\alpha \mathbb{Z}^l$ is homeomorphic to the mapping torus $M(\widetilde{\alpha})$ of the induced homeomorphism $\widetilde{\alpha}$ on $X_\Lambda$, i.e., $X_{\Lambda \times_\alpha \mathbb{Z}^l} \cong M(\widetilde{\alpha})$.
• The fundamental group extension $1 \to \pi_1(\Lambda) \to \pi_1(\Lambda \times_\alpha \mathbb{Z}^l) \to \mathbb{Z}^l \to 0$ holds, reflecting the structure of the crossed product.