Real and complex K-theory for higher rank graph algebras arising from cube complexes

TL;DR

This paper computes the real and complex K-theory of certain higher-rank graph C*-algebras derived from cube complexes, using spectral sequences, and classifies these algebras based on K-theoretic and combinatorial data.

Contribution

It introduces a method to compute both real and complex K-theory for higher-rank graph algebras from cube complexes, extending previous approaches.

Findings

Computed K-theory for families of higher-rank graph algebras

Classified algebras using K-theoretic and number-theoretic properties

Extended spectral sequence techniques to real K-theory

Abstract

Using the Evans spectral sequence and its counter-part for real $K$-theory, we compute both the real and complex $K$-theory of several infinite families of $C^*$-algebras based on higher-rank graphs of rank $3$ and $4$. The higher-rank graphs we consider arise from double-covers of cube complexes. By considering the real and complex $K$-theory together, we are able to carry these computations much further than might be possible considering complex $K$-theory alone. As these algebras are classified by $K$-theory, we are able to characterize the isomorphism classes of the graph algebras in terms of the combinatorial and number-theoretic properties of the construction ingredients.